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v0.5.1
v0.5.1
https://github.com/python/cpython
Tip revision: 68707af1fb87c8b3d128b71c863cd68195da198f authored by Georg Brandl on 23 February 2014, 07:46:00 UTC
merge
merge
Tip revision: 68707af
mpdecimal.c
/*
* Copyright (c) 2008-2016 Stefan Krah. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include "mpdecimal.h"
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <limits.h>
#include <math.h>
#include "basearith.h"
#include "bits.h"
#include "convolute.h"
#include "crt.h"
#include "memory.h"
#include "typearith.h"
#include "umodarith.h"
#ifdef PPRO
#if defined(_MSC_VER)
#include <float.h>
#pragma fenv_access(on)
#elif !defined(__OpenBSD__) && !defined(__NetBSD__)
/* C99 */
#include <fenv.h>
#pragma STDC FENV_ACCESS ON
#endif
#endif
#if defined(_MSC_VER)
#define ALWAYS_INLINE __forceinline
#elif defined(LEGACY_COMPILER)
#define ALWAYS_INLINE
#undef inline
#define inline
#else
#ifdef TEST_COVERAGE
#define ALWAYS_INLINE
#else
#define ALWAYS_INLINE inline __attribute__ ((always_inline))
#endif
#endif
#define MPD_NEWTONDIV_CUTOFF 1024L
#define MPD_NEW_STATIC(name, flags, exp, digits, len) \
mpd_uint_t name##_data[MPD_MINALLOC_MAX]; \
mpd_t name = {flags|MPD_STATIC|MPD_STATIC_DATA, exp, digits, \
len, MPD_MINALLOC_MAX, name##_data}
#define MPD_NEW_CONST(name, flags, exp, digits, len, alloc, initval) \
mpd_uint_t name##_data[alloc] = {initval}; \
mpd_t name = {flags|MPD_STATIC|MPD_CONST_DATA, exp, digits, \
len, alloc, name##_data}
#define MPD_NEW_SHARED(name, a) \
mpd_t name = {(a->flags&~MPD_DATAFLAGS)|MPD_STATIC|MPD_SHARED_DATA, \
a->exp, a->digits, a->len, a->alloc, a->data}
static mpd_uint_t data_one[1] = {1};
static mpd_uint_t data_zero[1] = {0};
static const mpd_t one = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_one};
static const mpd_t minus_one = {MPD_NEG|MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1,
data_one};
static const mpd_t zero = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_zero};
static inline void _mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx,
uint32_t *status);
static void _settriple(mpd_t *result, uint8_t sign, mpd_uint_t a,
mpd_ssize_t exp);
static inline mpd_ssize_t _mpd_real_size(mpd_uint_t *data, mpd_ssize_t size);
static int _mpd_cmp_abs(const mpd_t *a, const mpd_t *b);
static void _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status);
static inline void _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status);
static void _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a,
const mpd_t *b, uint32_t *status);
static inline void _mpd_qpow_uint(mpd_t *result, const mpd_t *base,
mpd_uint_t exp, uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status);
static mpd_uint_t mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n);
/******************************************************************************/
/* Version */
/******************************************************************************/
const char *
mpd_version(void)
{
return MPD_VERSION;
}
/******************************************************************************/
/* Performance critical inline functions */
/******************************************************************************/
#ifdef CONFIG_64
/* Digits in a word, primarily useful for the most significant word. */
ALWAYS_INLINE int
mpd_word_digits(mpd_uint_t word)
{
if (word < mpd_pow10[9]) {
if (word < mpd_pow10[4]) {
if (word < mpd_pow10[2]) {
return (word < mpd_pow10[1]) ? 1 : 2;
}
return (word < mpd_pow10[3]) ? 3 : 4;
}
if (word < mpd_pow10[6]) {
return (word < mpd_pow10[5]) ? 5 : 6;
}
if (word < mpd_pow10[8]) {
return (word < mpd_pow10[7]) ? 7 : 8;
}
return 9;
}
if (word < mpd_pow10[14]) {
if (word < mpd_pow10[11]) {
return (word < mpd_pow10[10]) ? 10 : 11;
}
if (word < mpd_pow10[13]) {
return (word < mpd_pow10[12]) ? 12 : 13;
}
return 14;
}
if (word < mpd_pow10[18]) {
if (word < mpd_pow10[16]) {
return (word < mpd_pow10[15]) ? 15 : 16;
}
return (word < mpd_pow10[17]) ? 17 : 18;
}
return (word < mpd_pow10[19]) ? 19 : 20;
}
#else
ALWAYS_INLINE int
mpd_word_digits(mpd_uint_t word)
{
if (word < mpd_pow10[4]) {
if (word < mpd_pow10[2]) {
return (word < mpd_pow10[1]) ? 1 : 2;
}
return (word < mpd_pow10[3]) ? 3 : 4;
}
if (word < mpd_pow10[6]) {
return (word < mpd_pow10[5]) ? 5 : 6;
}
if (word < mpd_pow10[8]) {
return (word < mpd_pow10[7]) ? 7 : 8;
}
return (word < mpd_pow10[9]) ? 9 : 10;
}
#endif
/* Adjusted exponent */
ALWAYS_INLINE mpd_ssize_t
mpd_adjexp(const mpd_t *dec)
{
return (dec->exp + dec->digits) - 1;
}
/* Etiny */
ALWAYS_INLINE mpd_ssize_t
mpd_etiny(const mpd_context_t *ctx)
{
return ctx->emin - (ctx->prec - 1);
}
/* Etop: used for folding down in IEEE clamping */
ALWAYS_INLINE mpd_ssize_t
mpd_etop(const mpd_context_t *ctx)
{
return ctx->emax - (ctx->prec - 1);
}
/* Most significant word */
ALWAYS_INLINE mpd_uint_t
mpd_msword(const mpd_t *dec)
{
assert(dec->len > 0);
return dec->data[dec->len-1];
}
/* Most significant digit of a word */
inline mpd_uint_t
mpd_msd(mpd_uint_t word)
{
int n;
n = mpd_word_digits(word);
return word / mpd_pow10[n-1];
}
/* Least significant digit of a word */
ALWAYS_INLINE mpd_uint_t
mpd_lsd(mpd_uint_t word)
{
return word % 10;
}
/* Coefficient size needed to store 'digits' */
ALWAYS_INLINE mpd_ssize_t
mpd_digits_to_size(mpd_ssize_t digits)
{
mpd_ssize_t q, r;
_mpd_idiv_word(&q, &r, digits, MPD_RDIGITS);
return (r == 0) ? q : q+1;
}
/* Number of digits in the exponent. Not defined for MPD_SSIZE_MIN. */
inline int
mpd_exp_digits(mpd_ssize_t exp)
{
exp = (exp < 0) ? -exp : exp;
return mpd_word_digits(exp);
}
/* Canonical */
ALWAYS_INLINE int
mpd_iscanonical(const mpd_t *dec UNUSED)
{
return 1;
}
/* Finite */
ALWAYS_INLINE int
mpd_isfinite(const mpd_t *dec)
{
return !(dec->flags & MPD_SPECIAL);
}
/* Infinite */
ALWAYS_INLINE int
mpd_isinfinite(const mpd_t *dec)
{
return dec->flags & MPD_INF;
}
/* NaN */
ALWAYS_INLINE int
mpd_isnan(const mpd_t *dec)
{
return dec->flags & (MPD_NAN|MPD_SNAN);
}
/* Negative */
ALWAYS_INLINE int
mpd_isnegative(const mpd_t *dec)
{
return dec->flags & MPD_NEG;
}
/* Positive */
ALWAYS_INLINE int
mpd_ispositive(const mpd_t *dec)
{
return !(dec->flags & MPD_NEG);
}
/* qNaN */
ALWAYS_INLINE int
mpd_isqnan(const mpd_t *dec)
{
return dec->flags & MPD_NAN;
}
/* Signed */
ALWAYS_INLINE int
mpd_issigned(const mpd_t *dec)
{
return dec->flags & MPD_NEG;
}
/* sNaN */
ALWAYS_INLINE int
mpd_issnan(const mpd_t *dec)
{
return dec->flags & MPD_SNAN;
}
/* Special */
ALWAYS_INLINE int
mpd_isspecial(const mpd_t *dec)
{
return dec->flags & MPD_SPECIAL;
}
/* Zero */
ALWAYS_INLINE int
mpd_iszero(const mpd_t *dec)
{
return !mpd_isspecial(dec) && mpd_msword(dec) == 0;
}
/* Test for zero when specials have been ruled out already */
ALWAYS_INLINE int
mpd_iszerocoeff(const mpd_t *dec)
{
return mpd_msword(dec) == 0;
}
/* Normal */
inline int
mpd_isnormal(const mpd_t *dec, const mpd_context_t *ctx)
{
if (mpd_isspecial(dec)) return 0;
if (mpd_iszerocoeff(dec)) return 0;
return mpd_adjexp(dec) >= ctx->emin;
}
/* Subnormal */
inline int
mpd_issubnormal(const mpd_t *dec, const mpd_context_t *ctx)
{
if (mpd_isspecial(dec)) return 0;
if (mpd_iszerocoeff(dec)) return 0;
return mpd_adjexp(dec) < ctx->emin;
}
/* Odd word */
ALWAYS_INLINE int
mpd_isoddword(mpd_uint_t word)
{
return word & 1;
}
/* Odd coefficient */
ALWAYS_INLINE int
mpd_isoddcoeff(const mpd_t *dec)
{
return mpd_isoddword(dec->data[0]);
}
/* 0 if dec is positive, 1 if dec is negative */
ALWAYS_INLINE uint8_t
mpd_sign(const mpd_t *dec)
{
return dec->flags & MPD_NEG;
}
/* 1 if dec is positive, -1 if dec is negative */
ALWAYS_INLINE int
mpd_arith_sign(const mpd_t *dec)
{
return 1 - 2 * mpd_isnegative(dec);
}
/* Radix */
ALWAYS_INLINE long
mpd_radix(void)
{
return 10;
}
/* Dynamic decimal */
ALWAYS_INLINE int
mpd_isdynamic(const mpd_t *dec)
{
return !(dec->flags & MPD_STATIC);
}
/* Static decimal */
ALWAYS_INLINE int
mpd_isstatic(const mpd_t *dec)
{
return dec->flags & MPD_STATIC;
}
/* Data of decimal is dynamic */
ALWAYS_INLINE int
mpd_isdynamic_data(const mpd_t *dec)
{
return !(dec->flags & MPD_DATAFLAGS);
}
/* Data of decimal is static */
ALWAYS_INLINE int
mpd_isstatic_data(const mpd_t *dec)
{
return dec->flags & MPD_STATIC_DATA;
}
/* Data of decimal is shared */
ALWAYS_INLINE int
mpd_isshared_data(const mpd_t *dec)
{
return dec->flags & MPD_SHARED_DATA;
}
/* Data of decimal is const */
ALWAYS_INLINE int
mpd_isconst_data(const mpd_t *dec)
{
return dec->flags & MPD_CONST_DATA;
}
/******************************************************************************/
/* Inline memory handling */
/******************************************************************************/
/* Fill destination with zeros */
ALWAYS_INLINE void
mpd_uint_zero(mpd_uint_t *dest, mpd_size_t len)
{
mpd_size_t i;
for (i = 0; i < len; i++) {
dest[i] = 0;
}
}
/* Free a decimal */
ALWAYS_INLINE void
mpd_del(mpd_t *dec)
{
if (mpd_isdynamic_data(dec)) {
mpd_free(dec->data);
}
if (mpd_isdynamic(dec)) {
mpd_free(dec);
}
}
/*
* Resize the coefficient. Existing data up to 'nwords' is left untouched.
* Return 1 on success, 0 otherwise.
*
* Input invariant: MPD_MINALLOC <= result->alloc.
*
* Case nwords == result->alloc:
* 'result' is unchanged. Return 1.
*
* Case nwords > result->alloc:
* Case realloc success:
* The value of 'result' does not change. Return 1.
* Case realloc failure:
* 'result' is NaN, status is updated with MPD_Malloc_error. Return 0.
*
* Case nwords < result->alloc:
* Case is_static_data or realloc failure [1]:
* 'result' is unchanged. Return 1.
* Case realloc success:
* The value of result is undefined (expected). Return 1.
*
*
* [1] In that case the old (now oversized) area is still valid.
*/
ALWAYS_INLINE int
mpd_qresize(mpd_t *result, mpd_ssize_t nwords, uint32_t *status)
{
assert(!mpd_isconst_data(result)); /* illegal operation for a const */
assert(!mpd_isshared_data(result)); /* illegal operation for a shared */
assert(MPD_MINALLOC <= result->alloc);
nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords;
if (nwords == result->alloc) {
return 1;
}
if (mpd_isstatic_data(result)) {
if (nwords > result->alloc) {
return mpd_switch_to_dyn(result, nwords, status);
}
return 1;
}
return mpd_realloc_dyn(result, nwords, status);
}
/* Same as mpd_qresize, but the complete coefficient (including the old
* memory area!) is initialized to zero. */
ALWAYS_INLINE int
mpd_qresize_zero(mpd_t *result, mpd_ssize_t nwords, uint32_t *status)
{
assert(!mpd_isconst_data(result)); /* illegal operation for a const */
assert(!mpd_isshared_data(result)); /* illegal operation for a shared */
assert(MPD_MINALLOC <= result->alloc);
nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords;
if (nwords != result->alloc) {
if (mpd_isstatic_data(result)) {
if (nwords > result->alloc) {
return mpd_switch_to_dyn_zero(result, nwords, status);
}
}
else if (!mpd_realloc_dyn(result, nwords, status)) {
return 0;
}
}
mpd_uint_zero(result->data, nwords);
return 1;
}
/*
* Reduce memory size for the coefficient to MPD_MINALLOC. In theory,
* realloc may fail even when reducing the memory size. But in that case
* the old memory area is always big enough, so checking for MPD_Malloc_error
* is not imperative.
*/
ALWAYS_INLINE void
mpd_minalloc(mpd_t *result)
{
assert(!mpd_isconst_data(result)); /* illegal operation for a const */
assert(!mpd_isshared_data(result)); /* illegal operation for a shared */
if (!mpd_isstatic_data(result) && result->alloc > MPD_MINALLOC) {
uint8_t err = 0;
result->data = mpd_realloc(result->data, MPD_MINALLOC,
sizeof *result->data, &err);
if (!err) {
result->alloc = MPD_MINALLOC;
}
}
}
int
mpd_resize(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx)
{
uint32_t status = 0;
if (!mpd_qresize(result, nwords, &status)) {
mpd_addstatus_raise(ctx, status);
return 0;
}
return 1;
}
int
mpd_resize_zero(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx)
{
uint32_t status = 0;
if (!mpd_qresize_zero(result, nwords, &status)) {
mpd_addstatus_raise(ctx, status);
return 0;
}
return 1;
}
/******************************************************************************/
/* Set attributes of a decimal */
/******************************************************************************/
/* Set digits. Assumption: result->len is initialized and > 0. */
inline void
mpd_setdigits(mpd_t *result)
{
mpd_ssize_t wdigits = mpd_word_digits(mpd_msword(result));
result->digits = wdigits + (result->len-1) * MPD_RDIGITS;
}
/* Set sign */
ALWAYS_INLINE void
mpd_set_sign(mpd_t *result, uint8_t sign)
{
result->flags &= ~MPD_NEG;
result->flags |= sign;
}
/* Copy sign from another decimal */
ALWAYS_INLINE void
mpd_signcpy(mpd_t *result, const mpd_t *a)
{
uint8_t sign = a->flags&MPD_NEG;
result->flags &= ~MPD_NEG;
result->flags |= sign;
}
/* Set infinity */
ALWAYS_INLINE void
mpd_set_infinity(mpd_t *result)
{
result->flags &= ~MPD_SPECIAL;
result->flags |= MPD_INF;
}
/* Set qNaN */
ALWAYS_INLINE void
mpd_set_qnan(mpd_t *result)
{
result->flags &= ~MPD_SPECIAL;
result->flags |= MPD_NAN;
}
/* Set sNaN */
ALWAYS_INLINE void
mpd_set_snan(mpd_t *result)
{
result->flags &= ~MPD_SPECIAL;
result->flags |= MPD_SNAN;
}
/* Set to negative */
ALWAYS_INLINE void
mpd_set_negative(mpd_t *result)
{
result->flags |= MPD_NEG;
}
/* Set to positive */
ALWAYS_INLINE void
mpd_set_positive(mpd_t *result)
{
result->flags &= ~MPD_NEG;
}
/* Set to dynamic */
ALWAYS_INLINE void
mpd_set_dynamic(mpd_t *result)
{
result->flags &= ~MPD_STATIC;
}
/* Set to static */
ALWAYS_INLINE void
mpd_set_static(mpd_t *result)
{
result->flags |= MPD_STATIC;
}
/* Set data to dynamic */
ALWAYS_INLINE void
mpd_set_dynamic_data(mpd_t *result)
{
result->flags &= ~MPD_DATAFLAGS;
}
/* Set data to static */
ALWAYS_INLINE void
mpd_set_static_data(mpd_t *result)
{
result->flags &= ~MPD_DATAFLAGS;
result->flags |= MPD_STATIC_DATA;
}
/* Set data to shared */
ALWAYS_INLINE void
mpd_set_shared_data(mpd_t *result)
{
result->flags &= ~MPD_DATAFLAGS;
result->flags |= MPD_SHARED_DATA;
}
/* Set data to const */
ALWAYS_INLINE void
mpd_set_const_data(mpd_t *result)
{
result->flags &= ~MPD_DATAFLAGS;
result->flags |= MPD_CONST_DATA;
}
/* Clear flags, preserving memory attributes. */
ALWAYS_INLINE void
mpd_clear_flags(mpd_t *result)
{
result->flags &= (MPD_STATIC|MPD_DATAFLAGS);
}
/* Set flags, preserving memory attributes. */
ALWAYS_INLINE void
mpd_set_flags(mpd_t *result, uint8_t flags)
{
result->flags &= (MPD_STATIC|MPD_DATAFLAGS);
result->flags |= flags;
}
/* Copy flags, preserving memory attributes of result. */
ALWAYS_INLINE void
mpd_copy_flags(mpd_t *result, const mpd_t *a)
{
uint8_t aflags = a->flags;
result->flags &= (MPD_STATIC|MPD_DATAFLAGS);
result->flags |= (aflags & ~(MPD_STATIC|MPD_DATAFLAGS));
}
/* Initialize a workcontext from ctx. Set traps, flags and newtrap to 0. */
static inline void
mpd_workcontext(mpd_context_t *workctx, const mpd_context_t *ctx)
{
workctx->prec = ctx->prec;
workctx->emax = ctx->emax;
workctx->emin = ctx->emin;
workctx->round = ctx->round;
workctx->traps = 0;
workctx->status = 0;
workctx->newtrap = 0;
workctx->clamp = ctx->clamp;
workctx->allcr = ctx->allcr;
}
/******************************************************************************/
/* Getting and setting parts of decimals */
/******************************************************************************/
/* Flip the sign of a decimal */
static inline void
_mpd_negate(mpd_t *dec)
{
dec->flags ^= MPD_NEG;
}
/* Set coefficient to zero */
void
mpd_zerocoeff(mpd_t *result)
{
mpd_minalloc(result);
result->digits = 1;
result->len = 1;
result->data[0] = 0;
}
/* Set the coefficient to all nines. */
void
mpd_qmaxcoeff(mpd_t *result, const mpd_context_t *ctx, uint32_t *status)
{
mpd_ssize_t len, r;
_mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS);
len = (r == 0) ? len : len+1;
if (!mpd_qresize(result, len, status)) {
return;
}
result->len = len;
result->digits = ctx->prec;
--len;
if (r > 0) {
result->data[len--] = mpd_pow10[r]-1;
}
for (; len >= 0; --len) {
result->data[len] = MPD_RADIX-1;
}
}
/*
* Cut off the most significant digits so that the rest fits in ctx->prec.
* Cannot fail.
*/
static void
_mpd_cap(mpd_t *result, const mpd_context_t *ctx)
{
uint32_t dummy;
mpd_ssize_t len, r;
if (result->len > 0 && result->digits > ctx->prec) {
_mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS);
len = (r == 0) ? len : len+1;
if (r != 0) {
result->data[len-1] %= mpd_pow10[r];
}
len = _mpd_real_size(result->data, len);
/* resize to fewer words cannot fail */
mpd_qresize(result, len, &dummy);
result->len = len;
mpd_setdigits(result);
}
if (mpd_iszero(result)) {
_settriple(result, mpd_sign(result), 0, result->exp);
}
}
/*
* Cut off the most significant digits of a NaN payload so that the rest
* fits in ctx->prec - ctx->clamp. Cannot fail.
*/
static void
_mpd_fix_nan(mpd_t *result, const mpd_context_t *ctx)
{
uint32_t dummy;
mpd_ssize_t prec;
mpd_ssize_t len, r;
prec = ctx->prec - ctx->clamp;
if (result->len > 0 && result->digits > prec) {
if (prec == 0) {
mpd_minalloc(result);
result->len = result->digits = 0;
}
else {
_mpd_idiv_word(&len, &r, prec, MPD_RDIGITS);
len = (r == 0) ? len : len+1;
if (r != 0) {
result->data[len-1] %= mpd_pow10[r];
}
len = _mpd_real_size(result->data, len);
/* resize to fewer words cannot fail */
mpd_qresize(result, len, &dummy);
result->len = len;
mpd_setdigits(result);
if (mpd_iszerocoeff(result)) {
/* NaN0 is not a valid representation */
result->len = result->digits = 0;
}
}
}
}
/*
* Get n most significant digits from a decimal, where 0 < n <= MPD_UINT_DIGITS.
* Assumes MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for 32 and 64 bit
* machines.
*
* The result of the operation will be in lo. If the operation is impossible,
* hi will be nonzero. This is used to indicate an error.
*/
static inline void
_mpd_get_msdigits(mpd_uint_t *hi, mpd_uint_t *lo, const mpd_t *dec,
unsigned int n)
{
mpd_uint_t r, tmp;
assert(0 < n && n <= MPD_RDIGITS+1);
_mpd_div_word(&tmp, &r, dec->digits, MPD_RDIGITS);
r = (r == 0) ? MPD_RDIGITS : r; /* digits in the most significant word */
*hi = 0;
*lo = dec->data[dec->len-1];
if (n <= r) {
*lo /= mpd_pow10[r-n];
}
else if (dec->len > 1) {
/* at this point 1 <= r < n <= MPD_RDIGITS+1 */
_mpd_mul_words(hi, lo, *lo, mpd_pow10[n-r]);
tmp = dec->data[dec->len-2] / mpd_pow10[MPD_RDIGITS-(n-r)];
*lo = *lo + tmp;
if (*lo < tmp) (*hi)++;
}
}
/******************************************************************************/
/* Gathering information about a decimal */
/******************************************************************************/
/* The real size of the coefficient without leading zero words. */
static inline mpd_ssize_t
_mpd_real_size(mpd_uint_t *data, mpd_ssize_t size)
{
while (size > 1 && data[size-1] == 0) {
size--;
}
return size;
}
/* Return number of trailing zeros. No errors are possible. */
mpd_ssize_t
mpd_trail_zeros(const mpd_t *dec)
{
mpd_uint_t word;
mpd_ssize_t i, tz = 0;
for (i=0; i < dec->len; ++i) {
if (dec->data[i] != 0) {
word = dec->data[i];
tz = i * MPD_RDIGITS;
while (word % 10 == 0) {
word /= 10;
tz++;
}
break;
}
}
return tz;
}
/* Integer: Undefined for specials */
static int
_mpd_isint(const mpd_t *dec)
{
mpd_ssize_t tz;
if (mpd_iszerocoeff(dec)) {
return 1;
}
tz = mpd_trail_zeros(dec);
return (dec->exp + tz >= 0);
}
/* Integer */
int
mpd_isinteger(const mpd_t *dec)
{
if (mpd_isspecial(dec)) {
return 0;
}
return _mpd_isint(dec);
}
/* Word is a power of 10 */
static int
mpd_word_ispow10(mpd_uint_t word)
{
int n;
n = mpd_word_digits(word);
if (word == mpd_pow10[n-1]) {
return 1;
}
return 0;
}
/* Coefficient is a power of 10 */
static int
mpd_coeff_ispow10(const mpd_t *dec)
{
if (mpd_word_ispow10(mpd_msword(dec))) {
if (_mpd_isallzero(dec->data, dec->len-1)) {
return 1;
}
}
return 0;
}
/* All digits of a word are nines */
static int
mpd_word_isallnine(mpd_uint_t word)
{
int n;
n = mpd_word_digits(word);
if (word == mpd_pow10[n]-1) {
return 1;
}
return 0;
}
/* All digits of the coefficient are nines */
static int
mpd_coeff_isallnine(const mpd_t *dec)
{
if (mpd_word_isallnine(mpd_msword(dec))) {
if (_mpd_isallnine(dec->data, dec->len-1)) {
return 1;
}
}
return 0;
}
/* Odd decimal: Undefined for non-integers! */
int
mpd_isodd(const mpd_t *dec)
{
mpd_uint_t q, r;
assert(mpd_isinteger(dec));
if (mpd_iszerocoeff(dec)) return 0;
if (dec->exp < 0) {
_mpd_div_word(&q, &r, -dec->exp, MPD_RDIGITS);
q = dec->data[q] / mpd_pow10[r];
return mpd_isoddword(q);
}
return dec->exp == 0 && mpd_isoddword(dec->data[0]);
}
/* Even: Undefined for non-integers! */
int
mpd_iseven(const mpd_t *dec)
{
return !mpd_isodd(dec);
}
/******************************************************************************/
/* Getting and setting decimals */
/******************************************************************************/
/* Internal function: Set a static decimal from a triple, no error checking. */
static void
_ssettriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp)
{
mpd_set_flags(result, sign);
result->exp = exp;
_mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX);
result->len = (result->data[1] == 0) ? 1 : 2;
mpd_setdigits(result);
}
/* Internal function: Set a decimal from a triple, no error checking. */
static void
_settriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp)
{
mpd_minalloc(result);
mpd_set_flags(result, sign);
result->exp = exp;
_mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX);
result->len = (result->data[1] == 0) ? 1 : 2;
mpd_setdigits(result);
}
/* Set a special number from a triple */
void
mpd_setspecial(mpd_t *result, uint8_t sign, uint8_t type)
{
mpd_minalloc(result);
result->flags &= ~(MPD_NEG|MPD_SPECIAL);
result->flags |= (sign|type);
result->exp = result->digits = result->len = 0;
}
/* Set result of NaN with an error status */
void
mpd_seterror(mpd_t *result, uint32_t flags, uint32_t *status)
{
mpd_minalloc(result);
mpd_set_qnan(result);
mpd_set_positive(result);
result->exp = result->digits = result->len = 0;
*status |= flags;
}
/* quietly set a static decimal from an mpd_ssize_t */
void
mpd_qsset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_uint_t u;
uint8_t sign = MPD_POS;
if (a < 0) {
if (a == MPD_SSIZE_MIN) {
u = (mpd_uint_t)MPD_SSIZE_MAX +
(-(MPD_SSIZE_MIN+MPD_SSIZE_MAX));
}
else {
u = -a;
}
sign = MPD_NEG;
}
else {
u = a;
}
_ssettriple(result, sign, u, 0);
mpd_qfinalize(result, ctx, status);
}
/* quietly set a static decimal from an mpd_uint_t */
void
mpd_qsset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx,
uint32_t *status)
{
_ssettriple(result, MPD_POS, a, 0);
mpd_qfinalize(result, ctx, status);
}
/* quietly set a static decimal from an int32_t */
void
mpd_qsset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qsset_ssize(result, a, ctx, status);
}
/* quietly set a static decimal from a uint32_t */
void
mpd_qsset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qsset_uint(result, a, ctx, status);
}
#ifdef CONFIG_64
/* quietly set a static decimal from an int64_t */
void
mpd_qsset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qsset_ssize(result, a, ctx, status);
}
/* quietly set a static decimal from a uint64_t */
void
mpd_qsset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qsset_uint(result, a, ctx, status);
}
#endif
/* quietly set a decimal from an mpd_ssize_t */
void
mpd_qset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_minalloc(result);
mpd_qsset_ssize(result, a, ctx, status);
}
/* quietly set a decimal from an mpd_uint_t */
void
mpd_qset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx,
uint32_t *status)
{
_settriple(result, MPD_POS, a, 0);
mpd_qfinalize(result, ctx, status);
}
/* quietly set a decimal from an int32_t */
void
mpd_qset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qset_ssize(result, a, ctx, status);
}
/* quietly set a decimal from a uint32_t */
void
mpd_qset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_qset_uint(result, a, ctx, status);
}
#if defined(CONFIG_32) && !defined(LEGACY_COMPILER)
/* set a decimal from a uint64_t */
static void
_c32setu64(mpd_t *result, uint64_t u, uint8_t sign, uint32_t *status)
{
mpd_uint_t w[3];
uint64_t q;
int i, len;
len = 0;
do {
q = u / MPD_RADIX;
w[len] = (mpd_uint_t)(u - q * MPD_RADIX);
u = q; len++;
} while (u != 0);
if (!mpd_qresize(result, len, status)) {
return;
}
for (i = 0; i < len; i++) {
result->data[i] = w[i];
}
mpd_set_sign(result, sign);
result->exp = 0;
result->len = len;
mpd_setdigits(result);
}
static void
_c32_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
_c32setu64(result, a, MPD_POS, status);
mpd_qfinalize(result, ctx, status);
}
/* set a decimal from an int64_t */
static void
_c32_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
uint64_t u;
uint8_t sign = MPD_POS;
if (a < 0) {
if (a == INT64_MIN) {
u = (uint64_t)INT64_MAX + (-(INT64_MIN+INT64_MAX));
}
else {
u = -a;
}
sign = MPD_NEG;
}
else {
u = a;
}
_c32setu64(result, u, sign, status);
mpd_qfinalize(result, ctx, status);
}
#endif /* CONFIG_32 && !LEGACY_COMPILER */
#ifndef LEGACY_COMPILER
/* quietly set a decimal from an int64_t */
void
mpd_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
#ifdef CONFIG_64
mpd_qset_ssize(result, a, ctx, status);
#else
_c32_qset_i64(result, a, ctx, status);
#endif
}
/* quietly set a decimal from a uint64_t */
void
mpd_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx,
uint32_t *status)
{
#ifdef CONFIG_64
mpd_qset_uint(result, a, ctx, status);
#else
_c32_qset_u64(result, a, ctx, status);
#endif
}
#endif /* !LEGACY_COMPILER */
/*
* Quietly get an mpd_uint_t from a decimal. Assumes
* MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for
* 32 and 64 bit machines.
*
* If the operation is impossible, MPD_Invalid_operation is set.
*/
static mpd_uint_t
_mpd_qget_uint(int use_sign, const mpd_t *a, uint32_t *status)
{
mpd_t tmp;
mpd_uint_t tmp_data[2];
mpd_uint_t lo, hi;
if (mpd_isspecial(a)) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
if (mpd_iszero(a)) {
return 0;
}
if (use_sign && mpd_isnegative(a)) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
if (a->digits+a->exp > MPD_RDIGITS+1) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
if (a->exp < 0) {
if (!_mpd_isint(a)) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
/* At this point a->digits+a->exp <= MPD_RDIGITS+1,
* so the shift fits. */
tmp.data = tmp_data;
tmp.flags = MPD_STATIC|MPD_STATIC_DATA;
tmp.alloc = 2;
mpd_qsshiftr(&tmp, a, -a->exp);
tmp.exp = 0;
a = &tmp;
}
_mpd_get_msdigits(&hi, &lo, a, MPD_RDIGITS+1);
if (hi) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
if (a->exp > 0) {
_mpd_mul_words(&hi, &lo, lo, mpd_pow10[a->exp]);
if (hi) {
*status |= MPD_Invalid_operation;
return MPD_UINT_MAX;
}
}
return lo;
}
/*
* Sets Invalid_operation for:
* - specials
* - negative numbers (except negative zero)
* - non-integers
* - overflow
*/
mpd_uint_t
mpd_qget_uint(const mpd_t *a, uint32_t *status)
{
return _mpd_qget_uint(1, a, status);
}
/* Same as above, but gets the absolute value, i.e. the sign is ignored. */
mpd_uint_t
mpd_qabs_uint(const mpd_t *a, uint32_t *status)
{
return _mpd_qget_uint(0, a, status);
}
/* quietly get an mpd_ssize_t from a decimal */
mpd_ssize_t
mpd_qget_ssize(const mpd_t *a, uint32_t *status)
{
mpd_uint_t u;
int isneg;
u = mpd_qabs_uint(a, status);
if (*status&MPD_Invalid_operation) {
return MPD_SSIZE_MAX;
}
isneg = mpd_isnegative(a);
if (u <= MPD_SSIZE_MAX) {
return isneg ? -((mpd_ssize_t)u) : (mpd_ssize_t)u;
}
else if (isneg && u+(MPD_SSIZE_MIN+MPD_SSIZE_MAX) == MPD_SSIZE_MAX) {
return MPD_SSIZE_MIN;
}
*status |= MPD_Invalid_operation;
return MPD_SSIZE_MAX;
}
#if defined(CONFIG_32) && !defined(LEGACY_COMPILER)
/*
* Quietly get a uint64_t from a decimal. If the operation is impossible,
* MPD_Invalid_operation is set.
*/
static uint64_t
_c32_qget_u64(int use_sign, const mpd_t *a, uint32_t *status)
{
MPD_NEW_STATIC(tmp,0,0,20,3);
mpd_context_t maxcontext;
uint64_t ret;
tmp_data[0] = 709551615;
tmp_data[1] = 446744073;
tmp_data[2] = 18;
if (mpd_isspecial(a)) {
*status |= MPD_Invalid_operation;
return UINT64_MAX;
}
if (mpd_iszero(a)) {
return 0;
}
if (use_sign && mpd_isnegative(a)) {
*status |= MPD_Invalid_operation;
return UINT64_MAX;
}
if (!_mpd_isint(a)) {
*status |= MPD_Invalid_operation;
return UINT64_MAX;
}
if (_mpd_cmp_abs(a, &tmp) > 0) {
*status |= MPD_Invalid_operation;
return UINT64_MAX;
}
mpd_maxcontext(&maxcontext);
mpd_qrescale(&tmp, a, 0, &maxcontext, &maxcontext.status);
maxcontext.status &= ~MPD_Rounded;
if (maxcontext.status != 0) {
*status |= (maxcontext.status|MPD_Invalid_operation); /* GCOV_NOT_REACHED */
return UINT64_MAX; /* GCOV_NOT_REACHED */
}
ret = 0;
switch (tmp.len) {
case 3:
ret += (uint64_t)tmp_data[2] * 1000000000000000000ULL;
case 2:
ret += (uint64_t)tmp_data[1] * 1000000000ULL;
case 1:
ret += tmp_data[0];
break;
default:
abort(); /* GCOV_NOT_REACHED */
}
return ret;
}
static int64_t
_c32_qget_i64(const mpd_t *a, uint32_t *status)
{
uint64_t u;
int isneg;
u = _c32_qget_u64(0, a, status);
if (*status&MPD_Invalid_operation) {
return INT64_MAX;
}
isneg = mpd_isnegative(a);
if (u <= INT64_MAX) {
return isneg ? -((int64_t)u) : (int64_t)u;
}
else if (isneg && u+(INT64_MIN+INT64_MAX) == INT64_MAX) {
return INT64_MIN;
}
*status |= MPD_Invalid_operation;
return INT64_MAX;
}
#endif /* CONFIG_32 && !LEGACY_COMPILER */
#ifdef CONFIG_64
/* quietly get a uint64_t from a decimal */
uint64_t
mpd_qget_u64(const mpd_t *a, uint32_t *status)
{
return mpd_qget_uint(a, status);
}
/* quietly get an int64_t from a decimal */
int64_t
mpd_qget_i64(const mpd_t *a, uint32_t *status)
{
return mpd_qget_ssize(a, status);
}
/* quietly get a uint32_t from a decimal */
uint32_t
mpd_qget_u32(const mpd_t *a, uint32_t *status)
{
uint64_t x = mpd_qget_uint(a, status);
if (*status&MPD_Invalid_operation) {
return UINT32_MAX;
}
if (x > UINT32_MAX) {
*status |= MPD_Invalid_operation;
return UINT32_MAX;
}
return (uint32_t)x;
}
/* quietly get an int32_t from a decimal */
int32_t
mpd_qget_i32(const mpd_t *a, uint32_t *status)
{
int64_t x = mpd_qget_ssize(a, status);
if (*status&MPD_Invalid_operation) {
return INT32_MAX;
}
if (x < INT32_MIN || x > INT32_MAX) {
*status |= MPD_Invalid_operation;
return INT32_MAX;
}
return (int32_t)x;
}
#else
#ifndef LEGACY_COMPILER
/* quietly get a uint64_t from a decimal */
uint64_t
mpd_qget_u64(const mpd_t *a, uint32_t *status)
{
return _c32_qget_u64(1, a, status);
}
/* quietly get an int64_t from a decimal */
int64_t
mpd_qget_i64(const mpd_t *a, uint32_t *status)
{
return _c32_qget_i64(a, status);
}
#endif
/* quietly get a uint32_t from a decimal */
uint32_t
mpd_qget_u32(const mpd_t *a, uint32_t *status)
{
return mpd_qget_uint(a, status);
}
/* quietly get an int32_t from a decimal */
int32_t
mpd_qget_i32(const mpd_t *a, uint32_t *status)
{
return mpd_qget_ssize(a, status);
}
#endif
/******************************************************************************/
/* Filtering input of functions, finalizing output of functions */
/******************************************************************************/
/*
* Check if the operand is NaN, copy to result and return 1 if this is
* the case. Copying can fail since NaNs are allowed to have a payload that
* does not fit in MPD_MINALLOC.
*/
int
mpd_qcheck_nan(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isnan(a)) {
*status |= mpd_issnan(a) ? MPD_Invalid_operation : 0;
mpd_qcopy(result, a, status);
mpd_set_qnan(result);
_mpd_fix_nan(result, ctx);
return 1;
}
return 0;
}
/*
* Check if either operand is NaN, copy to result and return 1 if this
* is the case. Copying can fail since NaNs are allowed to have a payload
* that does not fit in MPD_MINALLOC.
*/
int
mpd_qcheck_nans(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
if ((a->flags|b->flags)&(MPD_NAN|MPD_SNAN)) {
const mpd_t *choice = b;
if (mpd_issnan(a)) {
choice = a;
*status |= MPD_Invalid_operation;
}
else if (mpd_issnan(b)) {
*status |= MPD_Invalid_operation;
}
else if (mpd_isqnan(a)) {
choice = a;
}
mpd_qcopy(result, choice, status);
mpd_set_qnan(result);
_mpd_fix_nan(result, ctx);
return 1;
}
return 0;
}
/*
* Check if one of the operands is NaN, copy to result and return 1 if this
* is the case. Copying can fail since NaNs are allowed to have a payload
* that does not fit in MPD_MINALLOC.
*/
static int
mpd_qcheck_3nans(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c,
const mpd_context_t *ctx, uint32_t *status)
{
if ((a->flags|b->flags|c->flags)&(MPD_NAN|MPD_SNAN)) {
const mpd_t *choice = c;
if (mpd_issnan(a)) {
choice = a;
*status |= MPD_Invalid_operation;
}
else if (mpd_issnan(b)) {
choice = b;
*status |= MPD_Invalid_operation;
}
else if (mpd_issnan(c)) {
*status |= MPD_Invalid_operation;
}
else if (mpd_isqnan(a)) {
choice = a;
}
else if (mpd_isqnan(b)) {
choice = b;
}
mpd_qcopy(result, choice, status);
mpd_set_qnan(result);
_mpd_fix_nan(result, ctx);
return 1;
}
return 0;
}
/* Check if rounding digit 'rnd' leads to an increment. */
static inline int
_mpd_rnd_incr(const mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx)
{
int ld;
switch (ctx->round) {
case MPD_ROUND_DOWN: case MPD_ROUND_TRUNC:
return 0;
case MPD_ROUND_HALF_UP:
return (rnd >= 5);
case MPD_ROUND_HALF_EVEN:
return (rnd > 5) || ((rnd == 5) && mpd_isoddcoeff(dec));
case MPD_ROUND_CEILING:
return !(rnd == 0 || mpd_isnegative(dec));
case MPD_ROUND_FLOOR:
return !(rnd == 0 || mpd_ispositive(dec));
case MPD_ROUND_HALF_DOWN:
return (rnd > 5);
case MPD_ROUND_UP:
return !(rnd == 0);
case MPD_ROUND_05UP:
ld = (int)mpd_lsd(dec->data[0]);
return (!(rnd == 0) && (ld == 0 || ld == 5));
default:
/* Without a valid context, further results will be undefined. */
return 0; /* GCOV_NOT_REACHED */
}
}
/*
* Apply rounding to a decimal that has been right-shifted into a full
* precision decimal. If an increment leads to an overflow of the precision,
* adjust the coefficient and the exponent and check the new exponent for
* overflow.
*/
static inline void
_mpd_apply_round(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx,
uint32_t *status)
{
if (_mpd_rnd_incr(dec, rnd, ctx)) {
/* We have a number with exactly ctx->prec digits. The increment
* can only lead to an overflow if the decimal is all nines. In
* that case, the result is a power of ten with prec+1 digits.
*
* If the precision is a multiple of MPD_RDIGITS, this situation is
* detected by _mpd_baseincr returning a carry.
* If the precision is not a multiple of MPD_RDIGITS, we have to
* check if the result has one digit too many.
*/
mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len);
if (carry) {
dec->data[dec->len-1] = mpd_pow10[MPD_RDIGITS-1];
dec->exp += 1;
_mpd_check_exp(dec, ctx, status);
return;
}
mpd_setdigits(dec);
if (dec->digits > ctx->prec) {
mpd_qshiftr_inplace(dec, 1);
dec->exp += 1;
dec->digits = ctx->prec;
_mpd_check_exp(dec, ctx, status);
}
}
}
/*
* Apply rounding to a decimal. Allow overflow of the precision.
*/
static inline void
_mpd_apply_round_excess(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx,
uint32_t *status)
{
if (_mpd_rnd_incr(dec, rnd, ctx)) {
mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len);
if (carry) {
if (!mpd_qresize(dec, dec->len+1, status)) {
return;
}
dec->data[dec->len] = 1;
dec->len += 1;
}
mpd_setdigits(dec);
}
}
/*
* Apply rounding to a decimal that has been right-shifted into a decimal
* with full precision or less. Return failure if an increment would
* overflow the precision.
*/
static inline int
_mpd_apply_round_fit(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx,
uint32_t *status)
{
if (_mpd_rnd_incr(dec, rnd, ctx)) {
mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len);
if (carry) {
if (!mpd_qresize(dec, dec->len+1, status)) {
return 0;
}
dec->data[dec->len] = 1;
dec->len += 1;
}
mpd_setdigits(dec);
if (dec->digits > ctx->prec) {
mpd_seterror(dec, MPD_Invalid_operation, status);
return 0;
}
}
return 1;
}
/* Check a normal number for overflow, underflow, clamping. If the operand
is modified, it will be zero, special or (sub)normal with a coefficient
that fits into the current context precision. */
static inline void
_mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status)
{
mpd_ssize_t adjexp, etiny, shift;
int rnd;
adjexp = mpd_adjexp(dec);
if (adjexp > ctx->emax) {
if (mpd_iszerocoeff(dec)) {
dec->exp = ctx->emax;
if (ctx->clamp) {
dec->exp -= (ctx->prec-1);
}
mpd_zerocoeff(dec);
*status |= MPD_Clamped;
return;
}
switch (ctx->round) {
case MPD_ROUND_HALF_UP: case MPD_ROUND_HALF_EVEN:
case MPD_ROUND_HALF_DOWN: case MPD_ROUND_UP:
case MPD_ROUND_TRUNC:
mpd_setspecial(dec, mpd_sign(dec), MPD_INF);
break;
case MPD_ROUND_DOWN: case MPD_ROUND_05UP:
mpd_qmaxcoeff(dec, ctx, status);
dec->exp = ctx->emax - ctx->prec + 1;
break;
case MPD_ROUND_CEILING:
if (mpd_isnegative(dec)) {
mpd_qmaxcoeff(dec, ctx, status);
dec->exp = ctx->emax - ctx->prec + 1;
}
else {
mpd_setspecial(dec, MPD_POS, MPD_INF);
}
break;
case MPD_ROUND_FLOOR:
if (mpd_ispositive(dec)) {
mpd_qmaxcoeff(dec, ctx, status);
dec->exp = ctx->emax - ctx->prec + 1;
}
else {
mpd_setspecial(dec, MPD_NEG, MPD_INF);
}
break;
default: /* debug */
abort(); /* GCOV_NOT_REACHED */
}
*status |= MPD_Overflow|MPD_Inexact|MPD_Rounded;
} /* fold down */
else if (ctx->clamp && dec->exp > mpd_etop(ctx)) {
/* At this point adjexp=exp+digits-1 <= emax and exp > etop=emax-prec+1:
* (1) shift = exp -emax+prec-1 > 0
* (2) digits+shift = exp+digits-1 - emax + prec <= prec */
shift = dec->exp - mpd_etop(ctx);
if (!mpd_qshiftl(dec, dec, shift, status)) {
return;
}
dec->exp -= shift;
*status |= MPD_Clamped;
if (!mpd_iszerocoeff(dec) && adjexp < ctx->emin) {
/* Underflow is impossible, since exp < etiny=emin-prec+1
* and exp > etop=emax-prec+1 would imply emax < emin. */
*status |= MPD_Subnormal;
}
}
else if (adjexp < ctx->emin) {
etiny = mpd_etiny(ctx);
if (mpd_iszerocoeff(dec)) {
if (dec->exp < etiny) {
dec->exp = etiny;
mpd_zerocoeff(dec);
*status |= MPD_Clamped;
}
return;
}
*status |= MPD_Subnormal;
if (dec->exp < etiny) {
/* At this point adjexp=exp+digits-1 < emin and exp < etiny=emin-prec+1:
* (1) shift = emin-prec+1 - exp > 0
* (2) digits-shift = exp+digits-1 - emin + prec < prec */
shift = etiny - dec->exp;
rnd = (int)mpd_qshiftr_inplace(dec, shift);
dec->exp = etiny;
/* We always have a spare digit in case of an increment. */
_mpd_apply_round_excess(dec, rnd, ctx, status);
*status |= MPD_Rounded;
if (rnd) {
*status |= (MPD_Inexact|MPD_Underflow);
if (mpd_iszerocoeff(dec)) {
mpd_zerocoeff(dec);
*status |= MPD_Clamped;
}
}
}
/* Case exp >= etiny=emin-prec+1:
* (1) adjexp=exp+digits-1 < emin
* (2) digits < emin-exp+1 <= prec */
}
}
/* Transcendental functions do not always set Underflow reliably,
* since they only use as much precision as is necessary for correct
* rounding. If a result like 1.0000000000e-101 is finalized, there
* is no rounding digit that would trigger Underflow. But we can
* assume Inexact, so a short check suffices. */
static inline void
mpd_check_underflow(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status)
{
if (mpd_adjexp(dec) < ctx->emin && !mpd_iszero(dec) &&
dec->exp < mpd_etiny(ctx)) {
*status |= MPD_Underflow;
}
}
/* Check if a normal number must be rounded after the exponent has been checked. */
static inline void
_mpd_check_round(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status)
{
mpd_uint_t rnd;
mpd_ssize_t shift;
/* must handle specials: _mpd_check_exp() can produce infinities or NaNs */
if (mpd_isspecial(dec)) {
return;
}
if (dec->digits > ctx->prec) {
shift = dec->digits - ctx->prec;
rnd = mpd_qshiftr_inplace(dec, shift);
dec->exp += shift;
_mpd_apply_round(dec, rnd, ctx, status);
*status |= MPD_Rounded;
if (rnd) {
*status |= MPD_Inexact;
}
}
}
/* Finalize all operations. */
void
mpd_qfinalize(mpd_t *result, const mpd_context_t *ctx, uint32_t *status)
{
if (mpd_isspecial(result)) {
if (mpd_isnan(result)) {
_mpd_fix_nan(result, ctx);
}
return;
}
_mpd_check_exp(result, ctx, status);
_mpd_check_round(result, ctx, status);
}
/******************************************************************************/
/* Copying */
/******************************************************************************/
/* Internal function: Copy a decimal, share data with src: USE WITH CARE! */
static inline void
_mpd_copy_shared(mpd_t *dest, const mpd_t *src)
{
dest->flags = src->flags;
dest->exp = src->exp;
dest->digits = src->digits;
dest->len = src->len;
dest->alloc = src->alloc;
dest->data = src->data;
mpd_set_shared_data(dest);
}
/*
* Copy a decimal. In case of an error, status is set to MPD_Malloc_error.
*/
int
mpd_qcopy(mpd_t *result, const mpd_t *a, uint32_t *status)
{
if (result == a) return 1;
if (!mpd_qresize(result, a->len, status)) {
return 0;
}
mpd_copy_flags(result, a);
result->exp = a->exp;
result->digits = a->digits;
result->len = a->len;
memcpy(result->data, a->data, a->len * (sizeof *result->data));
return 1;
}
/*
* Copy to a decimal with a static buffer. The caller has to make sure that
* the buffer is big enough. Cannot fail.
*/
static void
mpd_qcopy_static(mpd_t *result, const mpd_t *a)
{
if (result == a) return;
memcpy(result->data, a->data, a->len * (sizeof *result->data));
mpd_copy_flags(result, a);
result->exp = a->exp;
result->digits = a->digits;
result->len = a->len;
}
/*
* Return a newly allocated copy of the operand. In case of an error,
* status is set to MPD_Malloc_error and the return value is NULL.
*/
mpd_t *
mpd_qncopy(const mpd_t *a)
{
mpd_t *result;
if ((result = mpd_qnew_size(a->len)) == NULL) {
return NULL;
}
memcpy(result->data, a->data, a->len * (sizeof *result->data));
mpd_copy_flags(result, a);
result->exp = a->exp;
result->digits = a->digits;
result->len = a->len;
return result;
}
/*
* Copy a decimal and set the sign to positive. In case of an error, the
* status is set to MPD_Malloc_error.
*/
int
mpd_qcopy_abs(mpd_t *result, const mpd_t *a, uint32_t *status)
{
if (!mpd_qcopy(result, a, status)) {
return 0;
}
mpd_set_positive(result);
return 1;
}
/*
* Copy a decimal and negate the sign. In case of an error, the
* status is set to MPD_Malloc_error.
*/
int
mpd_qcopy_negate(mpd_t *result, const mpd_t *a, uint32_t *status)
{
if (!mpd_qcopy(result, a, status)) {
return 0;
}
_mpd_negate(result);
return 1;
}
/*
* Copy a decimal, setting the sign of the first operand to the sign of the
* second operand. In case of an error, the status is set to MPD_Malloc_error.
*/
int
mpd_qcopy_sign(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status)
{
uint8_t sign_b = mpd_sign(b); /* result may equal b! */
if (!mpd_qcopy(result, a, status)) {
return 0;
}
mpd_set_sign(result, sign_b);
return 1;
}
/******************************************************************************/
/* Comparisons */
/******************************************************************************/
/*
* For all functions that compare two operands and return an int the usual
* convention applies to the return value:
*
* -1 if op1 < op2
* 0 if op1 == op2
* 1 if op1 > op2
*
* INT_MAX for error
*/
/* Convenience macro. If a and b are not equal, return from the calling
* function with the correct comparison value. */
#define CMP_EQUAL_OR_RETURN(a, b) \
if (a != b) { \
if (a < b) { \
return -1; \
} \
return 1; \
}
/*
* Compare the data of big and small. This function does the equivalent
* of first shifting small to the left and then comparing the data of
* big and small, except that no allocation for the left shift is needed.
*/
static int
_mpd_basecmp(mpd_uint_t *big, mpd_uint_t *small, mpd_size_t n, mpd_size_t m,
mpd_size_t shift)
{
#if defined(__GNUC__) && !defined(__INTEL_COMPILER) && !defined(__clang__)
/* spurious uninitialized warnings */
mpd_uint_t l=l, lprev=lprev, h=h;
#else
mpd_uint_t l, lprev, h;
#endif
mpd_uint_t q, r;
mpd_uint_t ph, x;
assert(m > 0 && n >= m && shift > 0);
_mpd_div_word(&q, &r, (mpd_uint_t)shift, MPD_RDIGITS);
if (r != 0) {
ph = mpd_pow10[r];
--m; --n;
_mpd_divmod_pow10(&h, &lprev, small[m--], MPD_RDIGITS-r);
if (h != 0) {
CMP_EQUAL_OR_RETURN(big[n], h)
--n;
}
for (; m != MPD_SIZE_MAX; m--,n--) {
_mpd_divmod_pow10(&h, &l, small[m], MPD_RDIGITS-r);
x = ph * lprev + h;
CMP_EQUAL_OR_RETURN(big[n], x)
lprev = l;
}
x = ph * lprev;
CMP_EQUAL_OR_RETURN(big[q], x)
}
else {
while (--m != MPD_SIZE_MAX) {
CMP_EQUAL_OR_RETURN(big[m+q], small[m])
}
}
return !_mpd_isallzero(big, q);
}
/* Compare two decimals with the same adjusted exponent. */
static int
_mpd_cmp_same_adjexp(const mpd_t *a, const mpd_t *b)
{
mpd_ssize_t shift, i;
if (a->exp != b->exp) {
/* Cannot wrap: a->exp + a->digits = b->exp + b->digits, so
* a->exp - b->exp = b->digits - a->digits. */
shift = a->exp - b->exp;
if (shift > 0) {
return -1 * _mpd_basecmp(b->data, a->data, b->len, a->len, shift);
}
else {
return _mpd_basecmp(a->data, b->data, a->len, b->len, -shift);
}
}
/*
* At this point adjexp(a) == adjexp(b) and a->exp == b->exp,
* so a->digits == b->digits, therefore a->len == b->len.
*/
for (i = a->len-1; i >= 0; --i) {
CMP_EQUAL_OR_RETURN(a->data[i], b->data[i])
}
return 0;
}
/* Compare two numerical values. */
static int
_mpd_cmp(const mpd_t *a, const mpd_t *b)
{
mpd_ssize_t adjexp_a, adjexp_b;
/* equal pointers */
if (a == b) {
return 0;
}
/* infinities */
if (mpd_isinfinite(a)) {
if (mpd_isinfinite(b)) {
return mpd_isnegative(b) - mpd_isnegative(a);
}
return mpd_arith_sign(a);
}
if (mpd_isinfinite(b)) {
return -mpd_arith_sign(b);
}
/* zeros */
if (mpd_iszerocoeff(a)) {
if (mpd_iszerocoeff(b)) {
return 0;
}
return -mpd_arith_sign(b);
}
if (mpd_iszerocoeff(b)) {
return mpd_arith_sign(a);
}
/* different signs */
if (mpd_sign(a) != mpd_sign(b)) {
return mpd_sign(b) - mpd_sign(a);
}
/* different adjusted exponents */
adjexp_a = mpd_adjexp(a);
adjexp_b = mpd_adjexp(b);
if (adjexp_a != adjexp_b) {
if (adjexp_a < adjexp_b) {
return -1 * mpd_arith_sign(a);
}
return mpd_arith_sign(a);
}
/* same adjusted exponents */
return _mpd_cmp_same_adjexp(a, b) * mpd_arith_sign(a);
}
/* Compare the absolutes of two numerical values. */
static int
_mpd_cmp_abs(const mpd_t *a, const mpd_t *b)
{
mpd_ssize_t adjexp_a, adjexp_b;
/* equal pointers */
if (a == b) {
return 0;
}
/* infinities */
if (mpd_isinfinite(a)) {
if (mpd_isinfinite(b)) {
return 0;
}
return 1;
}
if (mpd_isinfinite(b)) {
return -1;
}
/* zeros */
if (mpd_iszerocoeff(a)) {
if (mpd_iszerocoeff(b)) {
return 0;
}
return -1;
}
if (mpd_iszerocoeff(b)) {
return 1;
}
/* different adjusted exponents */
adjexp_a = mpd_adjexp(a);
adjexp_b = mpd_adjexp(b);
if (adjexp_a != adjexp_b) {
if (adjexp_a < adjexp_b) {
return -1;
}
return 1;
}
/* same adjusted exponents */
return _mpd_cmp_same_adjexp(a, b);
}
/* Compare two values and return an integer result. */
int
mpd_qcmp(const mpd_t *a, const mpd_t *b, uint32_t *status)
{
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_isnan(a) || mpd_isnan(b)) {
*status |= MPD_Invalid_operation;
return INT_MAX;
}
}
return _mpd_cmp(a, b);
}
/*
* Compare a and b, convert the usual integer result to a decimal and
* store it in 'result'. For convenience, the integer result of the comparison
* is returned. Comparisons involving NaNs return NaN/INT_MAX.
*/
int
mpd_qcompare(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return INT_MAX;
}
}
c = _mpd_cmp(a, b);
_settriple(result, (c < 0), (c != 0), 0);
return c;
}
/* Same as mpd_compare(), but signal for all NaNs, i.e. also for quiet NaNs. */
int
mpd_qcompare_signal(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
*status |= MPD_Invalid_operation;
return INT_MAX;
}
}
c = _mpd_cmp(a, b);
_settriple(result, (c < 0), (c != 0), 0);
return c;
}
/* Compare the operands using a total order. */
int
mpd_cmp_total(const mpd_t *a, const mpd_t *b)
{
mpd_t aa, bb;
int nan_a, nan_b;
int c;
if (mpd_sign(a) != mpd_sign(b)) {
return mpd_sign(b) - mpd_sign(a);
}
if (mpd_isnan(a)) {
c = 1;
if (mpd_isnan(b)) {
nan_a = (mpd_isqnan(a)) ? 1 : 0;
nan_b = (mpd_isqnan(b)) ? 1 : 0;
if (nan_b == nan_a) {
if (a->len > 0 && b->len > 0) {
_mpd_copy_shared(&aa, a);
_mpd_copy_shared(&bb, b);
aa.exp = bb.exp = 0;
/* compare payload */
c = _mpd_cmp_abs(&aa, &bb);
}
else {
c = (a->len > 0) - (b->len > 0);
}
}
else {
c = nan_a - nan_b;
}
}
}
else if (mpd_isnan(b)) {
c = -1;
}
else {
c = _mpd_cmp_abs(a, b);
if (c == 0 && a->exp != b->exp) {
c = (a->exp < b->exp) ? -1 : 1;
}
}
return c * mpd_arith_sign(a);
}
/*
* Compare a and b according to a total order, convert the usual integer result
* to a decimal and store it in 'result'. For convenience, the integer result
* of the comparison is returned.
*/
int
mpd_compare_total(mpd_t *result, const mpd_t *a, const mpd_t *b)
{
int c;
c = mpd_cmp_total(a, b);
_settriple(result, (c < 0), (c != 0), 0);
return c;
}
/* Compare the magnitude of the operands using a total order. */
int
mpd_cmp_total_mag(const mpd_t *a, const mpd_t *b)
{
mpd_t aa, bb;
_mpd_copy_shared(&aa, a);
_mpd_copy_shared(&bb, b);
mpd_set_positive(&aa);
mpd_set_positive(&bb);
return mpd_cmp_total(&aa, &bb);
}
/*
* Compare the magnitude of a and b according to a total order, convert the
* the usual integer result to a decimal and store it in 'result'.
* For convenience, the integer result of the comparison is returned.
*/
int
mpd_compare_total_mag(mpd_t *result, const mpd_t *a, const mpd_t *b)
{
int c;
c = mpd_cmp_total_mag(a, b);
_settriple(result, (c < 0), (c != 0), 0);
return c;
}
/* Determine an ordering for operands that are numerically equal. */
static inline int
_mpd_cmp_numequal(const mpd_t *a, const mpd_t *b)
{
int sign_a, sign_b;
int c;
sign_a = mpd_sign(a);
sign_b = mpd_sign(b);
if (sign_a != sign_b) {
c = sign_b - sign_a;
}
else {
c = (a->exp < b->exp) ? -1 : 1;
c *= mpd_arith_sign(a);
}
return c;
}
/******************************************************************************/
/* Shifting the coefficient */
/******************************************************************************/
/*
* Shift the coefficient of the operand to the left, no check for specials.
* Both operands may be the same pointer. If the result length has to be
* increased, mpd_qresize() might fail with MPD_Malloc_error.
*/
int
mpd_qshiftl(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status)
{
mpd_ssize_t size;
assert(!mpd_isspecial(a));
assert(n >= 0);
if (mpd_iszerocoeff(a) || n == 0) {
return mpd_qcopy(result, a, status);
}
size = mpd_digits_to_size(a->digits+n);
if (!mpd_qresize(result, size, status)) {
return 0; /* result is NaN */
}
_mpd_baseshiftl(result->data, a->data, size, a->len, n);
mpd_copy_flags(result, a);
result->exp = a->exp;
result->digits = a->digits+n;
result->len = size;
return 1;
}
/* Determine the rounding indicator if all digits of the coefficient are shifted
* out of the picture. */
static mpd_uint_t
_mpd_get_rnd(const mpd_uint_t *data, mpd_ssize_t len, int use_msd)
{
mpd_uint_t rnd = 0, rest = 0, word;
word = data[len-1];
/* special treatment for the most significant digit if shift == digits */
if (use_msd) {
_mpd_divmod_pow10(&rnd, &rest, word, mpd_word_digits(word)-1);
if (len > 1 && rest == 0) {
rest = !_mpd_isallzero(data, len-1);
}
}
else {
rest = !_mpd_isallzero(data, len);
}
return (rnd == 0 || rnd == 5) ? rnd + !!rest : rnd;
}
/*
* Same as mpd_qshiftr(), but 'result' is an mpd_t with a static coefficient.
* It is the caller's responsibility to ensure that the coefficient is big
* enough. The function cannot fail.
*/
static mpd_uint_t
mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n)
{
mpd_uint_t rnd;
mpd_ssize_t size;
assert(!mpd_isspecial(a));
assert(n >= 0);
if (mpd_iszerocoeff(a) || n == 0) {
mpd_qcopy_static(result, a);
return 0;
}
if (n >= a->digits) {
rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits));
mpd_zerocoeff(result);
}
else {
result->digits = a->digits-n;
size = mpd_digits_to_size(result->digits);
rnd = _mpd_baseshiftr(result->data, a->data, a->len, n);
result->len = size;
}
mpd_copy_flags(result, a);
result->exp = a->exp;
return rnd;
}
/*
* Inplace shift of the coefficient to the right, no check for specials.
* Returns the rounding indicator for mpd_rnd_incr().
* The function cannot fail.
*/
mpd_uint_t
mpd_qshiftr_inplace(mpd_t *result, mpd_ssize_t n)
{
uint32_t dummy;
mpd_uint_t rnd;
mpd_ssize_t size;
assert(!mpd_isspecial(result));
assert(n >= 0);
if (mpd_iszerocoeff(result) || n == 0) {
return 0;
}
if (n >= result->digits) {
rnd = _mpd_get_rnd(result->data, result->len, (n==result->digits));
mpd_zerocoeff(result);
}
else {
rnd = _mpd_baseshiftr(result->data, result->data, result->len, n);
result->digits -= n;
size = mpd_digits_to_size(result->digits);
/* reducing the size cannot fail */
mpd_qresize(result, size, &dummy);
result->len = size;
}
return rnd;
}
/*
* Shift the coefficient of the operand to the right, no check for specials.
* Both operands may be the same pointer. Returns the rounding indicator to
* be used by mpd_rnd_incr(). If the result length has to be increased,
* mpd_qcopy() or mpd_qresize() might fail with MPD_Malloc_error. In those
* cases, MPD_UINT_MAX is returned.
*/
mpd_uint_t
mpd_qshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status)
{
mpd_uint_t rnd;
mpd_ssize_t size;
assert(!mpd_isspecial(a));
assert(n >= 0);
if (mpd_iszerocoeff(a) || n == 0) {
if (!mpd_qcopy(result, a, status)) {
return MPD_UINT_MAX;
}
return 0;
}
if (n >= a->digits) {
rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits));
mpd_zerocoeff(result);
}
else {
result->digits = a->digits-n;
size = mpd_digits_to_size(result->digits);
if (result == a) {
rnd = _mpd_baseshiftr(result->data, a->data, a->len, n);
/* reducing the size cannot fail */
mpd_qresize(result, size, status);
}
else {
if (!mpd_qresize(result, size, status)) {
return MPD_UINT_MAX;
}
rnd = _mpd_baseshiftr(result->data, a->data, a->len, n);
}
result->len = size;
}
mpd_copy_flags(result, a);
result->exp = a->exp;
return rnd;
}
/******************************************************************************/
/* Miscellaneous operations */
/******************************************************************************/
/* Logical And */
void
mpd_qand(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
const mpd_t *big = a, *small = b;
mpd_uint_t x, y, z, xbit, ybit;
int k, mswdigits;
mpd_ssize_t i;
if (mpd_isspecial(a) || mpd_isspecial(b) ||
mpd_isnegative(a) || mpd_isnegative(b) ||
a->exp != 0 || b->exp != 0) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (b->digits > a->digits) {
big = b;
small = a;
}
if (!mpd_qresize(result, big->len, status)) {
return;
}
/* full words */
for (i = 0; i < small->len-1; i++) {
x = small->data[i];
y = big->data[i];
z = 0;
for (k = 0; k < MPD_RDIGITS; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit&ybit) ? mpd_pow10[k] : 0;
}
result->data[i] = z;
}
/* most significant word of small */
x = small->data[i];
y = big->data[i];
z = 0;
mswdigits = mpd_word_digits(x);
for (k = 0; k < mswdigits; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit&ybit) ? mpd_pow10[k] : 0;
}
result->data[i++] = z;
/* scan the rest of y for digits > 1 */
for (; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
}
/* scan the rest of big for digits > 1 */
for (; i < big->len; i++) {
y = big->data[i];
for (k = 0; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
}
}
mpd_clear_flags(result);
result->exp = 0;
result->len = _mpd_real_size(result->data, small->len);
mpd_qresize(result, result->len, status);
mpd_setdigits(result);
_mpd_cap(result, ctx);
return;
invalid_operation:
mpd_seterror(result, MPD_Invalid_operation, status);
}
/* Class of an operand. Returns a pointer to the constant name. */
const char *
mpd_class(const mpd_t *a, const mpd_context_t *ctx)
{
if (mpd_isnan(a)) {
if (mpd_isqnan(a))
return "NaN";
else
return "sNaN";
}
else if (mpd_ispositive(a)) {
if (mpd_isinfinite(a))
return "+Infinity";
else if (mpd_iszero(a))
return "+Zero";
else if (mpd_isnormal(a, ctx))
return "+Normal";
else
return "+Subnormal";
}
else {
if (mpd_isinfinite(a))
return "-Infinity";
else if (mpd_iszero(a))
return "-Zero";
else if (mpd_isnormal(a, ctx))
return "-Normal";
else
return "-Subnormal";
}
}
/* Logical Xor */
void
mpd_qinvert(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_uint_t x, z, xbit;
mpd_ssize_t i, digits, len;
mpd_ssize_t q, r;
int k;
if (mpd_isspecial(a) || mpd_isnegative(a) || a->exp != 0) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
digits = (a->digits < ctx->prec) ? ctx->prec : a->digits;
_mpd_idiv_word(&q, &r, digits, MPD_RDIGITS);
len = (r == 0) ? q : q+1;
if (!mpd_qresize(result, len, status)) {
return;
}
for (i = 0; i < len; i++) {
x = (i < a->len) ? a->data[i] : 0;
z = 0;
for (k = 0; k < MPD_RDIGITS; k++) {
xbit = x % 10;
x /= 10;
if (xbit > 1) {
goto invalid_operation;
}
z += !xbit ? mpd_pow10[k] : 0;
}
result->data[i] = z;
}
mpd_clear_flags(result);
result->exp = 0;
result->len = _mpd_real_size(result->data, len);
mpd_qresize(result, result->len, status);
mpd_setdigits(result);
_mpd_cap(result, ctx);
return;
invalid_operation:
mpd_seterror(result, MPD_Invalid_operation, status);
}
/* Exponent of the magnitude of the most significant digit of the operand. */
void
mpd_qlogb(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
mpd_setspecial(result, MPD_POS, MPD_INF);
}
else if (mpd_iszerocoeff(a)) {
mpd_setspecial(result, MPD_NEG, MPD_INF);
*status |= MPD_Division_by_zero;
}
else {
mpd_qset_ssize(result, mpd_adjexp(a), ctx, status);
}
}
/* Logical Or */
void
mpd_qor(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
const mpd_t *big = a, *small = b;
mpd_uint_t x, y, z, xbit, ybit;
int k, mswdigits;
mpd_ssize_t i;
if (mpd_isspecial(a) || mpd_isspecial(b) ||
mpd_isnegative(a) || mpd_isnegative(b) ||
a->exp != 0 || b->exp != 0) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (b->digits > a->digits) {
big = b;
small = a;
}
if (!mpd_qresize(result, big->len, status)) {
return;
}
/* full words */
for (i = 0; i < small->len-1; i++) {
x = small->data[i];
y = big->data[i];
z = 0;
for (k = 0; k < MPD_RDIGITS; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit|ybit) ? mpd_pow10[k] : 0;
}
result->data[i] = z;
}
/* most significant word of small */
x = small->data[i];
y = big->data[i];
z = 0;
mswdigits = mpd_word_digits(x);
for (k = 0; k < mswdigits; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit|ybit) ? mpd_pow10[k] : 0;
}
/* scan for digits > 1 and copy the rest of y */
for (; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
z += ybit*mpd_pow10[k];
}
result->data[i++] = z;
/* scan for digits > 1 and copy the rest of big */
for (; i < big->len; i++) {
y = big->data[i];
for (k = 0; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
}
result->data[i] = big->data[i];
}
mpd_clear_flags(result);
result->exp = 0;
result->len = _mpd_real_size(result->data, big->len);
mpd_qresize(result, result->len, status);
mpd_setdigits(result);
_mpd_cap(result, ctx);
return;
invalid_operation:
mpd_seterror(result, MPD_Invalid_operation, status);
}
/*
* Rotate the coefficient of 'a' by 'b' digits. 'b' must be an integer with
* exponent 0.
*/
void
mpd_qrotate(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
MPD_NEW_STATIC(tmp,0,0,0,0);
MPD_NEW_STATIC(big,0,0,0,0);
MPD_NEW_STATIC(small,0,0,0,0);
mpd_ssize_t n, lshift, rshift;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
}
if (b->exp != 0 || mpd_isinfinite(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
n = mpd_qget_ssize(b, &workstatus);
if (workstatus&MPD_Invalid_operation) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (n > ctx->prec || n < -ctx->prec) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(a)) {
mpd_qcopy(result, a, status);
return;
}
if (n >= 0) {
lshift = n;
rshift = ctx->prec-n;
}
else {
lshift = ctx->prec+n;
rshift = -n;
}
if (a->digits > ctx->prec) {
if (!mpd_qcopy(&tmp, a, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
_mpd_cap(&tmp, ctx);
a = &tmp;
}
if (!mpd_qshiftl(&big, a, lshift, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
_mpd_cap(&big, ctx);
if (mpd_qshiftr(&small, a, rshift, status) == MPD_UINT_MAX) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
_mpd_qadd(result, &big, &small, ctx, status);
finish:
mpd_del(&tmp);
mpd_del(&big);
mpd_del(&small);
}
/*
* b must be an integer with exponent 0 and in the range +-2*(emax + prec).
* XXX: In my opinion +-(2*emax + prec) would be more sensible.
* The result is a with the value of b added to its exponent.
*/
void
mpd_qscaleb(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_uint_t n, maxjump;
#ifndef LEGACY_COMPILER
int64_t exp;
#else
mpd_uint_t x;
int x_sign, n_sign;
mpd_ssize_t exp;
#endif
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
}
if (b->exp != 0 || mpd_isinfinite(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
n = mpd_qabs_uint(b, &workstatus);
/* the spec demands this */
maxjump = 2 * (mpd_uint_t)(ctx->emax + ctx->prec);
if (n > maxjump || workstatus&MPD_Invalid_operation) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(a)) {
mpd_qcopy(result, a, status);
return;
}
#ifndef LEGACY_COMPILER
exp = a->exp + (int64_t)n * mpd_arith_sign(b);
exp = (exp > MPD_EXP_INF) ? MPD_EXP_INF : exp;
exp = (exp < MPD_EXP_CLAMP) ? MPD_EXP_CLAMP : exp;
#else
x = (a->exp < 0) ? -a->exp : a->exp;
x_sign = (a->exp < 0) ? 1 : 0;
n_sign = mpd_isnegative(b) ? 1 : 0;
if (x_sign == n_sign) {
x = x + n;
if (x < n) x = MPD_UINT_MAX;
}
else {
x_sign = (x >= n) ? x_sign : n_sign;
x = (x >= n) ? x - n : n - x;
}
if (!x_sign && x > MPD_EXP_INF) x = MPD_EXP_INF;
if (x_sign && x > -MPD_EXP_CLAMP) x = -MPD_EXP_CLAMP;
exp = x_sign ? -((mpd_ssize_t)x) : (mpd_ssize_t)x;
#endif
mpd_qcopy(result, a, status);
result->exp = (mpd_ssize_t)exp;
mpd_qfinalize(result, ctx, status);
}
/*
* Shift the coefficient by n digits, positive n is a left shift. In the case
* of a left shift, the result is decapitated to fit the context precision. If
* you don't want that, use mpd_shiftl().
*/
void
mpd_qshiftn(mpd_t *result, const mpd_t *a, mpd_ssize_t n, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
mpd_qcopy(result, a, status);
return;
}
if (n >= 0 && n <= ctx->prec) {
mpd_qshiftl(result, a, n, status);
_mpd_cap(result, ctx);
}
else if (n < 0 && n >= -ctx->prec) {
if (!mpd_qcopy(result, a, status)) {
return;
}
_mpd_cap(result, ctx);
mpd_qshiftr_inplace(result, -n);
}
else {
mpd_seterror(result, MPD_Invalid_operation, status);
}
}
/*
* Same as mpd_shiftn(), but the shift is specified by the decimal b, which
* must be an integer with a zero exponent. Infinities remain infinities.
*/
void
mpd_qshift(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx,
uint32_t *status)
{
uint32_t workstatus = 0;
mpd_ssize_t n;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
}
if (b->exp != 0 || mpd_isinfinite(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
n = mpd_qget_ssize(b, &workstatus);
if (workstatus&MPD_Invalid_operation) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (n > ctx->prec || n < -ctx->prec) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(a)) {
mpd_qcopy(result, a, status);
return;
}
if (n >= 0) {
mpd_qshiftl(result, a, n, status);
_mpd_cap(result, ctx);
}
else {
if (!mpd_qcopy(result, a, status)) {
return;
}
_mpd_cap(result, ctx);
mpd_qshiftr_inplace(result, -n);
}
}
/* Logical Xor */
void
mpd_qxor(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
const mpd_t *big = a, *small = b;
mpd_uint_t x, y, z, xbit, ybit;
int k, mswdigits;
mpd_ssize_t i;
if (mpd_isspecial(a) || mpd_isspecial(b) ||
mpd_isnegative(a) || mpd_isnegative(b) ||
a->exp != 0 || b->exp != 0) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (b->digits > a->digits) {
big = b;
small = a;
}
if (!mpd_qresize(result, big->len, status)) {
return;
}
/* full words */
for (i = 0; i < small->len-1; i++) {
x = small->data[i];
y = big->data[i];
z = 0;
for (k = 0; k < MPD_RDIGITS; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit^ybit) ? mpd_pow10[k] : 0;
}
result->data[i] = z;
}
/* most significant word of small */
x = small->data[i];
y = big->data[i];
z = 0;
mswdigits = mpd_word_digits(x);
for (k = 0; k < mswdigits; k++) {
xbit = x % 10;
x /= 10;
ybit = y % 10;
y /= 10;
if (xbit > 1 || ybit > 1) {
goto invalid_operation;
}
z += (xbit^ybit) ? mpd_pow10[k] : 0;
}
/* scan for digits > 1 and copy the rest of y */
for (; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
z += ybit*mpd_pow10[k];
}
result->data[i++] = z;
/* scan for digits > 1 and copy the rest of big */
for (; i < big->len; i++) {
y = big->data[i];
for (k = 0; k < MPD_RDIGITS; k++) {
ybit = y % 10;
y /= 10;
if (ybit > 1) {
goto invalid_operation;
}
}
result->data[i] = big->data[i];
}
mpd_clear_flags(result);
result->exp = 0;
result->len = _mpd_real_size(result->data, big->len);
mpd_qresize(result, result->len, status);
mpd_setdigits(result);
_mpd_cap(result, ctx);
return;
invalid_operation:
mpd_seterror(result, MPD_Invalid_operation, status);
}
/******************************************************************************/
/* Arithmetic operations */
/******************************************************************************/
/*
* The absolute value of a. If a is negative, the result is the same
* as the result of the minus operation. Otherwise, the result is the
* result of the plus operation.
*/
void
mpd_qabs(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
}
if (mpd_isnegative(a)) {
mpd_qminus(result, a, ctx, status);
}
else {
mpd_qplus(result, a, ctx, status);
}
}
static inline void
_mpd_ptrswap(const mpd_t **a, const mpd_t **b)
{
const mpd_t *t = *a;
*a = *b;
*b = t;
}
/* Add or subtract infinities. */
static void
_mpd_qaddsub_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b,
uint32_t *status)
{
if (mpd_isinfinite(a)) {
if (mpd_sign(a) != sign_b && mpd_isinfinite(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
else {
mpd_setspecial(result, mpd_sign(a), MPD_INF);
}
return;
}
assert(mpd_isinfinite(b));
mpd_setspecial(result, sign_b, MPD_INF);
}
/* Add or subtract non-special numbers. */
static void
_mpd_qaddsub(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b,
const mpd_context_t *ctx, uint32_t *status)
{
const mpd_t *big, *small;
MPD_NEW_STATIC(big_aligned,0,0,0,0);
MPD_NEW_CONST(tiny,0,0,1,1,1,1);
mpd_uint_t carry;
mpd_ssize_t newsize, shift;
mpd_ssize_t exp, i;
int swap = 0;
/* compare exponents */
big = a; small = b;
if (big->exp != small->exp) {
if (small->exp > big->exp) {
_mpd_ptrswap(&big, &small);
swap++;
}
/* align the coefficients */
if (!mpd_iszerocoeff(big)) {
exp = big->exp - 1;
exp += (big->digits > ctx->prec) ? 0 : big->digits-ctx->prec-1;
if (mpd_adjexp(small) < exp) {
/*
* Avoid huge shifts by substituting a value for small that is
* guaranteed to produce the same results.
*
* adjexp(small) < exp if and only if:
*
* bdigits <= prec AND
* bdigits+shift >= prec+2+sdigits AND
* exp = bexp+bdigits-prec-2
*
* 1234567000000000 -> bdigits + shift
* ----------XX1234 -> sdigits
* ----------X1 -> tiny-digits
* |- prec -|
*
* OR
*
* bdigits > prec AND
* shift > sdigits AND
* exp = bexp-1
*
* 1234567892100000 -> bdigits + shift
* ----------XX1234 -> sdigits
* ----------X1 -> tiny-digits
* |- prec -|
*
* If tiny is zero, adding or subtracting is a no-op.
* Otherwise, adding tiny generates a non-zero digit either
* below the rounding digit or the least significant digit
* of big. When subtracting, tiny is in the same position as
* the carry that would be generated by subtracting sdigits.
*/
mpd_copy_flags(&tiny, small);
tiny.exp = exp;
tiny.digits = 1;
tiny.len = 1;
tiny.data[0] = mpd_iszerocoeff(small) ? 0 : 1;
small = &tiny;
}
/* This cannot wrap: the difference is positive and <= maxprec */
shift = big->exp - small->exp;
if (!mpd_qshiftl(&big_aligned, big, shift, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
big = &big_aligned;
}
}
result->exp = small->exp;
/* compare length of coefficients */
if (big->len < small->len) {
_mpd_ptrswap(&big, &small);
swap++;
}
newsize = big->len;
if (!mpd_qresize(result, newsize, status)) {
goto finish;
}
if (mpd_sign(a) == sign_b) {
carry = _mpd_baseadd(result->data, big->data, small->data,
big->len, small->len);
if (carry) {
newsize = big->len + 1;
if (!mpd_qresize(result, newsize, status)) {
goto finish;
}
result->data[newsize-1] = carry;
}
result->len = newsize;
mpd_set_flags(result, sign_b);
}
else {
if (big->len == small->len) {
for (i=big->len-1; i >= 0; --i) {
if (big->data[i] != small->data[i]) {
if (big->data[i] < small->data[i]) {
_mpd_ptrswap(&big, &small);
swap++;
}
break;
}
}
}
_mpd_basesub(result->data, big->data, small->data,
big->len, small->len);
newsize = _mpd_real_size(result->data, big->len);
/* resize to smaller cannot fail */
(void)mpd_qresize(result, newsize, status);
result->len = newsize;
sign_b = (swap & 1) ? sign_b : mpd_sign(a);
mpd_set_flags(result, sign_b);
if (mpd_iszerocoeff(result)) {
mpd_set_positive(result);
if (ctx->round == MPD_ROUND_FLOOR) {
mpd_set_negative(result);
}
}
}
mpd_setdigits(result);
finish:
mpd_del(&big_aligned);
}
/* Add a and b. No specials, no finalizing. */
static void
_mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
_mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status);
}
/* Subtract b from a. No specials, no finalizing. */
static void
_mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
_mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status);
}
/* Add a and b. */
void
mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
_mpd_qaddsub_inf(result, a, b, mpd_sign(b), status);
return;
}
_mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status);
mpd_qfinalize(result, ctx, status);
}
/* Add a and b. Set NaN/Invalid_operation if the result is inexact. */
static void
_mpd_qadd_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_qadd(result, a, b, ctx, &workstatus);
*status |= workstatus;
if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
}
/* Subtract b from a. */
void
mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
_mpd_qaddsub_inf(result, a, b, !mpd_sign(b), status);
return;
}
_mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status);
mpd_qfinalize(result, ctx, status);
}
/* Subtract b from a. Set NaN/Invalid_operation if the result is inexact. */
static void
_mpd_qsub_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_qsub(result, a, b, ctx, &workstatus);
*status |= workstatus;
if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
}
/* Add decimal and mpd_ssize_t. */
void
mpd_qadd_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_ssize(&bb, b, &maxcontext, status);
mpd_qadd(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Add decimal and mpd_uint_t. */
void
mpd_qadd_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_uint(&bb, b, &maxcontext, status);
mpd_qadd(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Subtract mpd_ssize_t from decimal. */
void
mpd_qsub_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_ssize(&bb, b, &maxcontext, status);
mpd_qsub(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Subtract mpd_uint_t from decimal. */
void
mpd_qsub_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_uint(&bb, b, &maxcontext, status);
mpd_qsub(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Add decimal and int32_t. */
void
mpd_qadd_i32(mpd_t *result, const mpd_t *a, int32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qadd_ssize(result, a, b, ctx, status);
}
/* Add decimal and uint32_t. */
void
mpd_qadd_u32(mpd_t *result, const mpd_t *a, uint32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qadd_uint(result, a, b, ctx, status);
}
#ifdef CONFIG_64
/* Add decimal and int64_t. */
void
mpd_qadd_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qadd_ssize(result, a, b, ctx, status);
}
/* Add decimal and uint64_t. */
void
mpd_qadd_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qadd_uint(result, a, b, ctx, status);
}
#elif !defined(LEGACY_COMPILER)
/* Add decimal and int64_t. */
void
mpd_qadd_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_i64(&bb, b, &maxcontext, status);
mpd_qadd(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Add decimal and uint64_t. */
void
mpd_qadd_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_u64(&bb, b, &maxcontext, status);
mpd_qadd(result, a, &bb, ctx, status);
mpd_del(&bb);
}
#endif
/* Subtract int32_t from decimal. */
void
mpd_qsub_i32(mpd_t *result, const mpd_t *a, int32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qsub_ssize(result, a, b, ctx, status);
}
/* Subtract uint32_t from decimal. */
void
mpd_qsub_u32(mpd_t *result, const mpd_t *a, uint32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qsub_uint(result, a, b, ctx, status);
}
#ifdef CONFIG_64
/* Subtract int64_t from decimal. */
void
mpd_qsub_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qsub_ssize(result, a, b, ctx, status);
}
/* Subtract uint64_t from decimal. */
void
mpd_qsub_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qsub_uint(result, a, b, ctx, status);
}
#elif !defined(LEGACY_COMPILER)
/* Subtract int64_t from decimal. */
void
mpd_qsub_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_i64(&bb, b, &maxcontext, status);
mpd_qsub(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Subtract uint64_t from decimal. */
void
mpd_qsub_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_u64(&bb, b, &maxcontext, status);
mpd_qsub(result, a, &bb, ctx, status);
mpd_del(&bb);
}
#endif
/* Divide infinities. */
static void
_mpd_qdiv_inf(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
if (mpd_isinfinite(a)) {
if (mpd_isinfinite(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF);
return;
}
assert(mpd_isinfinite(b));
_settriple(result, mpd_sign(a)^mpd_sign(b), 0, mpd_etiny(ctx));
*status |= MPD_Clamped;
}
enum {NO_IDEAL_EXP, SET_IDEAL_EXP};
/* Divide a by b. */
static void
_mpd_qdiv(int action, mpd_t *q, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
MPD_NEW_STATIC(aligned,0,0,0,0);
mpd_uint_t ld;
mpd_ssize_t shift, exp, tz;
mpd_ssize_t newsize;
mpd_ssize_t ideal_exp;
mpd_uint_t rem;
uint8_t sign_a = mpd_sign(a);
uint8_t sign_b = mpd_sign(b);
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(q, a, b, ctx, status)) {
return;
}
_mpd_qdiv_inf(q, a, b, ctx, status);
return;
}
if (mpd_iszerocoeff(b)) {
if (mpd_iszerocoeff(a)) {
mpd_seterror(q, MPD_Division_undefined, status);
}
else {
mpd_setspecial(q, sign_a^sign_b, MPD_INF);
*status |= MPD_Division_by_zero;
}
return;
}
if (mpd_iszerocoeff(a)) {
exp = a->exp - b->exp;
_settriple(q, sign_a^sign_b, 0, exp);
mpd_qfinalize(q, ctx, status);
return;
}
shift = (b->digits - a->digits) + ctx->prec + 1;
ideal_exp = a->exp - b->exp;
exp = ideal_exp - shift;
if (shift > 0) {
if (!mpd_qshiftl(&aligned, a, shift, status)) {
mpd_seterror(q, MPD_Malloc_error, status);
goto finish;
}
a = &aligned;
}
else if (shift < 0) {
shift = -shift;
if (!mpd_qshiftl(&aligned, b, shift, status)) {
mpd_seterror(q, MPD_Malloc_error, status);
goto finish;
}
b = &aligned;
}
newsize = a->len - b->len + 1;
if ((q != b && q != a) || (q == b && newsize > b->len)) {
if (!mpd_qresize(q, newsize, status)) {
mpd_seterror(q, MPD_Malloc_error, status);
goto finish;
}
}
if (b->len == 1) {
rem = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]);
}
else if (b->len <= MPD_NEWTONDIV_CUTOFF) {
int ret = _mpd_basedivmod(q->data, NULL, a->data, b->data,
a->len, b->len);
if (ret < 0) {
mpd_seterror(q, MPD_Malloc_error, status);
goto finish;
}
rem = ret;
}
else {
MPD_NEW_STATIC(r,0,0,0,0);
_mpd_base_ndivmod(q, &r, a, b, status);
if (mpd_isspecial(q) || mpd_isspecial(&r)) {
mpd_setspecial(q, MPD_POS, MPD_NAN);
mpd_del(&r);
goto finish;
}
rem = !mpd_iszerocoeff(&r);
mpd_del(&r);
newsize = q->len;
}
newsize = _mpd_real_size(q->data, newsize);
/* resize to smaller cannot fail */
mpd_qresize(q, newsize, status);
mpd_set_flags(q, sign_a^sign_b);
q->len = newsize;
mpd_setdigits(q);
shift = ideal_exp - exp;
if (rem) {
ld = mpd_lsd(q->data[0]);
if (ld == 0 || ld == 5) {
q->data[0] += 1;
}
}
else if (action == SET_IDEAL_EXP && shift > 0) {
tz = mpd_trail_zeros(q);
shift = (tz > shift) ? shift : tz;
mpd_qshiftr_inplace(q, shift);
exp += shift;
}
q->exp = exp;
finish:
mpd_del(&aligned);
mpd_qfinalize(q, ctx, status);
}
/* Divide a by b. */
void
mpd_qdiv(mpd_t *q, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
_mpd_qdiv(SET_IDEAL_EXP, q, a, b, ctx, status);
}
/* Internal function. */
static void
_mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
MPD_NEW_STATIC(aligned,0,0,0,0);
mpd_ssize_t qsize, rsize;
mpd_ssize_t ideal_exp, expdiff, shift;
uint8_t sign_a = mpd_sign(a);
uint8_t sign_ab = mpd_sign(a)^mpd_sign(b);
ideal_exp = (a->exp > b->exp) ? b->exp : a->exp;
if (mpd_iszerocoeff(a)) {
if (!mpd_qcopy(r, a, status)) {
goto nanresult; /* GCOV_NOT_REACHED */
}
r->exp = ideal_exp;
_settriple(q, sign_ab, 0, 0);
return;
}
expdiff = mpd_adjexp(a) - mpd_adjexp(b);
if (expdiff < 0) {
if (a->exp > b->exp) {
/* positive and less than b->digits - a->digits */
shift = a->exp - b->exp;
if (!mpd_qshiftl(r, a, shift, status)) {
goto nanresult;
}
r->exp = ideal_exp;
}
else {
if (!mpd_qcopy(r, a, status)) {
goto nanresult;
}
}
_settriple(q, sign_ab, 0, 0);
return;
}
if (expdiff > ctx->prec) {
*status |= MPD_Division_impossible;
goto nanresult;
}
/*
* At this point we have:
* (1) 0 <= a->exp + a->digits - b->exp - b->digits <= prec
* (2) a->exp - b->exp >= b->digits - a->digits
* (3) a->exp - b->exp <= prec + b->digits - a->digits
*/
if (a->exp != b->exp) {
shift = a->exp - b->exp;
if (shift > 0) {
/* by (3), after the shift a->digits <= prec + b->digits */
if (!mpd_qshiftl(&aligned, a, shift, status)) {
goto nanresult;
}
a = &aligned;
}
else {
shift = -shift;
/* by (2), after the shift b->digits <= a->digits */
if (!mpd_qshiftl(&aligned, b, shift, status)) {
goto nanresult;
}
b = &aligned;
}
}
qsize = a->len - b->len + 1;
if (!(q == a && qsize < a->len) && !(q == b && qsize < b->len)) {
if (!mpd_qresize(q, qsize, status)) {
goto nanresult;
}
}
rsize = b->len;
if (!(r == a && rsize < a->len)) {
if (!mpd_qresize(r, rsize, status)) {
goto nanresult;
}
}
if (b->len == 1) {
if (a->len == 1) {
_mpd_div_word(&q->data[0], &r->data[0], a->data[0], b->data[0]);
}
else {
r->data[0] = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]);
}
}
else if (b->len <= MPD_NEWTONDIV_CUTOFF) {
int ret;
ret = _mpd_basedivmod(q->data, r->data, a->data, b->data,
a->len, b->len);
if (ret == -1) {
*status |= MPD_Malloc_error;
goto nanresult;
}
}
else {
_mpd_base_ndivmod(q, r, a, b, status);
if (mpd_isspecial(q) || mpd_isspecial(r)) {
goto nanresult;
}
qsize = q->len;
rsize = r->len;
}
qsize = _mpd_real_size(q->data, qsize);
/* resize to smaller cannot fail */
mpd_qresize(q, qsize, status);
q->len = qsize;
mpd_setdigits(q);
mpd_set_flags(q, sign_ab);
q->exp = 0;
if (q->digits > ctx->prec) {
*status |= MPD_Division_impossible;
goto nanresult;
}
rsize = _mpd_real_size(r->data, rsize);
/* resize to smaller cannot fail */
mpd_qresize(r, rsize, status);
r->len = rsize;
mpd_setdigits(r);
mpd_set_flags(r, sign_a);
r->exp = ideal_exp;
out:
mpd_del(&aligned);
return;
nanresult:
mpd_setspecial(q, MPD_POS, MPD_NAN);
mpd_setspecial(r, MPD_POS, MPD_NAN);
goto out;
}
/* Integer division with remainder. */
void
mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint8_t sign = mpd_sign(a)^mpd_sign(b);
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(q, a, b, ctx, status)) {
mpd_qcopy(r, q, status);
return;
}
if (mpd_isinfinite(a)) {
if (mpd_isinfinite(b)) {
mpd_setspecial(q, MPD_POS, MPD_NAN);
}
else {
mpd_setspecial(q, sign, MPD_INF);
}
mpd_setspecial(r, MPD_POS, MPD_NAN);
*status |= MPD_Invalid_operation;
return;
}
if (mpd_isinfinite(b)) {
if (!mpd_qcopy(r, a, status)) {
mpd_seterror(q, MPD_Malloc_error, status);
return;
}
mpd_qfinalize(r, ctx, status);
_settriple(q, sign, 0, 0);
return;
}
/* debug */
abort(); /* GCOV_NOT_REACHED */
}
if (mpd_iszerocoeff(b)) {
if (mpd_iszerocoeff(a)) {
mpd_setspecial(q, MPD_POS, MPD_NAN);
mpd_setspecial(r, MPD_POS, MPD_NAN);
*status |= MPD_Division_undefined;
}
else {
mpd_setspecial(q, sign, MPD_INF);
mpd_setspecial(r, MPD_POS, MPD_NAN);
*status |= (MPD_Division_by_zero|MPD_Invalid_operation);
}
return;
}
_mpd_qdivmod(q, r, a, b, ctx, status);
mpd_qfinalize(q, ctx, status);
mpd_qfinalize(r, ctx, status);
}
void
mpd_qdivint(mpd_t *q, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
MPD_NEW_STATIC(r,0,0,0,0);
uint8_t sign = mpd_sign(a)^mpd_sign(b);
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(q, a, b, ctx, status)) {
return;
}
if (mpd_isinfinite(a) && mpd_isinfinite(b)) {
mpd_seterror(q, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(a)) {
mpd_setspecial(q, sign, MPD_INF);
return;
}
if (mpd_isinfinite(b)) {
_settriple(q, sign, 0, 0);
return;
}
/* debug */
abort(); /* GCOV_NOT_REACHED */
}
if (mpd_iszerocoeff(b)) {
if (mpd_iszerocoeff(a)) {
mpd_seterror(q, MPD_Division_undefined, status);
}
else {
mpd_setspecial(q, sign, MPD_INF);
*status |= MPD_Division_by_zero;
}
return;
}
_mpd_qdivmod(q, &r, a, b, ctx, status);
mpd_del(&r);
mpd_qfinalize(q, ctx, status);
}
/* Divide decimal by mpd_ssize_t. */
void
mpd_qdiv_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_ssize(&bb, b, &maxcontext, status);
mpd_qdiv(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Divide decimal by mpd_uint_t. */
void
mpd_qdiv_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_uint(&bb, b, &maxcontext, status);
mpd_qdiv(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Divide decimal by int32_t. */
void
mpd_qdiv_i32(mpd_t *result, const mpd_t *a, int32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qdiv_ssize(result, a, b, ctx, status);
}
/* Divide decimal by uint32_t. */
void
mpd_qdiv_u32(mpd_t *result, const mpd_t *a, uint32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qdiv_uint(result, a, b, ctx, status);
}
#ifdef CONFIG_64
/* Divide decimal by int64_t. */
void
mpd_qdiv_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qdiv_ssize(result, a, b, ctx, status);
}
/* Divide decimal by uint64_t. */
void
mpd_qdiv_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qdiv_uint(result, a, b, ctx, status);
}
#elif !defined(LEGACY_COMPILER)
/* Divide decimal by int64_t. */
void
mpd_qdiv_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_i64(&bb, b, &maxcontext, status);
mpd_qdiv(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Divide decimal by uint64_t. */
void
mpd_qdiv_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_u64(&bb, b, &maxcontext, status);
mpd_qdiv(result, a, &bb, ctx, status);
mpd_del(&bb);
}
#endif
/* Pad the result with trailing zeros if it has fewer digits than prec. */
static void
_mpd_zeropad(mpd_t *result, const mpd_context_t *ctx, uint32_t *status)
{
if (!mpd_isspecial(result) && !mpd_iszero(result) &&
result->digits < ctx->prec) {
mpd_ssize_t shift = ctx->prec - result->digits;
mpd_qshiftl(result, result, shift, status);
result->exp -= shift;
}
}
/* Check if the result is guaranteed to be one. */
static int
_mpd_qexp_check_one(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
MPD_NEW_CONST(lim,0,-(ctx->prec+1),1,1,1,9);
MPD_NEW_SHARED(aa, a);
mpd_set_positive(&aa);
/* abs(a) <= 9 * 10**(-prec-1) */
if (_mpd_cmp(&aa, &lim) <= 0) {
_settriple(result, 0, 1, 0);
*status |= MPD_Rounded|MPD_Inexact;
return 1;
}
return 0;
}
/*
* Get the number of iterations for the Horner scheme in _mpd_qexp().
*/
static inline mpd_ssize_t
_mpd_get_exp_iterations(const mpd_t *r, mpd_ssize_t p)
{
mpd_ssize_t log10pbyr; /* lower bound for log10(p / abs(r)) */
mpd_ssize_t n;
assert(p >= 10);
assert(!mpd_iszero(r));
assert(-p < mpd_adjexp(r) && mpd_adjexp(r) <= -1);
#ifdef CONFIG_64
if (p > (mpd_ssize_t)(1ULL<<52)) {
return MPD_SSIZE_MAX;
}
#endif
/*
* Lower bound for log10(p / abs(r)): adjexp(p) - (adjexp(r) + 1)
* At this point (for CONFIG_64, CONFIG_32 is not problematic):
* 1) 10 <= p <= 2**52
* 2) -p < adjexp(r) <= -1
* 3) 1 <= log10pbyr <= 2**52 + 14
*/
log10pbyr = (mpd_word_digits(p)-1) - (mpd_adjexp(r)+1);
/*
* The numerator in the paper is 1.435 * p - 1.182, calculated
* exactly. We compensate for rounding errors by using 1.43503.
* ACL2 proofs:
* 1) exp-iter-approx-lower-bound: The term below evaluated
* in 53-bit floating point arithmetic is greater than or
* equal to the exact term used in the paper.
* 2) exp-iter-approx-upper-bound: The term below is less than
* or equal to 3/2 * p <= 3/2 * 2**52.
*/
n = (mpd_ssize_t)ceil((1.43503*(double)p - 1.182) / (double)log10pbyr);
return n >= 3 ? n : 3;
}
/*
* Internal function, specials have been dealt with. Apart from Overflow
* and Underflow, two cases must be considered for the error of the result:
*
* 1) abs(a) <= 9 * 10**(-prec-1) ==> result == 1
*
* Absolute error: abs(1 - e**x) < 10**(-prec)
* -------------------------------------------
*
* 2) abs(a) > 9 * 10**(-prec-1)
*
* Relative error: abs(result - e**x) < 0.5 * 10**(-prec) * e**x
* -------------------------------------------------------------
*
* The algorithm is from Hull&Abrham, Variable Precision Exponential Function,
* ACM Transactions on Mathematical Software, Vol. 12, No. 2, June 1986.
*
* Main differences:
*
* - The number of iterations for the Horner scheme is calculated using
* 53-bit floating point arithmetic.
*
* - In the error analysis for ER (relative error accumulated in the
* evaluation of the truncated series) the reduced operand r may
* have any number of digits.
* ACL2 proof: exponent-relative-error
*
* - The analysis for early abortion has been adapted for the mpd_t
* ranges.
*/
static void
_mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_STATIC(tmp,0,0,0,0);
MPD_NEW_STATIC(sum,0,0,0,0);
MPD_NEW_CONST(word,0,0,1,1,1,1);
mpd_ssize_t j, n, t;
assert(!mpd_isspecial(a));
if (mpd_iszerocoeff(a)) {
_settriple(result, MPD_POS, 1, 0);
return;
}
/*
* We are calculating e^x = e^(r*10^t) = (e^r)^(10^t), where abs(r) < 1 and t >= 0.
*
* If t > 0, we have:
*
* (1) 0.1 <= r < 1, so e^0.1 <= e^r. If t > MAX_T, overflow occurs:
*
* MAX-EMAX+1 < log10(e^(0.1*10*t)) <= log10(e^(r*10^t)) < adjexp(e^(r*10^t))+1
*
* (2) -1 < r <= -0.1, so e^r <= e^-0.1. If t > MAX_T, underflow occurs:
*
* adjexp(e^(r*10^t)) <= log10(e^(r*10^t)) <= log10(e^(-0.1*10^t)) < MIN-ETINY
*/
#if defined(CONFIG_64)
#define MPD_EXP_MAX_T 19
#elif defined(CONFIG_32)
#define MPD_EXP_MAX_T 10
#endif
t = a->digits + a->exp;
t = (t > 0) ? t : 0;
if (t > MPD_EXP_MAX_T) {
if (mpd_ispositive(a)) {
mpd_setspecial(result, MPD_POS, MPD_INF);
*status |= MPD_Overflow|MPD_Inexact|MPD_Rounded;
}
else {
_settriple(result, MPD_POS, 0, mpd_etiny(ctx));
*status |= (MPD_Inexact|MPD_Rounded|MPD_Subnormal|
MPD_Underflow|MPD_Clamped);
}
return;
}
/* abs(a) <= 9 * 10**(-prec-1) */
if (_mpd_qexp_check_one(result, a, ctx, status)) {
return;
}
mpd_maxcontext(&workctx);
workctx.prec = ctx->prec + t + 2;
workctx.prec = (workctx.prec < 10) ? 10 : workctx.prec;
workctx.round = MPD_ROUND_HALF_EVEN;
if (!mpd_qcopy(result, a, status)) {
return;
}
result->exp -= t;
/*
* At this point:
* 1) 9 * 10**(-prec-1) < abs(a)
* 2) 9 * 10**(-prec-t-1) < abs(r)
* 3) log10(9) - prec - t - 1 < log10(abs(r)) < adjexp(abs(r)) + 1
* 4) - prec - t - 2 < adjexp(abs(r)) <= -1
*/
n = _mpd_get_exp_iterations(result, workctx.prec);
if (n == MPD_SSIZE_MAX) {
mpd_seterror(result, MPD_Invalid_operation, status); /* GCOV_UNLIKELY */
return; /* GCOV_UNLIKELY */
}
_settriple(&sum, MPD_POS, 1, 0);
for (j = n-1; j >= 1; j--) {
word.data[0] = j;
mpd_setdigits(&word);
mpd_qdiv(&tmp, result, &word, &workctx, &workctx.status);
mpd_qfma(&sum, &sum, &tmp, &one, &workctx, &workctx.status);
}
#ifdef CONFIG_64
_mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status);
#else
if (t <= MPD_MAX_POW10) {
_mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status);
}
else {
t -= MPD_MAX_POW10;
_mpd_qpow_uint(&tmp, &sum, mpd_pow10[MPD_MAX_POW10], MPD_POS,
&workctx, status);
_mpd_qpow_uint(result, &tmp, mpd_pow10[t], MPD_POS, &workctx, status);
}
#endif
mpd_del(&tmp);
mpd_del(&sum);
*status |= (workctx.status&MPD_Errors);
*status |= (MPD_Inexact|MPD_Rounded);
}
/* exp(a) */
void
mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
if (mpd_isnegative(a)) {
_settriple(result, MPD_POS, 0, 0);
}
else {
mpd_setspecial(result, MPD_POS, MPD_INF);
}
return;
}
if (mpd_iszerocoeff(a)) {
_settriple(result, MPD_POS, 1, 0);
return;
}
workctx = *ctx;
workctx.round = MPD_ROUND_HALF_EVEN;
if (ctx->allcr) {
MPD_NEW_STATIC(t1, 0,0,0,0);
MPD_NEW_STATIC(t2, 0,0,0,0);
MPD_NEW_STATIC(ulp, 0,0,0,0);
MPD_NEW_STATIC(aa, 0,0,0,0);
mpd_ssize_t prec;
mpd_ssize_t ulpexp;
uint32_t workstatus;
if (result == a) {
if (!mpd_qcopy(&aa, a, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
a = &aa;
}
workctx.clamp = 0;
prec = ctx->prec + 3;
while (1) {
workctx.prec = prec;
workstatus = 0;
_mpd_qexp(result, a, &workctx, &workstatus);
*status |= workstatus;
ulpexp = result->exp + result->digits - workctx.prec;
if (workstatus & MPD_Underflow) {
/* The effective work precision is result->digits. */
ulpexp = result->exp;
}
_ssettriple(&ulp, MPD_POS, 1, ulpexp);
/*
* At this point [1]:
* 1) abs(result - e**x) < 0.5 * 10**(-prec) * e**x
* 2) result - ulp < e**x < result + ulp
* 3) result - ulp < result < result + ulp
*
* If round(result-ulp)==round(result+ulp), then
* round(result)==round(e**x). Therefore the result
* is correctly rounded.
*
* [1] If abs(a) <= 9 * 10**(-prec-1), use the absolute
* error for a similar argument.
*/
workctx.prec = ctx->prec;
mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status);
mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status);
if (mpd_isspecial(result) || mpd_iszerocoeff(result) ||
mpd_qcmp(&t1, &t2, status) == 0) {
workctx.clamp = ctx->clamp;
_mpd_zeropad(result, &workctx, status);
mpd_check_underflow(result, &workctx, status);
mpd_qfinalize(result, &workctx, status);
break;
}
prec += MPD_RDIGITS;
}
mpd_del(&t1);
mpd_del(&t2);
mpd_del(&ulp);
mpd_del(&aa);
}
else {
_mpd_qexp(result, a, &workctx, status);
_mpd_zeropad(result, &workctx, status);
mpd_check_underflow(result, &workctx, status);
mpd_qfinalize(result, &workctx, status);
}
}
/* Fused multiply-add: (a * b) + c, with a single final rounding. */
void
mpd_qfma(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_t *cc = NULL;
if (result == c) {
if ((cc = mpd_qncopy(c)) == NULL) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
c = cc;
}
_mpd_qmul(result, a, b, ctx, &workstatus);
if (!(workstatus&MPD_Invalid_operation)) {
mpd_qadd(result, result, c, ctx, &workstatus);
}
if (cc) mpd_del(cc);
*status |= workstatus;
}
/*
* Schedule the optimal precision increase for the Newton iteration.
* v := input operand
* z_0 := initial approximation
* initprec := natural number such that abs(log(v) - z_0) < 10**-initprec
* maxprec := target precision
*
* For convenience the output klist contains the elements in reverse order:
* klist := [k_n-1, ..., k_0], where
* 1) k_0 <= initprec and
* 2) abs(log(v) - result) < 10**(-2*k_n-1 + 1) <= 10**-maxprec.
*/
static inline int
ln_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], mpd_ssize_t maxprec,
mpd_ssize_t initprec)
{
mpd_ssize_t k;
int i;
assert(maxprec >= 2 && initprec >= 2);
if (maxprec <= initprec) return -1;
i = 0; k = maxprec;
do {
k = (k+2) / 2;
klist[i++] = k;
} while (k > initprec);
return i-1;
}
/* The constants have been verified with both decimal.py and mpfr. */
#ifdef CONFIG_64
#if MPD_RDIGITS != 19
#error "mpdecimal.c: MPD_RDIGITS must be 19."
#endif
static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = {
6983716328982174407ULL, 9089704281976336583ULL, 1515961135648465461ULL,
4416816335727555703ULL, 2900988039194170265ULL, 2307925037472986509ULL,
107598438319191292ULL, 3466624107184669231ULL, 4450099781311469159ULL,
9807828059751193854ULL, 7713456862091670584ULL, 1492198849978748873ULL,
6528728696511086257ULL, 2385392051446341972ULL, 8692180205189339507ULL,
6518769751037497088ULL, 2375253577097505395ULL, 9095610299291824318ULL,
982748238504564801ULL, 5438635917781170543ULL, 7547331541421808427ULL,
752371033310119785ULL, 3171643095059950878ULL, 9785265383207606726ULL,
2932258279850258550ULL, 5497347726624257094ULL, 2976979522110718264ULL,
9221477656763693866ULL, 1979650047149510504ULL, 6674183485704422507ULL,
9702766860595249671ULL, 9278096762712757753ULL, 9314848524948644871ULL,
6826928280848118428ULL, 754403708474699401ULL, 230105703089634572ULL,
1929203337658714166ULL, 7589402567763113569ULL, 4208241314695689016ULL,
2922455440575892572ULL, 9356734206705811364ULL, 2684916746550586856ULL,
644507064800027750ULL, 9476834636167921018ULL, 5659121373450747856ULL,
2835522011480466371ULL, 6470806855677432162ULL, 7141748003688084012ULL,
9619404400222105101ULL, 5504893431493939147ULL, 6674744042432743651ULL,
2287698219886746543ULL, 7773262884616336622ULL, 1985283935053089653ULL,
4680843799894826233ULL, 8168948290720832555ULL, 8067566662873690987ULL,
6248633409525465082ULL, 9829834196778404228ULL, 3524802359972050895ULL,
3327900967572609677ULL, 110148862877297603ULL, 179914546843642076ULL,
2302585092994045684ULL
};
#else
#if MPD_RDIGITS != 9
#error "mpdecimal.c: MPD_RDIGITS must be 9."
#endif
static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = {
401682692UL, 708474699UL, 720754403UL, 30896345UL, 602301057UL, 765871416UL,
192920333UL, 763113569UL, 589402567UL, 956890167UL, 82413146UL, 589257242UL,
245544057UL, 811364292UL, 734206705UL, 868569356UL, 167465505UL, 775026849UL,
706480002UL, 18064450UL, 636167921UL, 569476834UL, 734507478UL, 156591213UL,
148046637UL, 283552201UL, 677432162UL, 470806855UL, 880840126UL, 417480036UL,
210510171UL, 940440022UL, 939147961UL, 893431493UL, 436515504UL, 440424327UL,
654366747UL, 821988674UL, 622228769UL, 884616336UL, 537773262UL, 350530896UL,
319852839UL, 989482623UL, 468084379UL, 720832555UL, 168948290UL, 736909878UL,
675666628UL, 546508280UL, 863340952UL, 404228624UL, 834196778UL, 508959829UL,
23599720UL, 967735248UL, 96757260UL, 603332790UL, 862877297UL, 760110148UL,
468436420UL, 401799145UL, 299404568UL, 230258509UL
};
#endif
/* _mpd_ln10 is used directly for precisions smaller than MINALLOC_MAX*RDIGITS.
Otherwise, it serves as the initial approximation for calculating ln(10). */
static const mpd_t _mpd_ln10 = {
MPD_STATIC|MPD_CONST_DATA, -(MPD_MINALLOC_MAX*MPD_RDIGITS-1),
MPD_MINALLOC_MAX*MPD_RDIGITS, MPD_MINALLOC_MAX, MPD_MINALLOC_MAX,
(mpd_uint_t *)mpd_ln10_data
};
/*
* Set 'result' to log(10).
* Ulp error: abs(result - log(10)) < ulp(log(10))
* Relative error: abs(result - log(10)) < 5 * 10**-prec * log(10)
*
* NOTE: The relative error is not derived from the ulp error, but
* calculated separately using the fact that 23/10 < log(10) < 24/10.
*/
void
mpd_qln10(mpd_t *result, mpd_ssize_t prec, uint32_t *status)
{
mpd_context_t varcontext, maxcontext;
MPD_NEW_STATIC(tmp, 0,0,0,0);
MPD_NEW_CONST(static10, 0,0,2,1,1,10);
mpd_ssize_t klist[MPD_MAX_PREC_LOG2];
mpd_uint_t rnd;
mpd_ssize_t shift;
int i;
assert(prec >= 1);
shift = MPD_MINALLOC_MAX*MPD_RDIGITS-prec;
shift = shift < 0 ? 0 : shift;
rnd = mpd_qshiftr(result, &_mpd_ln10, shift, status);
if (rnd == MPD_UINT_MAX) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
result->exp = -(result->digits-1);
mpd_maxcontext(&maxcontext);
if (prec < MPD_MINALLOC_MAX*MPD_RDIGITS) {
maxcontext.prec = prec;
_mpd_apply_round_excess(result, rnd, &maxcontext, status);
*status |= (MPD_Inexact|MPD_Rounded);
return;
}
mpd_maxcontext(&varcontext);
varcontext.round = MPD_ROUND_TRUNC;
i = ln_schedule_prec(klist, prec+2, -result->exp);
for (; i >= 0; i--) {
varcontext.prec = 2*klist[i]+3;
result->flags ^= MPD_NEG;
_mpd_qexp(&tmp, result, &varcontext, status);
result->flags ^= MPD_NEG;
mpd_qmul(&tmp, &static10, &tmp, &varcontext, status);
mpd_qsub(&tmp, &tmp, &one, &maxcontext, status);
mpd_qadd(result, result, &tmp, &maxcontext, status);
if (mpd_isspecial(result)) {
break;
}
}
mpd_del(&tmp);
maxcontext.prec = prec;
mpd_qfinalize(result, &maxcontext, status);
}
/*
* Initial approximations for the ln() iteration. The values have the
* following properties (established with both decimal.py and mpfr):
*
* Index 0 - 400, logarithms of x in [1.00, 5.00]:
* abs(lnapprox[i] * 10**-3 - log((i+100)/100)) < 10**-2
* abs(lnapprox[i] * 10**-3 - log((i+1+100)/100)) < 10**-2
*
* Index 401 - 899, logarithms of x in (0.500, 0.999]:
* abs(-lnapprox[i] * 10**-3 - log((i+100)/1000)) < 10**-2
* abs(-lnapprox[i] * 10**-3 - log((i+1+100)/1000)) < 10**-2
*/
static const uint16_t lnapprox[900] = {
/* index 0 - 400: log((i+100)/100) * 1000 */
0, 10, 20, 30, 39, 49, 58, 68, 77, 86, 95, 104, 113, 122, 131, 140, 148, 157,
166, 174, 182, 191, 199, 207, 215, 223, 231, 239, 247, 255, 262, 270, 278,
285, 293, 300, 308, 315, 322, 329, 336, 344, 351, 358, 365, 372, 378, 385,
392, 399, 406, 412, 419, 425, 432, 438, 445, 451, 457, 464, 470, 476, 482,
489, 495, 501, 507, 513, 519, 525, 531, 536, 542, 548, 554, 560, 565, 571,
577, 582, 588, 593, 599, 604, 610, 615, 621, 626, 631, 637, 642, 647, 652,
658, 663, 668, 673, 678, 683, 688, 693, 698, 703, 708, 713, 718, 723, 728,
732, 737, 742, 747, 751, 756, 761, 766, 770, 775, 779, 784, 788, 793, 798,
802, 806, 811, 815, 820, 824, 829, 833, 837, 842, 846, 850, 854, 859, 863,
867, 871, 876, 880, 884, 888, 892, 896, 900, 904, 908, 912, 916, 920, 924,
928, 932, 936, 940, 944, 948, 952, 956, 959, 963, 967, 971, 975, 978, 982,
986, 990, 993, 997, 1001, 1004, 1008, 1012, 1015, 1019, 1022, 1026, 1030,
1033, 1037, 1040, 1044, 1047, 1051, 1054, 1058, 1061, 1065, 1068, 1072, 1075,
1078, 1082, 1085, 1089, 1092, 1095, 1099, 1102, 1105, 1109, 1112, 1115, 1118,
1122, 1125, 1128, 1131, 1135, 1138, 1141, 1144, 1147, 1151, 1154, 1157, 1160,
1163, 1166, 1169, 1172, 1176, 1179, 1182, 1185, 1188, 1191, 1194, 1197, 1200,
1203, 1206, 1209, 1212, 1215, 1218, 1221, 1224, 1227, 1230, 1233, 1235, 1238,
1241, 1244, 1247, 1250, 1253, 1256, 1258, 1261, 1264, 1267, 1270, 1273, 1275,
1278, 1281, 1284, 1286, 1289, 1292, 1295, 1297, 1300, 1303, 1306, 1308, 1311,
1314, 1316, 1319, 1322, 1324, 1327, 1330, 1332, 1335, 1338, 1340, 1343, 1345,
1348, 1351, 1353, 1356, 1358, 1361, 1364, 1366, 1369, 1371, 1374, 1376, 1379,
1381, 1384, 1386, 1389, 1391, 1394, 1396, 1399, 1401, 1404, 1406, 1409, 1411,
1413, 1416, 1418, 1421, 1423, 1426, 1428, 1430, 1433, 1435, 1437, 1440, 1442,
1445, 1447, 1449, 1452, 1454, 1456, 1459, 1461, 1463, 1466, 1468, 1470, 1472,
1475, 1477, 1479, 1482, 1484, 1486, 1488, 1491, 1493, 1495, 1497, 1500, 1502,
1504, 1506, 1509, 1511, 1513, 1515, 1517, 1520, 1522, 1524, 1526, 1528, 1530,
1533, 1535, 1537, 1539, 1541, 1543, 1545, 1548, 1550, 1552, 1554, 1556, 1558,
1560, 1562, 1564, 1567, 1569, 1571, 1573, 1575, 1577, 1579, 1581, 1583, 1585,
1587, 1589, 1591, 1593, 1595, 1597, 1599, 1601, 1603, 1605, 1607, 1609,
/* index 401 - 899: -log((i+100)/1000) * 1000 */
691, 689, 687, 685, 683, 681, 679, 677, 675, 673, 671, 669, 668, 666, 664,
662, 660, 658, 656, 654, 652, 650, 648, 646, 644, 642, 641, 639, 637, 635,
633, 631, 629, 627, 626, 624, 622, 620, 618, 616, 614, 612, 611, 609, 607,
605, 603, 602, 600, 598, 596, 594, 592, 591, 589, 587, 585, 583, 582, 580,
578, 576, 574, 573, 571, 569, 567, 566, 564, 562, 560, 559, 557, 555, 553,
552, 550, 548, 546, 545, 543, 541, 540, 538, 536, 534, 533, 531, 529, 528,
526, 524, 523, 521, 519, 518, 516, 514, 512, 511, 509, 508, 506, 504, 502,
501, 499, 498, 496, 494, 493, 491, 489, 488, 486, 484, 483, 481, 480, 478,
476, 475, 473, 472, 470, 468, 467, 465, 464, 462, 460, 459, 457, 456, 454,
453, 451, 449, 448, 446, 445, 443, 442, 440, 438, 437, 435, 434, 432, 431,
429, 428, 426, 425, 423, 422, 420, 419, 417, 416, 414, 412, 411, 410, 408,
406, 405, 404, 402, 400, 399, 398, 396, 394, 393, 392, 390, 389, 387, 386,
384, 383, 381, 380, 378, 377, 375, 374, 372, 371, 370, 368, 367, 365, 364,
362, 361, 360, 358, 357, 355, 354, 352, 351, 350, 348, 347, 345, 344, 342,
341, 340, 338, 337, 336, 334, 333, 331, 330, 328, 327, 326, 324, 323, 322,
320, 319, 318, 316, 315, 313, 312, 311, 309, 308, 306, 305, 304, 302, 301,
300, 298, 297, 296, 294, 293, 292, 290, 289, 288, 286, 285, 284, 282, 281,
280, 278, 277, 276, 274, 273, 272, 270, 269, 268, 267, 265, 264, 263, 261,
260, 259, 258, 256, 255, 254, 252, 251, 250, 248, 247, 246, 245, 243, 242,
241, 240, 238, 237, 236, 234, 233, 232, 231, 229, 228, 227, 226, 224, 223,
222, 221, 219, 218, 217, 216, 214, 213, 212, 211, 210, 208, 207, 206, 205,
203, 202, 201, 200, 198, 197, 196, 195, 194, 192, 191, 190, 189, 188, 186,
185, 184, 183, 182, 180, 179, 178, 177, 176, 174, 173, 172, 171, 170, 168,
167, 166, 165, 164, 162, 161, 160, 159, 158, 157, 156, 154, 153, 152, 151,
150, 148, 147, 146, 145, 144, 143, 142, 140, 139, 138, 137, 136, 135, 134,
132, 131, 130, 129, 128, 127, 126, 124, 123, 122, 121, 120, 119, 118, 116,
115, 114, 113, 112, 111, 110, 109, 108, 106, 105, 104, 103, 102, 101, 100,
99, 98, 97, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 84, 83, 82, 81, 80, 79,
78, 77, 76, 75, 74, 73, 72, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59,
58, 57, 56, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39,
38, 37, 36, 35, 34, 33, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19,
18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
};
/*
* Internal ln() function that does not check for specials, zero or one.
* Relative error: abs(result - log(a)) < 0.1 * 10**-prec * abs(log(a))
*/
static void
_mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t varcontext, maxcontext;
mpd_t *z = (mpd_t *) result;
MPD_NEW_STATIC(v,0,0,0,0);
MPD_NEW_STATIC(vtmp,0,0,0,0);
MPD_NEW_STATIC(tmp,0,0,0,0);
mpd_ssize_t klist[MPD_MAX_PREC_LOG2];
mpd_ssize_t maxprec, shift, t;
mpd_ssize_t a_digits, a_exp;
mpd_uint_t dummy, x;
int i;
assert(!mpd_isspecial(a) && !mpd_iszerocoeff(a));
/*
* We are calculating ln(a) = ln(v * 10^t) = ln(v) + t*ln(10),
* where 0.5 < v <= 5.
*/
if (!mpd_qcopy(&v, a, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
/* Initial approximation: we have at least one non-zero digit */
_mpd_get_msdigits(&dummy, &x, &v, 3);
if (x < 10) x *= 10;
if (x < 100) x *= 10;
x -= 100;
/* a may equal z */
a_digits = a->digits;
a_exp = a->exp;
mpd_minalloc(z);
mpd_clear_flags(z);
z->data[0] = lnapprox[x];
z->len = 1;
z->exp = -3;
mpd_setdigits(z);
if (x <= 400) {
/* Reduce the input operand to 1.00 <= v <= 5.00. Let y = x + 100,
* so 100 <= y <= 500. Since y contains the most significant digits
* of v, y/100 <= v < (y+1)/100 and abs(z - log(v)) < 10**-2. */
v.exp = -(a_digits - 1);
t = a_exp + a_digits - 1;
}
else {
/* Reduce the input operand to 0.500 < v <= 0.999. Let y = x + 100,
* so 500 < y <= 999. Since y contains the most significant digits
* of v, y/1000 <= v < (y+1)/1000 and abs(z - log(v)) < 10**-2. */
v.exp = -a_digits;
t = a_exp + a_digits;
mpd_set_negative(z);
}
mpd_maxcontext(&maxcontext);
mpd_maxcontext(&varcontext);
varcontext.round = MPD_ROUND_TRUNC;
maxprec = ctx->prec + 2;
if (t == 0 && (x <= 15 || x >= 800)) {
/* 0.900 <= v <= 1.15: Estimate the magnitude of the logarithm.
* If ln(v) will underflow, skip the loop. Otherwise, adjust the
* precision upwards in order to obtain a sufficient number of
* significant digits.
*
* Case v > 1:
* abs((v-1)/10) < abs((v-1)/v) < abs(ln(v)) < abs(v-1)
* Case v < 1:
* abs(v-1) < abs(ln(v)) < abs((v-1)/v) < abs((v-1)*10)
*/
int cmp = _mpd_cmp(&v, &one);
/* Upper bound (assume v > 1): abs(v-1), unrounded */
_mpd_qsub(&tmp, &v, &one, &maxcontext, &maxcontext.status);
if (maxcontext.status & MPD_Errors) {
mpd_seterror(result, MPD_Malloc_error, status);
goto finish;
}
if (cmp < 0) {
/* v < 1: abs((v-1)*10) */
tmp.exp += 1;
}
if (mpd_adjexp(&tmp) < mpd_etiny(ctx)) {
/* The upper bound is less than etiny: Underflow to zero */
_settriple(result, (cmp<0), 1, mpd_etiny(ctx)-1);
goto finish;
}
/* Lower bound: abs((v-1)/10) or abs(v-1) */
tmp.exp -= 1;
if (mpd_adjexp(&tmp) < 0) {
/* Absolute error of the loop: abs(z - log(v)) < 10**-p. If
* p = ctx->prec+2-adjexp(lower), then the relative error of
* the result is (using 10**adjexp(x) <= abs(x)):
*
* abs(z - log(v)) / abs(log(v)) < 10**-p / abs(log(v))
* <= 10**(-ctx->prec-2)
*/
maxprec = maxprec - mpd_adjexp(&tmp);
}
}
i = ln_schedule_prec(klist, maxprec, 2);
for (; i >= 0; i--) {
varcontext.prec = 2*klist[i]+3;
z->flags ^= MPD_NEG;
_mpd_qexp(&tmp, z, &varcontext, status);
z->flags ^= MPD_NEG;
if (v.digits > varcontext.prec) {
shift = v.digits - varcontext.prec;
mpd_qshiftr(&vtmp, &v, shift, status);
vtmp.exp += shift;
mpd_qmul(&tmp, &vtmp, &tmp, &varcontext, status);
}
else {
mpd_qmul(&tmp, &v, &tmp, &varcontext, status);
}
mpd_qsub(&tmp, &tmp, &one, &maxcontext, status);
mpd_qadd(z, z, &tmp, &maxcontext, status);
if (mpd_isspecial(z)) {
break;
}
}
/*
* Case t == 0:
* t * log(10) == 0, the result does not change and the analysis
* above applies. If v < 0.900 or v > 1.15, the relative error is
* less than 10**(-ctx.prec-1).
* Case t != 0:
* z := approx(log(v))
* y := approx(log(10))
* p := maxprec = ctx->prec + 2
* Absolute errors:
* 1) abs(z - log(v)) < 10**-p
* 2) abs(y - log(10)) < 10**-p
* The multiplication is exact, so:
* 3) abs(t*y - t*log(10)) < t*10**-p
* The sum is exact, so:
* 4) abs((z + t*y) - (log(v) + t*log(10))) < (abs(t) + 1) * 10**-p
* Bounds for log(v) and log(10):
* 5) -7/10 < log(v) < 17/10
* 6) 23/10 < log(10) < 24/10
* Using 4), 5), 6) and t != 0, the relative error is:
*
* 7) relerr < ((abs(t) + 1)*10**-p) / abs(log(v) + t*log(10))
* < 0.5 * 10**(-p + 1) = 0.5 * 10**(-ctx->prec-1)
*/
mpd_qln10(&v, maxprec+1, status);
mpd_qmul_ssize(&tmp, &v, t, &maxcontext, status);
mpd_qadd(result, &tmp, z, &maxcontext, status);
finish:
*status |= (MPD_Inexact|MPD_Rounded);
mpd_del(&v);
mpd_del(&vtmp);
mpd_del(&tmp);
}
/* ln(a) */
void
mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
mpd_ssize_t adjexp, t;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
mpd_setspecial(result, MPD_POS, MPD_INF);
return;
}
if (mpd_iszerocoeff(a)) {
mpd_setspecial(result, MPD_NEG, MPD_INF);
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (_mpd_cmp(a, &one) == 0) {
_settriple(result, MPD_POS, 0, 0);
return;
}
/*
* Check if the result will overflow (0 < x, x != 1):
* 1) log10(x) < 0 iff adjexp(x) < 0
* 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y)
* 3) 0 < x /\ x != 1 ==> 2 * abs(log10(x)) < abs(log(x))
* 4) adjexp(x) <= log10(x) < adjexp(x) + 1
*
* Case adjexp(x) >= 0:
* 5) 2 * adjexp(x) < abs(log(x))
* Case adjexp(x) > 0:
* 6) adjexp(2 * adjexp(x)) <= adjexp(abs(log(x)))
* Case adjexp(x) == 0:
* mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered)
*
* Case adjexp(x) < 0:
* 7) 2 * (-adjexp(x) - 1) < abs(log(x))
* Case adjexp(x) < -1:
* 8) adjexp(2 * (-adjexp(x) - 1)) <= adjexp(abs(log(x)))
* Case adjexp(x) == -1:
* mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered)
*/
adjexp = mpd_adjexp(a);
t = (adjexp < 0) ? -adjexp-1 : adjexp;
t *= 2;
if (mpd_exp_digits(t)-1 > ctx->emax) {
*status |= MPD_Overflow|MPD_Inexact|MPD_Rounded;
mpd_setspecial(result, (adjexp<0), MPD_INF);
return;
}
workctx = *ctx;
workctx.round = MPD_ROUND_HALF_EVEN;
if (ctx->allcr) {
MPD_NEW_STATIC(t1, 0,0,0,0);
MPD_NEW_STATIC(t2, 0,0,0,0);
MPD_NEW_STATIC(ulp, 0,0,0,0);
MPD_NEW_STATIC(aa, 0,0,0,0);
mpd_ssize_t prec;
if (result == a) {
if (!mpd_qcopy(&aa, a, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
a = &aa;
}
workctx.clamp = 0;
prec = ctx->prec + 3;
while (1) {
workctx.prec = prec;
_mpd_qln(result, a, &workctx, status);
_ssettriple(&ulp, MPD_POS, 1,
result->exp + result->digits-workctx.prec);
workctx.prec = ctx->prec;
mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status);
mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status);
if (mpd_isspecial(result) || mpd_iszerocoeff(result) ||
mpd_qcmp(&t1, &t2, status) == 0) {
workctx.clamp = ctx->clamp;
mpd_check_underflow(result, &workctx, status);
mpd_qfinalize(result, &workctx, status);
break;
}
prec += MPD_RDIGITS;
}
mpd_del(&t1);
mpd_del(&t2);
mpd_del(&ulp);
mpd_del(&aa);
}
else {
_mpd_qln(result, a, &workctx, status);
mpd_check_underflow(result, &workctx, status);
mpd_qfinalize(result, &workctx, status);
}
}
/*
* Internal log10() function that does not check for specials, zero or one.
* Case SKIP_FINALIZE:
* Relative error: abs(result - log10(a)) < 0.1 * 10**-prec * abs(log10(a))
* Case DO_FINALIZE:
* Ulp error: abs(result - log10(a)) < ulp(log10(a))
*/
enum {SKIP_FINALIZE, DO_FINALIZE};
static void
_mpd_qlog10(int action, mpd_t *result, const mpd_t *a,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_STATIC(ln10,0,0,0,0);
mpd_maxcontext(&workctx);
workctx.prec = ctx->prec + 3;
/* relative error: 0.1 * 10**(-p-3). The specific underflow shortcut
* in _mpd_qln() does not change the final result. */
_mpd_qln(result, a, &workctx, status);
/* relative error: 5 * 10**(-p-3) */
mpd_qln10(&ln10, workctx.prec, status);
if (action == DO_FINALIZE) {
workctx = *ctx;
workctx.round = MPD_ROUND_HALF_EVEN;
}
/* SKIP_FINALIZE: relative error: 5 * 10**(-p-3) */
_mpd_qdiv(NO_IDEAL_EXP, result, result, &ln10, &workctx, status);
mpd_del(&ln10);
}
/* log10(a) */
void
mpd_qlog10(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
mpd_ssize_t adjexp, t;
workctx = *ctx;
workctx.round = MPD_ROUND_HALF_EVEN;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
mpd_setspecial(result, MPD_POS, MPD_INF);
return;
}
if (mpd_iszerocoeff(a)) {
mpd_setspecial(result, MPD_NEG, MPD_INF);
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_coeff_ispow10(a)) {
uint8_t sign = 0;
adjexp = mpd_adjexp(a);
if (adjexp < 0) {
sign = 1;
adjexp = -adjexp;
}
_settriple(result, sign, adjexp, 0);
mpd_qfinalize(result, &workctx, status);
return;
}
/*
* Check if the result will overflow (0 < x, x != 1):
* 1) log10(x) < 0 iff adjexp(x) < 0
* 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y)
* 3) adjexp(x) <= log10(x) < adjexp(x) + 1
*
* Case adjexp(x) >= 0:
* 4) adjexp(x) <= abs(log10(x))
* Case adjexp(x) > 0:
* 5) adjexp(adjexp(x)) <= adjexp(abs(log10(x)))
* Case adjexp(x) == 0:
* mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered)
*
* Case adjexp(x) < 0:
* 6) -adjexp(x) - 1 < abs(log10(x))
* Case adjexp(x) < -1:
* 7) adjexp(-adjexp(x) - 1) <= adjexp(abs(log(x)))
* Case adjexp(x) == -1:
* mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered)
*/
adjexp = mpd_adjexp(a);
t = (adjexp < 0) ? -adjexp-1 : adjexp;
if (mpd_exp_digits(t)-1 > ctx->emax) {
*status |= MPD_Overflow|MPD_Inexact|MPD_Rounded;
mpd_setspecial(result, (adjexp<0), MPD_INF);
return;
}
if (ctx->allcr) {
MPD_NEW_STATIC(t1, 0,0,0,0);
MPD_NEW_STATIC(t2, 0,0,0,0);
MPD_NEW_STATIC(ulp, 0,0,0,0);
MPD_NEW_STATIC(aa, 0,0,0,0);
mpd_ssize_t prec;
if (result == a) {
if (!mpd_qcopy(&aa, a, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
a = &aa;
}
workctx.clamp = 0;
prec = ctx->prec + 3;
while (1) {
workctx.prec = prec;
_mpd_qlog10(SKIP_FINALIZE, result, a, &workctx, status);
_ssettriple(&ulp, MPD_POS, 1,
result->exp + result->digits-workctx.prec);
workctx.prec = ctx->prec;
mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status);
mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status);
if (mpd_isspecial(result) || mpd_iszerocoeff(result) ||
mpd_qcmp(&t1, &t2, status) == 0) {
workctx.clamp = ctx->clamp;
mpd_check_underflow(result, &workctx, status);
mpd_qfinalize(result, &workctx, status);
break;
}
prec += MPD_RDIGITS;
}
mpd_del(&t1);
mpd_del(&t2);
mpd_del(&ulp);
mpd_del(&aa);
}
else {
_mpd_qlog10(DO_FINALIZE, result, a, &workctx, status);
mpd_check_underflow(result, &workctx, status);
}
}
/*
* Maximum of the two operands. Attention: If one operand is a quiet NaN and the
* other is numeric, the numeric operand is returned. This may not be what one
* expects.
*/
void
mpd_qmax(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isqnan(a) && !mpd_isnan(b)) {
mpd_qcopy(result, b, status);
}
else if (mpd_isqnan(b) && !mpd_isnan(a)) {
mpd_qcopy(result, a, status);
}
else if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
else {
c = _mpd_cmp(a, b);
if (c == 0) {
c = _mpd_cmp_numequal(a, b);
}
if (c < 0) {
mpd_qcopy(result, b, status);
}
else {
mpd_qcopy(result, a, status);
}
}
mpd_qfinalize(result, ctx, status);
}
/*
* Maximum magnitude: Same as mpd_max(), but compares the operands with their
* sign ignored.
*/
void
mpd_qmax_mag(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isqnan(a) && !mpd_isnan(b)) {
mpd_qcopy(result, b, status);
}
else if (mpd_isqnan(b) && !mpd_isnan(a)) {
mpd_qcopy(result, a, status);
}
else if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
else {
c = _mpd_cmp_abs(a, b);
if (c == 0) {
c = _mpd_cmp_numequal(a, b);
}
if (c < 0) {
mpd_qcopy(result, b, status);
}
else {
mpd_qcopy(result, a, status);
}
}
mpd_qfinalize(result, ctx, status);
}
/*
* Minimum of the two operands. Attention: If one operand is a quiet NaN and the
* other is numeric, the numeric operand is returned. This may not be what one
* expects.
*/
void
mpd_qmin(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isqnan(a) && !mpd_isnan(b)) {
mpd_qcopy(result, b, status);
}
else if (mpd_isqnan(b) && !mpd_isnan(a)) {
mpd_qcopy(result, a, status);
}
else if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
else {
c = _mpd_cmp(a, b);
if (c == 0) {
c = _mpd_cmp_numequal(a, b);
}
if (c < 0) {
mpd_qcopy(result, a, status);
}
else {
mpd_qcopy(result, b, status);
}
}
mpd_qfinalize(result, ctx, status);
}
/*
* Minimum magnitude: Same as mpd_min(), but compares the operands with their
* sign ignored.
*/
void
mpd_qmin_mag(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_isqnan(a) && !mpd_isnan(b)) {
mpd_qcopy(result, b, status);
}
else if (mpd_isqnan(b) && !mpd_isnan(a)) {
mpd_qcopy(result, a, status);
}
else if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
else {
c = _mpd_cmp_abs(a, b);
if (c == 0) {
c = _mpd_cmp_numequal(a, b);
}
if (c < 0) {
mpd_qcopy(result, a, status);
}
else {
mpd_qcopy(result, b, status);
}
}
mpd_qfinalize(result, ctx, status);
}
/* Minimum space needed for the result array in _karatsuba_rec(). */
static inline mpd_size_t
_kmul_resultsize(mpd_size_t la, mpd_size_t lb)
{
mpd_size_t n, m;
n = add_size_t(la, lb);
n = add_size_t(n, 1);
m = (la+1)/2 + 1;
m = mul_size_t(m, 3);
return (m > n) ? m : n;
}
/* Work space needed in _karatsuba_rec(). lim >= 4 */
static inline mpd_size_t
_kmul_worksize(mpd_size_t n, mpd_size_t lim)
{
mpd_size_t m;
if (n <= lim) {
return 0;
}
m = (n+1)/2 + 1;
return add_size_t(mul_size_t(m, 2), _kmul_worksize(m, lim));
}
#define MPD_KARATSUBA_BASECASE 16 /* must be >= 4 */
/*
* Add the product of a and b to c.
* c must be _kmul_resultsize(la, lb) in size.
* w is used as a work array and must be _kmul_worksize(a, lim) in size.
* Roman E. Maeder, Storage Allocation for the Karatsuba Integer Multiplication
* Algorithm. In "Design and implementation of symbolic computation systems",
* Springer, 1993, ISBN 354057235X, 9783540572350.
*/
static void
_karatsuba_rec(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b,
mpd_uint_t *w, mpd_size_t la, mpd_size_t lb)
{
mpd_size_t m, lt;
assert(la >= lb && lb > 0);
assert(la <= MPD_KARATSUBA_BASECASE || w != NULL);
if (la <= MPD_KARATSUBA_BASECASE) {
_mpd_basemul(c, a, b, la, lb);
return;
}
m = (la+1)/2; /* ceil(la/2) */
/* lb <= m < la */
if (lb <= m) {
/* lb can now be larger than la-m */
if (lb > la-m) {
lt = lb + lb + 1; /* space needed for result array */
mpd_uint_zero(w, lt); /* clear result array */
_karatsuba_rec(w, b, a+m, w+lt, lb, la-m); /* b*ah */
}
else {
lt = (la-m) + (la-m) + 1; /* space needed for result array */
mpd_uint_zero(w, lt); /* clear result array */
_karatsuba_rec(w, a+m, b, w+lt, la-m, lb); /* ah*b */
}
_mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */
lt = m + m + 1; /* space needed for the result array */
mpd_uint_zero(w, lt); /* clear result array */
_karatsuba_rec(w, a, b, w+lt, m, lb); /* al*b */
_mpd_baseaddto(c, w, m+lb); /* add al*b */
return;
}
/* la >= lb > m */
memcpy(w, a, m * sizeof *w);
w[m] = 0;
_mpd_baseaddto(w, a+m, la-m);
memcpy(w+(m+1), b, m * sizeof *w);
w[m+1+m] = 0;
_mpd_baseaddto(w+(m+1), b+m, lb-m);
_karatsuba_rec(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1);
lt = (la-m) + (la-m) + 1;
mpd_uint_zero(w, lt);
_karatsuba_rec(w, a+m, b+m, w+lt, la-m, lb-m);
_mpd_baseaddto(c+2*m, w, (la-m) + (lb-m));
_mpd_basesubfrom(c+m, w, (la-m) + (lb-m));
lt = m + m + 1;
mpd_uint_zero(w, lt);
_karatsuba_rec(w, a, b, w+lt, m, m);
_mpd_baseaddto(c, w, m+m);
_mpd_basesubfrom(c+m, w, m+m);
return;
}
/*
* Multiply u and v, using Karatsuba multiplication. Returns a pointer
* to the result or NULL in case of failure (malloc error).
* Conditions: ulen >= vlen, ulen >= 4
*/
static mpd_uint_t *
_mpd_kmul(const mpd_uint_t *u, const mpd_uint_t *v,
mpd_size_t ulen, mpd_size_t vlen,
mpd_size_t *rsize)
{
mpd_uint_t *result = NULL, *w = NULL;
mpd_size_t m;
assert(ulen >= 4);
assert(ulen >= vlen);
*rsize = _kmul_resultsize(ulen, vlen);
if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) {
return NULL;
}
m = _kmul_worksize(ulen, MPD_KARATSUBA_BASECASE);
if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) {
mpd_free(result);
return NULL;
}
_karatsuba_rec(result, u, v, w, ulen, vlen);
if (w) mpd_free(w);
return result;
}
/*
* Determine the minimum length for the number theoretic transform. Valid
* transform lengths are 2**n or 3*2**n, where 2**n <= MPD_MAXTRANSFORM_2N.
* The function finds the shortest length m such that rsize <= m.
*/
static inline mpd_size_t
_mpd_get_transform_len(mpd_size_t rsize)
{
mpd_size_t log2rsize;
mpd_size_t x, step;
assert(rsize >= 4);
log2rsize = mpd_bsr(rsize);
if (rsize <= 1024) {
/* 2**n is faster in this range. */
x = ((mpd_size_t)1)<<log2rsize;
return (rsize == x) ? x : x<<1;
}
else if (rsize <= MPD_MAXTRANSFORM_2N) {
x = ((mpd_size_t)1)<<log2rsize;
if (rsize == x) return x;
step = x>>1;
x += step;
return (rsize <= x) ? x : x + step;
}
else if (rsize <= MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2) {
return MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2;
}
else if (rsize <= 3*MPD_MAXTRANSFORM_2N) {
return 3*MPD_MAXTRANSFORM_2N;
}
else {
return MPD_SIZE_MAX;
}
}
#ifdef PPRO
#ifndef _MSC_VER
static inline unsigned short
_mpd_get_control87(void)
{
unsigned short cw;
__asm__ __volatile__ ("fnstcw %0" : "=m" (cw));
return cw;
}
static inline void
_mpd_set_control87(unsigned short cw)
{
__asm__ __volatile__ ("fldcw %0" : : "m" (cw));
}
#endif
static unsigned int
mpd_set_fenv(void)
{
unsigned int cw;
#ifdef _MSC_VER
unsigned int flags =
_EM_INVALID|_EM_DENORMAL|_EM_ZERODIVIDE|_EM_OVERFLOW|
_EM_UNDERFLOW|_EM_INEXACT|_RC_CHOP|_PC_64;
unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC;
unsigned int dummy;
__control87_2(0, 0, &cw, NULL);
__control87_2(flags, mask, &dummy, NULL);
#else
cw = _mpd_get_control87();
_mpd_set_control87(cw|0xF3F);
#endif
return cw;
}
static void
mpd_restore_fenv(unsigned int cw)
{
#ifdef _MSC_VER
unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC;
unsigned int dummy;
__control87_2(cw, mask, &dummy, NULL);
#else
_mpd_set_control87((unsigned short)cw);
#endif
}
#endif /* PPRO */
/*
* Multiply u and v, using the fast number theoretic transform. Returns
* a pointer to the result or NULL in case of failure (malloc error).
*/
static mpd_uint_t *
_mpd_fntmul(const mpd_uint_t *u, const mpd_uint_t *v,
mpd_size_t ulen, mpd_size_t vlen,
mpd_size_t *rsize)
{
mpd_uint_t *c1 = NULL, *c2 = NULL, *c3 = NULL, *vtmp = NULL;
mpd_size_t n;
#ifdef PPRO
unsigned int cw;
cw = mpd_set_fenv();
#endif
*rsize = add_size_t(ulen, vlen);
if ((n = _mpd_get_transform_len(*rsize)) == MPD_SIZE_MAX) {
goto malloc_error;
}
if ((c1 = mpd_calloc(n, sizeof *c1)) == NULL) {
goto malloc_error;
}
if ((c2 = mpd_calloc(n, sizeof *c2)) == NULL) {
goto malloc_error;
}
if ((c3 = mpd_calloc(n, sizeof *c3)) == NULL) {
goto malloc_error;
}
memcpy(c1, u, ulen * (sizeof *c1));
memcpy(c2, u, ulen * (sizeof *c2));
memcpy(c3, u, ulen * (sizeof *c3));
if (u == v) {
if (!fnt_autoconvolute(c1, n, P1) ||
!fnt_autoconvolute(c2, n, P2) ||
!fnt_autoconvolute(c3, n, P3)) {
goto malloc_error;
}
}
else {
if ((vtmp = mpd_calloc(n, sizeof *vtmp)) == NULL) {
goto malloc_error;
}
memcpy(vtmp, v, vlen * (sizeof *vtmp));
if (!fnt_convolute(c1, vtmp, n, P1)) {
mpd_free(vtmp);
goto malloc_error;
}
memcpy(vtmp, v, vlen * (sizeof *vtmp));
mpd_uint_zero(vtmp+vlen, n-vlen);
if (!fnt_convolute(c2, vtmp, n, P2)) {
mpd_free(vtmp);
goto malloc_error;
}
memcpy(vtmp, v, vlen * (sizeof *vtmp));
mpd_uint_zero(vtmp+vlen, n-vlen);
if (!fnt_convolute(c3, vtmp, n, P3)) {
mpd_free(vtmp);
goto malloc_error;
}
mpd_free(vtmp);
}
crt3(c1, c2, c3, *rsize);
out:
#ifdef PPRO
mpd_restore_fenv(cw);
#endif
if (c2) mpd_free(c2);
if (c3) mpd_free(c3);
return c1;
malloc_error:
if (c1) mpd_free(c1);
c1 = NULL;
goto out;
}
/*
* Karatsuba multiplication with FNT/basemul as the base case.
*/
static int
_karatsuba_rec_fnt(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b,
mpd_uint_t *w, mpd_size_t la, mpd_size_t lb)
{
mpd_size_t m, lt;
assert(la >= lb && lb > 0);
assert(la <= 3*(MPD_MAXTRANSFORM_2N/2) || w != NULL);
if (la <= 3*(MPD_MAXTRANSFORM_2N/2)) {
if (lb <= 192) {
_mpd_basemul(c, b, a, lb, la);
}
else {
mpd_uint_t *result;
mpd_size_t dummy;
if ((result = _mpd_fntmul(a, b, la, lb, &dummy)) == NULL) {
return 0;
}
memcpy(c, result, (la+lb) * (sizeof *result));
mpd_free(result);
}
return 1;
}
m = (la+1)/2; /* ceil(la/2) */
/* lb <= m < la */
if (lb <= m) {
/* lb can now be larger than la-m */
if (lb > la-m) {
lt = lb + lb + 1; /* space needed for result array */
mpd_uint_zero(w, lt); /* clear result array */
if (!_karatsuba_rec_fnt(w, b, a+m, w+lt, lb, la-m)) { /* b*ah */
return 0; /* GCOV_UNLIKELY */
}
}
else {
lt = (la-m) + (la-m) + 1; /* space needed for result array */
mpd_uint_zero(w, lt); /* clear result array */
if (!_karatsuba_rec_fnt(w, a+m, b, w+lt, la-m, lb)) { /* ah*b */
return 0; /* GCOV_UNLIKELY */
}
}
_mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */
lt = m + m + 1; /* space needed for the result array */
mpd_uint_zero(w, lt); /* clear result array */
if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, lb)) { /* al*b */
return 0; /* GCOV_UNLIKELY */
}
_mpd_baseaddto(c, w, m+lb); /* add al*b */
return 1;
}
/* la >= lb > m */
memcpy(w, a, m * sizeof *w);
w[m] = 0;
_mpd_baseaddto(w, a+m, la-m);
memcpy(w+(m+1), b, m * sizeof *w);
w[m+1+m] = 0;
_mpd_baseaddto(w+(m+1), b+m, lb-m);
if (!_karatsuba_rec_fnt(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1)) {
return 0; /* GCOV_UNLIKELY */
}
lt = (la-m) + (la-m) + 1;
mpd_uint_zero(w, lt);
if (!_karatsuba_rec_fnt(w, a+m, b+m, w+lt, la-m, lb-m)) {
return 0; /* GCOV_UNLIKELY */
}
_mpd_baseaddto(c+2*m, w, (la-m) + (lb-m));
_mpd_basesubfrom(c+m, w, (la-m) + (lb-m));
lt = m + m + 1;
mpd_uint_zero(w, lt);
if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, m)) {
return 0; /* GCOV_UNLIKELY */
}
_mpd_baseaddto(c, w, m+m);
_mpd_basesubfrom(c+m, w, m+m);
return 1;
}
/*
* Multiply u and v, using Karatsuba multiplication with the FNT as the
* base case. Returns a pointer to the result or NULL in case of failure
* (malloc error). Conditions: ulen >= vlen, ulen >= 4.
*/
static mpd_uint_t *
_mpd_kmul_fnt(const mpd_uint_t *u, const mpd_uint_t *v,
mpd_size_t ulen, mpd_size_t vlen,
mpd_size_t *rsize)
{
mpd_uint_t *result = NULL, *w = NULL;
mpd_size_t m;
assert(ulen >= 4);
assert(ulen >= vlen);
*rsize = _kmul_resultsize(ulen, vlen);
if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) {
return NULL;
}
m = _kmul_worksize(ulen, 3*(MPD_MAXTRANSFORM_2N/2));
if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) {
mpd_free(result); /* GCOV_UNLIKELY */
return NULL; /* GCOV_UNLIKELY */
}
if (!_karatsuba_rec_fnt(result, u, v, w, ulen, vlen)) {
mpd_free(result);
result = NULL;
}
if (w) mpd_free(w);
return result;
}
/* Deal with the special cases of multiplying infinities. */
static void
_mpd_qmul_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status)
{
if (mpd_isinfinite(a)) {
if (mpd_iszero(b)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
else {
mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF);
}
return;
}
assert(mpd_isinfinite(b));
if (mpd_iszero(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
else {
mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF);
}
}
/*
* Internal function: Multiply a and b. _mpd_qmul deals with specials but
* does NOT finalize the result. This is for use in mpd_fma().
*/
static inline void
_mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
const mpd_t *big = a, *small = b;
mpd_uint_t *rdata = NULL;
mpd_uint_t rbuf[MPD_MINALLOC_MAX];
mpd_size_t rsize, i;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
_mpd_qmul_inf(result, a, b, status);
return;
}
if (small->len > big->len) {
_mpd_ptrswap(&big, &small);
}
rsize = big->len + small->len;
if (big->len == 1) {
_mpd_singlemul(result->data, big->data[0], small->data[0]);
goto finish;
}
if (rsize <= (mpd_size_t)MPD_MINALLOC_MAX) {
if (big->len == 2) {
_mpd_mul_2_le2(rbuf, big->data, small->data, small->len);
}
else {
mpd_uint_zero(rbuf, rsize);
if (small->len == 1) {
_mpd_shortmul(rbuf, big->data, big->len, small->data[0]);
}
else {
_mpd_basemul(rbuf, small->data, big->data, small->len, big->len);
}
}
if (!mpd_qresize(result, rsize, status)) {
return;
}
for(i = 0; i < rsize; i++) {
result->data[i] = rbuf[i];
}
goto finish;
}
if (small->len <= 256) {
rdata = mpd_calloc(rsize, sizeof *rdata);
if (rdata != NULL) {
if (small->len == 1) {
_mpd_shortmul(rdata, big->data, big->len, small->data[0]);
}
else {
_mpd_basemul(rdata, small->data, big->data, small->len, big->len);
}
}
}
else if (rsize <= 1024) {
rdata = _mpd_kmul(big->data, small->data, big->len, small->len, &rsize);
}
else if (rsize <= 3*MPD_MAXTRANSFORM_2N) {
rdata = _mpd_fntmul(big->data, small->data, big->len, small->len, &rsize);
}
else {
rdata = _mpd_kmul_fnt(big->data, small->data, big->len, small->len, &rsize);
}
if (rdata == NULL) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
if (mpd_isdynamic_data(result)) {
mpd_free(result->data);
}
result->data = rdata;
result->alloc = rsize;
mpd_set_dynamic_data(result);
finish:
mpd_set_flags(result, mpd_sign(a)^mpd_sign(b));
result->exp = big->exp + small->exp;
result->len = _mpd_real_size(result->data, rsize);
/* resize to smaller cannot fail */
mpd_qresize(result, result->len, status);
mpd_setdigits(result);
}
/* Multiply a and b. */
void
mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
_mpd_qmul(result, a, b, ctx, status);
mpd_qfinalize(result, ctx, status);
}
/* Multiply a and b. Set NaN/Invalid_operation if the result is inexact. */
static void
_mpd_qmul_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_qmul(result, a, b, ctx, &workstatus);
*status |= workstatus;
if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
}
/* Multiply decimal and mpd_ssize_t. */
void
mpd_qmul_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_ssize(&bb, b, &maxcontext, status);
mpd_qmul(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Multiply decimal and mpd_uint_t. */
void
mpd_qmul_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qsset_uint(&bb, b, &maxcontext, status);
mpd_qmul(result, a, &bb, ctx, status);
mpd_del(&bb);
}
void
mpd_qmul_i32(mpd_t *result, const mpd_t *a, int32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qmul_ssize(result, a, b, ctx, status);
}
void
mpd_qmul_u32(mpd_t *result, const mpd_t *a, uint32_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qmul_uint(result, a, b, ctx, status);
}
#ifdef CONFIG_64
void
mpd_qmul_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qmul_ssize(result, a, b, ctx, status);
}
void
mpd_qmul_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_qmul_uint(result, a, b, ctx, status);
}
#elif !defined(LEGACY_COMPILER)
/* Multiply decimal and int64_t. */
void
mpd_qmul_i64(mpd_t *result, const mpd_t *a, int64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_i64(&bb, b, &maxcontext, status);
mpd_qmul(result, a, &bb, ctx, status);
mpd_del(&bb);
}
/* Multiply decimal and uint64_t. */
void
mpd_qmul_u64(mpd_t *result, const mpd_t *a, uint64_t b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(bb,0,0,0,0);
mpd_maxcontext(&maxcontext);
mpd_qset_u64(&bb, b, &maxcontext, status);
mpd_qmul(result, a, &bb, ctx, status);
mpd_del(&bb);
}
#endif
/* Like the minus operator. */
void
mpd_qminus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
}
if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) {
mpd_qcopy_abs(result, a, status);
}
else {
mpd_qcopy_negate(result, a, status);
}
mpd_qfinalize(result, ctx, status);
}
/* Like the plus operator. */
void
mpd_qplus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
}
if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) {
mpd_qcopy_abs(result, a, status);
}
else {
mpd_qcopy(result, a, status);
}
mpd_qfinalize(result, ctx, status);
}
/* The largest representable number that is smaller than the operand. */
void
mpd_qnext_minus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1);
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
assert(mpd_isinfinite(a));
if (mpd_isnegative(a)) {
mpd_qcopy(result, a, status);
return;
}
else {
mpd_clear_flags(result);
mpd_qmaxcoeff(result, ctx, status);
if (mpd_isnan(result)) {
return;
}
result->exp = mpd_etop(ctx);
return;
}
}
mpd_workcontext(&workctx, ctx);
workctx.round = MPD_ROUND_FLOOR;
if (!mpd_qcopy(result, a, status)) {
return;
}
mpd_qfinalize(result, &workctx, &workctx.status);
if (workctx.status&(MPD_Inexact|MPD_Errors)) {
*status |= (workctx.status&MPD_Errors);
return;
}
workctx.status = 0;
mpd_qsub(result, a, &tiny, &workctx, &workctx.status);
*status |= (workctx.status&MPD_Errors);
}
/* The smallest representable number that is larger than the operand. */
void
mpd_qnext_plus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1);
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
assert(mpd_isinfinite(a));
if (mpd_ispositive(a)) {
mpd_qcopy(result, a, status);
}
else {
mpd_clear_flags(result);
mpd_qmaxcoeff(result, ctx, status);
if (mpd_isnan(result)) {
return;
}
mpd_set_flags(result, MPD_NEG);
result->exp = mpd_etop(ctx);
}
return;
}
mpd_workcontext(&workctx, ctx);
workctx.round = MPD_ROUND_CEILING;
if (!mpd_qcopy(result, a, status)) {
return;
}
mpd_qfinalize(result, &workctx, &workctx.status);
if (workctx.status & (MPD_Inexact|MPD_Errors)) {
*status |= (workctx.status&MPD_Errors);
return;
}
workctx.status = 0;
mpd_qadd(result, a, &tiny, &workctx, &workctx.status);
*status |= (workctx.status&MPD_Errors);
}
/*
* The number closest to the first operand that is in the direction towards
* the second operand.
*/
void
mpd_qnext_toward(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
int c;
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
c = _mpd_cmp(a, b);
if (c == 0) {
mpd_qcopy_sign(result, a, b, status);
return;
}
if (c < 0) {
mpd_qnext_plus(result, a, ctx, status);
}
else {
mpd_qnext_minus(result, a, ctx, status);
}
if (mpd_isinfinite(result)) {
*status |= (MPD_Overflow|MPD_Rounded|MPD_Inexact);
}
else if (mpd_adjexp(result) < ctx->emin) {
*status |= (MPD_Underflow|MPD_Subnormal|MPD_Rounded|MPD_Inexact);
if (mpd_iszero(result)) {
*status |= MPD_Clamped;
}
}
}
/*
* Internal function: Integer power with mpd_uint_t exponent. The function
* can fail with MPD_Malloc_error.
*
* The error is equal to the error incurred in k-1 multiplications. Assuming
* the upper bound for the relative error in each operation:
*
* abs(err) = 5 * 10**-prec
* result = x**k * (1 + err)**(k-1)
*/
static inline void
_mpd_qpow_uint(mpd_t *result, const mpd_t *base, mpd_uint_t exp,
uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_uint_t n;
if (exp == 0) {
_settriple(result, resultsign, 1, 0); /* GCOV_NOT_REACHED */
return; /* GCOV_NOT_REACHED */
}
if (!mpd_qcopy(result, base, status)) {
return;
}
n = mpd_bits[mpd_bsr(exp)];
while (n >>= 1) {
mpd_qmul(result, result, result, ctx, &workstatus);
if (exp & n) {
mpd_qmul(result, result, base, ctx, &workstatus);
}
if (mpd_isspecial(result) ||
(mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) {
break;
}
}
*status |= workstatus;
mpd_set_sign(result, resultsign);
}
/*
* Internal function: Integer power with mpd_t exponent, tbase and texp
* are modified!! Function can fail with MPD_Malloc_error.
*
* The error is equal to the error incurred in k multiplications. Assuming
* the upper bound for the relative error in each operation:
*
* abs(err) = 5 * 10**-prec
* result = x**k * (1 + err)**k
*/
static inline void
_mpd_qpow_mpd(mpd_t *result, mpd_t *tbase, mpd_t *texp, uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_context_t maxctx;
MPD_NEW_CONST(two,0,0,1,1,1,2);
mpd_maxcontext(&maxctx);
/* resize to smaller cannot fail */
mpd_qcopy(result, &one, status);
while (!mpd_iszero(texp)) {
if (mpd_isodd(texp)) {
mpd_qmul(result, result, tbase, ctx, &workstatus);
*status |= workstatus;
if (mpd_isspecial(result) ||
(mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) {
break;
}
}
mpd_qmul(tbase, tbase, tbase, ctx, &workstatus);
mpd_qdivint(texp, texp, &two, &maxctx, &workstatus);
if (mpd_isnan(tbase) || mpd_isnan(texp)) {
mpd_seterror(result, workstatus&MPD_Errors, status);
return;
}
}
mpd_set_sign(result, resultsign);
}
/*
* The power function for integer exponents. Relative error _before_ the
* final rounding to prec:
* abs(result - base**exp) < 0.1 * 10**-prec * abs(base**exp)
*/
static void
_mpd_qpow_int(mpd_t *result, const mpd_t *base, const mpd_t *exp,
uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_STATIC(tbase,0,0,0,0);
MPD_NEW_STATIC(texp,0,0,0,0);
mpd_ssize_t n;
mpd_workcontext(&workctx, ctx);
workctx.prec += (exp->digits + exp->exp + 2);
workctx.round = MPD_ROUND_HALF_EVEN;
workctx.clamp = 0;
if (mpd_isnegative(exp)) {
workctx.prec += 1;
mpd_qdiv(&tbase, &one, base, &workctx, status);
if (*status&MPD_Errors) {
mpd_setspecial(result, MPD_POS, MPD_NAN);
goto finish;
}
}
else {
if (!mpd_qcopy(&tbase, base, status)) {
mpd_setspecial(result, MPD_POS, MPD_NAN);
goto finish;
}
}
n = mpd_qabs_uint(exp, &workctx.status);
if (workctx.status&MPD_Invalid_operation) {
if (!mpd_qcopy(&texp, exp, status)) {
mpd_setspecial(result, MPD_POS, MPD_NAN); /* GCOV_UNLIKELY */
goto finish; /* GCOV_UNLIKELY */
}
_mpd_qpow_mpd(result, &tbase, &texp, resultsign, &workctx, status);
}
else {
_mpd_qpow_uint(result, &tbase, n, resultsign, &workctx, status);
}
if (mpd_isinfinite(result)) {
/* for ROUND_DOWN, ROUND_FLOOR, etc. */
_settriple(result, resultsign, 1, MPD_EXP_INF);
}
finish:
mpd_del(&tbase);
mpd_del(&texp);
mpd_qfinalize(result, ctx, status);
}
/*
* If the exponent is infinite and base equals one, the result is one
* with a coefficient of length prec. Otherwise, result is undefined.
* Return the value of the comparison against one.
*/
static int
_qcheck_pow_one_inf(mpd_t *result, const mpd_t *base, uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_ssize_t shift;
int cmp;
if ((cmp = _mpd_cmp(base, &one)) == 0) {
shift = ctx->prec-1;
mpd_qshiftl(result, &one, shift, status);
result->exp = -shift;
mpd_set_flags(result, resultsign);
*status |= (MPD_Inexact|MPD_Rounded);
}
return cmp;
}
/*
* If abs(base) equals one, calculate the correct power of one result.
* Otherwise, result is undefined. Return the value of the comparison
* against 1.
*
* This is an internal function that does not check for specials.
*/
static int
_qcheck_pow_one(mpd_t *result, const mpd_t *base, const mpd_t *exp,
uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_ssize_t shift;
int cmp;
if ((cmp = _mpd_cmp_abs(base, &one)) == 0) {
if (_mpd_isint(exp)) {
if (mpd_isnegative(exp)) {
_settriple(result, resultsign, 1, 0);
return 0;
}
/* 1.000**3 = 1.000000000 */
mpd_qmul_ssize(result, exp, -base->exp, ctx, &workstatus);
if (workstatus&MPD_Errors) {
*status |= (workstatus&MPD_Errors);
return 0;
}
/* digits-1 after exponentiation */
shift = mpd_qget_ssize(result, &workstatus);
/* shift is MPD_SSIZE_MAX if result is too large */
if (shift > ctx->prec-1) {
shift = ctx->prec-1;
*status |= MPD_Rounded;
}
}
else if (mpd_ispositive(base)) {
shift = ctx->prec-1;
*status |= (MPD_Inexact|MPD_Rounded);
}
else {
return -2; /* GCOV_NOT_REACHED */
}
if (!mpd_qshiftl(result, &one, shift, status)) {
return 0;
}
result->exp = -shift;
mpd_set_flags(result, resultsign);
}
return cmp;
}
/*
* Detect certain over/underflow of x**y.
* ACL2 proof: pow-bounds.lisp.
*
* Symbols:
*
* e: EXP_INF or EXP_CLAMP
* x: base
* y: exponent
*
* omega(e) = log10(abs(e))
* zeta(x) = log10(abs(log10(x)))
* theta(y) = log10(abs(y))
*
* Upper and lower bounds:
*
* ub_omega(e) = ceil(log10(abs(e)))
* lb_theta(y) = floor(log10(abs(y)))
*
* | floor(log10(floor(abs(log10(x))))) if x < 1/10 or x >= 10
* lb_zeta(x) = | floor(log10(abs(x-1)/10)) if 1/10 <= x < 1
* | floor(log10(abs((x-1)/100))) if 1 < x < 10
*
* ub_omega(e) and lb_theta(y) are obviously upper and lower bounds
* for omega(e) and theta(y).
*
* lb_zeta is a lower bound for zeta(x):
*
* x < 1/10 or x >= 10:
*
* abs(log10(x)) >= 1, so the outer log10 is well defined. Since log10
* is strictly increasing, the end result is a lower bound.
*
* 1/10 <= x < 1:
*
* We use: log10(x) <= (x-1)/log(10)
* abs(log10(x)) >= abs(x-1)/log(10)
* abs(log10(x)) >= abs(x-1)/10
*
* 1 < x < 10:
*
* We use: (x-1)/(x*log(10)) < log10(x)
* abs((x-1)/100) < abs(log10(x))
*
* XXX: abs((x-1)/10) would work, need ACL2 proof.
*
*
* Let (0 < x < 1 and y < 0) or (x > 1 and y > 0). (H1)
* Let ub_omega(exp_inf) < lb_zeta(x) + lb_theta(y) (H2)
*
* Then:
* log10(abs(exp_inf)) < log10(abs(log10(x))) + log10(abs(y)). (1)
* exp_inf < log10(x) * y (2)
* 10**exp_inf < x**y (3)
*
* Let (0 < x < 1 and y > 0) or (x > 1 and y < 0). (H3)
* Let ub_omega(exp_clamp) < lb_zeta(x) + lb_theta(y) (H4)
*
* Then:
* log10(abs(exp_clamp)) < log10(abs(log10(x))) + log10(abs(y)). (4)
* log10(x) * y < exp_clamp (5)
* x**y < 10**exp_clamp (6)
*
*/
static mpd_ssize_t
_lower_bound_zeta(const mpd_t *x, uint32_t *status)
{
mpd_context_t maxctx;
MPD_NEW_STATIC(scratch,0,0,0,0);
mpd_ssize_t t, u;
t = mpd_adjexp(x);
if (t > 0) {
/* x >= 10 -> floor(log10(floor(abs(log10(x))))) */
return mpd_exp_digits(t) - 1;
}
else if (t < -1) {
/* x < 1/10 -> floor(log10(floor(abs(log10(x))))) */
return mpd_exp_digits(t+1) - 1;
}
else {
mpd_maxcontext(&maxctx);
mpd_qsub(&scratch, x, &one, &maxctx, status);
if (mpd_isspecial(&scratch)) {
mpd_del(&scratch);
return MPD_SSIZE_MAX;
}
u = mpd_adjexp(&scratch);
mpd_del(&scratch);
/* t == -1, 1/10 <= x < 1 -> floor(log10(abs(x-1)/10))
* t == 0, 1 < x < 10 -> floor(log10(abs(x-1)/100)) */
return (t == 0) ? u-2 : u-1;
}
}
/*
* Detect cases of certain overflow/underflow in the power function.
* Assumptions: x != 1, y != 0. The proof above is for positive x.
* If x is negative and y is an odd integer, x**y == -(abs(x)**y),
* so the analysis does not change.
*/
static int
_qcheck_pow_bounds(mpd_t *result, const mpd_t *x, const mpd_t *y,
uint8_t resultsign,
const mpd_context_t *ctx, uint32_t *status)
{
MPD_NEW_SHARED(abs_x, x);
mpd_ssize_t ub_omega, lb_zeta, lb_theta;
uint8_t sign;
mpd_set_positive(&abs_x);
lb_theta = mpd_adjexp(y);
lb_zeta = _lower_bound_zeta(&abs_x, status);
if (lb_zeta == MPD_SSIZE_MAX) {
mpd_seterror(result, MPD_Malloc_error, status);
return 1;
}
sign = (mpd_adjexp(&abs_x) < 0) ^ mpd_sign(y);
if (sign == 0) {
/* (0 < |x| < 1 and y < 0) or (|x| > 1 and y > 0) */
ub_omega = mpd_exp_digits(ctx->emax);
if (ub_omega < lb_zeta + lb_theta) {
_settriple(result, resultsign, 1, MPD_EXP_INF);
mpd_qfinalize(result, ctx, status);
return 1;
}
}
else {
/* (0 < |x| < 1 and y > 0) or (|x| > 1 and y < 0). */
ub_omega = mpd_exp_digits(mpd_etiny(ctx));
if (ub_omega < lb_zeta + lb_theta) {
_settriple(result, resultsign, 1, mpd_etiny(ctx)-1);
mpd_qfinalize(result, ctx, status);
return 1;
}
}
return 0;
}
/*
* TODO: Implement algorithm for computing exact powers from decimal.py.
* In order to prevent infinite loops, this has to be called before
* using Ziv's strategy for correct rounding.
*/
/*
static int
_mpd_qpow_exact(mpd_t *result, const mpd_t *base, const mpd_t *exp,
const mpd_context_t *ctx, uint32_t *status)
{
return 0;
}
*/
/*
* The power function for real exponents.
* Relative error: abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1)
*/
static void
_mpd_qpow_real(mpd_t *result, const mpd_t *base, const mpd_t *exp,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_STATIC(texp,0,0,0,0);
if (!mpd_qcopy(&texp, exp, status)) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
mpd_maxcontext(&workctx);
workctx.prec = (base->digits > ctx->prec) ? base->digits : ctx->prec;
workctx.prec += (4 + MPD_EXPDIGITS);
workctx.round = MPD_ROUND_HALF_EVEN;
workctx.allcr = ctx->allcr;
/*
* extra := MPD_EXPDIGITS = MPD_EXP_MAX_T
* wp := prec + 4 + extra
* abs(err) < 5 * 10**-wp
* y := log(base) * exp
* Calculate:
* 1) e**(y * (1 + err)**2) * (1 + err)
* = e**y * e**(y * (2*err + err**2)) * (1 + err)
* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
* Relative error of the underlined term:
* 2) abs(e**(y * (2*err + err**2)) - 1)
* Case abs(y) >= 10**extra:
* 3) adjexp(y)+1 > log10(abs(y)) >= extra
* This triggers the Overflow/Underflow shortcut in _mpd_qexp(),
* so no further analysis is necessary.
* Case abs(y) < 10**extra:
* 4) abs(y * (2*err + err**2)) < 1/5 * 10**(-prec - 2)
* Use (see _mpd_qexp):
* 5) abs(x) <= 9/10 * 10**-p ==> abs(e**x - 1) < 10**-p
* With 2), 4) and 5):
* 6) abs(e**(y * (2*err + err**2)) - 1) < 10**(-prec - 2)
* The complete relative error of 1) is:
* 7) abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1)
*/
mpd_qln(result, base, &workctx, &workctx.status);
mpd_qmul(result, result, &texp, &workctx, &workctx.status);
mpd_qexp(result, result, &workctx, status);
mpd_del(&texp);
*status |= (workctx.status&MPD_Errors);
*status |= (MPD_Inexact|MPD_Rounded);
}
/* The power function: base**exp */
void
mpd_qpow(mpd_t *result, const mpd_t *base, const mpd_t *exp,
const mpd_context_t *ctx, uint32_t *status)
{
uint8_t resultsign = 0;
int intexp = 0;
int cmp;
if (mpd_isspecial(base) || mpd_isspecial(exp)) {
if (mpd_qcheck_nans(result, base, exp, ctx, status)) {
return;
}
}
if (mpd_isinteger(exp)) {
intexp = 1;
resultsign = mpd_isnegative(base) && mpd_isodd(exp);
}
if (mpd_iszero(base)) {
if (mpd_iszero(exp)) {
mpd_seterror(result, MPD_Invalid_operation, status);
}
else if (mpd_isnegative(exp)) {
mpd_setspecial(result, resultsign, MPD_INF);
}
else {
_settriple(result, resultsign, 0, 0);
}
return;
}
if (mpd_isnegative(base)) {
if (!intexp || mpd_isinfinite(exp)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
}
if (mpd_isinfinite(exp)) {
/* power of one */
cmp = _qcheck_pow_one_inf(result, base, resultsign, ctx, status);
if (cmp == 0) {
return;
}
else {
cmp *= mpd_arith_sign(exp);
if (cmp < 0) {
_settriple(result, resultsign, 0, 0);
}
else {
mpd_setspecial(result, resultsign, MPD_INF);
}
}
return;
}
if (mpd_isinfinite(base)) {
if (mpd_iszero(exp)) {
_settriple(result, resultsign, 1, 0);
}
else if (mpd_isnegative(exp)) {
_settriple(result, resultsign, 0, 0);
}
else {
mpd_setspecial(result, resultsign, MPD_INF);
}
return;
}
if (mpd_iszero(exp)) {
_settriple(result, resultsign, 1, 0);
return;
}
if (_qcheck_pow_one(result, base, exp, resultsign, ctx, status) == 0) {
return;
}
if (_qcheck_pow_bounds(result, base, exp, resultsign, ctx, status)) {
return;
}
if (intexp) {
_mpd_qpow_int(result, base, exp, resultsign, ctx, status);
}
else {
_mpd_qpow_real(result, base, exp, ctx, status);
if (!mpd_isspecial(result) && _mpd_cmp(result, &one) == 0) {
mpd_ssize_t shift = ctx->prec-1;
mpd_qshiftl(result, &one, shift, status);
result->exp = -shift;
}
if (mpd_isinfinite(result)) {
/* for ROUND_DOWN, ROUND_FLOOR, etc. */
_settriple(result, MPD_POS, 1, MPD_EXP_INF);
}
mpd_qfinalize(result, ctx, status);
}
}
/*
* Internal function: Integer powmod with mpd_uint_t exponent, base is modified!
* Function can fail with MPD_Malloc_error.
*/
static inline void
_mpd_qpowmod_uint(mpd_t *result, mpd_t *base, mpd_uint_t exp,
const mpd_t *mod, uint32_t *status)
{
mpd_context_t maxcontext;
mpd_maxcontext(&maxcontext);
/* resize to smaller cannot fail */
mpd_qcopy(result, &one, status);
while (exp > 0) {
if (exp & 1) {
_mpd_qmul_exact(result, result, base, &maxcontext, status);
mpd_qrem(result, result, mod, &maxcontext, status);
}
_mpd_qmul_exact(base, base, base, &maxcontext, status);
mpd_qrem(base, base, mod, &maxcontext, status);
exp >>= 1;
}
}
/* The powmod function: (base**exp) % mod */
void
mpd_qpowmod(mpd_t *result, const mpd_t *base, const mpd_t *exp,
const mpd_t *mod,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(tbase,0,0,0,0);
MPD_NEW_STATIC(texp,0,0,0,0);
MPD_NEW_STATIC(tmod,0,0,0,0);
MPD_NEW_STATIC(tmp,0,0,0,0);
MPD_NEW_CONST(two,0,0,1,1,1,2);
mpd_ssize_t tbase_exp, texp_exp;
mpd_ssize_t i;
mpd_t t;
mpd_uint_t r;
uint8_t sign;
if (mpd_isspecial(base) || mpd_isspecial(exp) || mpd_isspecial(mod)) {
if (mpd_qcheck_3nans(result, base, exp, mod, ctx, status)) {
return;
}
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (!_mpd_isint(base) || !_mpd_isint(exp) || !_mpd_isint(mod)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_iszerocoeff(mod)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mod->digits+mod->exp > ctx->prec) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
sign = (mpd_isnegative(base)) && (mpd_isodd(exp));
if (mpd_iszerocoeff(exp)) {
if (mpd_iszerocoeff(base)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
r = (_mpd_cmp_abs(mod, &one)==0) ? 0 : 1;
_settriple(result, sign, r, 0);
return;
}
if (mpd_isnegative(exp)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_iszerocoeff(base)) {
_settriple(result, sign, 0, 0);
return;
}
mpd_maxcontext(&maxcontext);
mpd_qrescale(&tmod, mod, 0, &maxcontext, &maxcontext.status);
if (maxcontext.status&MPD_Errors) {
mpd_seterror(result, maxcontext.status&MPD_Errors, status);
goto out;
}
maxcontext.status = 0;
mpd_set_positive(&tmod);
mpd_qround_to_int(&tbase, base, &maxcontext, status);
mpd_set_positive(&tbase);
tbase_exp = tbase.exp;
tbase.exp = 0;
mpd_qround_to_int(&texp, exp, &maxcontext, status);
texp_exp = texp.exp;
texp.exp = 0;
/* base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo */
mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status);
mpd_qshiftl(result, &one, tbase_exp, status);
mpd_qrem(result, result, &tmod, &maxcontext, status);
_mpd_qmul_exact(&tbase, &tbase, result, &maxcontext, status);
mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status);
if (mpd_isspecial(&tbase) ||
mpd_isspecial(&texp) ||
mpd_isspecial(&tmod)) {
goto mpd_errors;
}
for (i = 0; i < texp_exp; i++) {
_mpd_qpowmod_uint(&tmp, &tbase, 10, &tmod, status);
t = tmp;
tmp = tbase;
tbase = t;
}
if (mpd_isspecial(&tbase)) {
goto mpd_errors; /* GCOV_UNLIKELY */
}
/* resize to smaller cannot fail */
mpd_qcopy(result, &one, status);
while (mpd_isfinite(&texp) && !mpd_iszero(&texp)) {
if (mpd_isodd(&texp)) {
_mpd_qmul_exact(result, result, &tbase, &maxcontext, status);
mpd_qrem(result, result, &tmod, &maxcontext, status);
}
_mpd_qmul_exact(&tbase, &tbase, &tbase, &maxcontext, status);
mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status);
mpd_qdivint(&texp, &texp, &two, &maxcontext, status);
}
if (mpd_isspecial(&texp) || mpd_isspecial(&tbase) ||
mpd_isspecial(&tmod) || mpd_isspecial(result)) {
/* MPD_Malloc_error */
goto mpd_errors;
}
else {
mpd_set_sign(result, sign);
}
out:
mpd_del(&tbase);
mpd_del(&texp);
mpd_del(&tmod);
mpd_del(&tmp);
return;
mpd_errors:
mpd_setspecial(result, MPD_POS, MPD_NAN);
goto out;
}
void
mpd_qquantize(mpd_t *result, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
uint32_t workstatus = 0;
mpd_ssize_t b_exp = b->exp;
mpd_ssize_t expdiff, shift;
mpd_uint_t rnd;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(result, a, b, ctx, status)) {
return;
}
if (mpd_isinfinite(a) && mpd_isinfinite(b)) {
mpd_qcopy(result, a, status);
return;
}
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (b->exp > ctx->emax || b->exp < mpd_etiny(ctx)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (mpd_iszero(a)) {
_settriple(result, mpd_sign(a), 0, b->exp);
mpd_qfinalize(result, ctx, status);
return;
}
expdiff = a->exp - b->exp;
if (a->digits + expdiff > ctx->prec) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (expdiff >= 0) {
shift = expdiff;
if (!mpd_qshiftl(result, a, shift, status)) {
return;
}
result->exp = b_exp;
}
else {
/* At this point expdiff < 0 and a->digits+expdiff <= prec,
* so the shift before an increment will fit in prec. */
shift = -expdiff;
rnd = mpd_qshiftr(result, a, shift, status);
if (rnd == MPD_UINT_MAX) {
return;
}
result->exp = b_exp;
if (!_mpd_apply_round_fit(result, rnd, ctx, status)) {
return;
}
workstatus |= MPD_Rounded;
if (rnd) {
workstatus |= MPD_Inexact;
}
}
if (mpd_adjexp(result) > ctx->emax ||
mpd_adjexp(result) < mpd_etiny(ctx)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
*status |= workstatus;
mpd_qfinalize(result, ctx, status);
}
void
mpd_qreduce(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_ssize_t shift, maxexp, maxshift;
uint8_t sign_a = mpd_sign(a);
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
mpd_qcopy(result, a, status);
return;
}
if (!mpd_qcopy(result, a, status)) {
return;
}
mpd_qfinalize(result, ctx, status);
if (mpd_isspecial(result)) {
return;
}
if (mpd_iszero(result)) {
_settriple(result, sign_a, 0, 0);
return;
}
shift = mpd_trail_zeros(result);
maxexp = (ctx->clamp) ? mpd_etop(ctx) : ctx->emax;
/* After the finalizing above result->exp <= maxexp. */
maxshift = maxexp - result->exp;
shift = (shift > maxshift) ? maxshift : shift;
mpd_qshiftr_inplace(result, shift);
result->exp += shift;
}
void
mpd_qrem(mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx,
uint32_t *status)
{
MPD_NEW_STATIC(q,0,0,0,0);
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(r, a, b, ctx, status)) {
return;
}
if (mpd_isinfinite(a)) {
mpd_seterror(r, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(b)) {
mpd_qcopy(r, a, status);
mpd_qfinalize(r, ctx, status);
return;
}
/* debug */
abort(); /* GCOV_NOT_REACHED */
}
if (mpd_iszerocoeff(b)) {
if (mpd_iszerocoeff(a)) {
mpd_seterror(r, MPD_Division_undefined, status);
}
else {
mpd_seterror(r, MPD_Invalid_operation, status);
}
return;
}
_mpd_qdivmod(&q, r, a, b, ctx, status);
mpd_del(&q);
mpd_qfinalize(r, ctx, status);
}
void
mpd_qrem_near(mpd_t *r, const mpd_t *a, const mpd_t *b,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_context_t workctx;
MPD_NEW_STATIC(btmp,0,0,0,0);
MPD_NEW_STATIC(q,0,0,0,0);
mpd_ssize_t expdiff, qdigits;
int cmp, isodd, allnine;
if (mpd_isspecial(a) || mpd_isspecial(b)) {
if (mpd_qcheck_nans(r, a, b, ctx, status)) {
return;
}
if (mpd_isinfinite(a)) {
mpd_seterror(r, MPD_Invalid_operation, status);
return;
}
if (mpd_isinfinite(b)) {
mpd_qcopy(r, a, status);
mpd_qfinalize(r, ctx, status);
return;
}
/* debug */
abort(); /* GCOV_NOT_REACHED */
}
if (mpd_iszerocoeff(b)) {
if (mpd_iszerocoeff(a)) {
mpd_seterror(r, MPD_Division_undefined, status);
}
else {
mpd_seterror(r, MPD_Invalid_operation, status);
}
return;
}
if (r == b) {
if (!mpd_qcopy(&btmp, b, status)) {
mpd_seterror(r, MPD_Malloc_error, status);
return;
}
b = &btmp;
}
_mpd_qdivmod(&q, r, a, b, ctx, status);
if (mpd_isnan(&q) || mpd_isnan(r)) {
goto finish;
}
if (mpd_iszerocoeff(r)) {
goto finish;
}
expdiff = mpd_adjexp(b) - mpd_adjexp(r);
if (-1 <= expdiff && expdiff <= 1) {
allnine = mpd_coeff_isallnine(&q);
qdigits = q.digits;
isodd = mpd_isodd(&q);
mpd_maxcontext(&workctx);
if (mpd_sign(a) == mpd_sign(b)) {
/* sign(r) == sign(b) */
_mpd_qsub(&q, r, b, &workctx, &workctx.status);
}
else {
/* sign(r) != sign(b) */
_mpd_qadd(&q, r, b, &workctx, &workctx.status);
}
if (workctx.status&MPD_Errors) {
mpd_seterror(r, workctx.status&MPD_Errors, status);
goto finish;
}
cmp = _mpd_cmp_abs(&q, r);
if (cmp < 0 || (cmp == 0 && isodd)) {
/* abs(r) > abs(b)/2 or abs(r) == abs(b)/2 and isodd(quotient) */
if (allnine && qdigits == ctx->prec) {
/* abs(quotient) + 1 == 10**prec */
mpd_seterror(r, MPD_Division_impossible, status);
goto finish;
}
mpd_qcopy(r, &q, status);
}
}
finish:
mpd_del(&btmp);
mpd_del(&q);
mpd_qfinalize(r, ctx, status);
}
static void
_mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_ssize_t expdiff, shift;
mpd_uint_t rnd;
if (mpd_isspecial(a)) {
mpd_qcopy(result, a, status);
return;
}
if (mpd_iszero(a)) {
_settriple(result, mpd_sign(a), 0, exp);
return;
}
expdiff = a->exp - exp;
if (expdiff >= 0) {
shift = expdiff;
if (a->digits + shift > MPD_MAX_PREC+1) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (!mpd_qshiftl(result, a, shift, status)) {
return;
}
result->exp = exp;
}
else {
shift = -expdiff;
rnd = mpd_qshiftr(result, a, shift, status);
if (rnd == MPD_UINT_MAX) {
return;
}
result->exp = exp;
_mpd_apply_round_excess(result, rnd, ctx, status);
*status |= MPD_Rounded;
if (rnd) {
*status |= MPD_Inexact;
}
}
if (mpd_issubnormal(result, ctx)) {
*status |= MPD_Subnormal;
}
}
/*
* Rescale a number so that it has exponent 'exp'. Does not regard context
* precision, emax, emin, but uses the rounding mode. Special numbers are
* quietly copied. Restrictions:
*
* MPD_MIN_ETINY <= exp <= MPD_MAX_EMAX+1
* result->digits <= MPD_MAX_PREC+1
*/
void
mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp,
const mpd_context_t *ctx, uint32_t *status)
{
if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
_mpd_qrescale(result, a, exp, ctx, status);
}
/*
* Same as mpd_qrescale, but with relaxed restrictions. The result of this
* function should only be used for formatting a number and never as input
* for other operations.
*
* MPD_MIN_ETINY-MPD_MAX_PREC <= exp <= MPD_MAX_EMAX+1
* result->digits <= MPD_MAX_PREC+1
*/
void
mpd_qrescale_fmt(mpd_t *result, const mpd_t *a, mpd_ssize_t exp,
const mpd_context_t *ctx, uint32_t *status)
{
if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY-MPD_MAX_PREC) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
_mpd_qrescale(result, a, exp, ctx, status);
}
/* Round to an integer according to 'action' and ctx->round. */
enum {TO_INT_EXACT, TO_INT_SILENT, TO_INT_TRUNC};
static void
_mpd_qround_to_integral(int action, mpd_t *result, const mpd_t *a,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_uint_t rnd;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
mpd_qcopy(result, a, status);
return;
}
if (a->exp >= 0) {
mpd_qcopy(result, a, status);
return;
}
if (mpd_iszerocoeff(a)) {
_settriple(result, mpd_sign(a), 0, 0);
return;
}
rnd = mpd_qshiftr(result, a, -a->exp, status);
if (rnd == MPD_UINT_MAX) {
return;
}
result->exp = 0;
if (action == TO_INT_EXACT || action == TO_INT_SILENT) {
_mpd_apply_round_excess(result, rnd, ctx, status);
if (action == TO_INT_EXACT) {
*status |= MPD_Rounded;
if (rnd) {
*status |= MPD_Inexact;
}
}
}
}
void
mpd_qround_to_intx(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
(void)_mpd_qround_to_integral(TO_INT_EXACT, result, a, ctx, status);
}
void
mpd_qround_to_int(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
(void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, ctx, status);
}
void
mpd_qtrunc(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
(void)_mpd_qround_to_integral(TO_INT_TRUNC, result, a, ctx, status);
}
void
mpd_qfloor(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx = *ctx;
workctx.round = MPD_ROUND_FLOOR;
(void)_mpd_qround_to_integral(TO_INT_SILENT, result, a,
&workctx, status);
}
void
mpd_qceil(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx = *ctx;
workctx.round = MPD_ROUND_CEILING;
(void)_mpd_qround_to_integral(TO_INT_SILENT, result, a,
&workctx, status);
}
int
mpd_same_quantum(const mpd_t *a, const mpd_t *b)
{
if (mpd_isspecial(a) || mpd_isspecial(b)) {
return ((mpd_isnan(a) && mpd_isnan(b)) ||
(mpd_isinfinite(a) && mpd_isinfinite(b)));
}
return a->exp == b->exp;
}
/* Schedule the increase in precision for the Newton iteration. */
static inline int
recpr_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2],
mpd_ssize_t maxprec, mpd_ssize_t initprec)
{
mpd_ssize_t k;
int i;
assert(maxprec > 0 && initprec > 0);
if (maxprec <= initprec) return -1;
i = 0; k = maxprec;
do {
k = (k+1) / 2;
klist[i++] = k;
} while (k > initprec);
return i-1;
}
/*
* Initial approximation for the reciprocal:
* k_0 := MPD_RDIGITS-2
* z_0 := 10**(-k_0) * floor(10**(2*k_0 + 2) / floor(v * 10**(k_0 + 2)))
* Absolute error:
* |1/v - z_0| < 10**(-k_0)
* ACL2 proof: maxerror-inverse-approx
*/
static void
_mpd_qreciprocal_approx(mpd_t *z, const mpd_t *v, uint32_t *status)
{
mpd_uint_t p10data[2] = {0, mpd_pow10[MPD_RDIGITS-2]};
mpd_uint_t dummy, word;
int n;
assert(v->exp == -v->digits);
_mpd_get_msdigits(&dummy, &word, v, MPD_RDIGITS);
n = mpd_word_digits(word);
word *= mpd_pow10[MPD_RDIGITS-n];
mpd_qresize(z, 2, status);
(void)_mpd_shortdiv(z->data, p10data, 2, word);
mpd_clear_flags(z);
z->exp = -(MPD_RDIGITS-2);
z->len = (z->data[1] == 0) ? 1 : 2;
mpd_setdigits(z);
}
/*
* Reciprocal, calculated with Newton's Method. Assumption: result != a.
* NOTE: The comments in the function show that certain operations are
* exact. The proof for the maximum error is too long to fit in here.
* ACL2 proof: maxerror-inverse-complete
*/
static void
_mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t varcontext, maxcontext;
mpd_t *z = result; /* current approximation */
mpd_t *v; /* a, normalized to a number between 0.1 and 1 */
MPD_NEW_SHARED(vtmp, a); /* v shares data with a */
MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */
MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */
MPD_NEW_CONST(two,0,0,1,1,1,2); /* const 2 */
mpd_ssize_t klist[MPD_MAX_PREC_LOG2];
mpd_ssize_t adj, maxprec, initprec;
uint8_t sign = mpd_sign(a);
int i;
assert(result != a);
v = &vtmp;
mpd_clear_flags(v);
adj = v->digits + v->exp;
v->exp = -v->digits;
/* Initial approximation */
_mpd_qreciprocal_approx(z, v, status);
mpd_maxcontext(&varcontext);
mpd_maxcontext(&maxcontext);
varcontext.round = maxcontext.round = MPD_ROUND_TRUNC;
varcontext.emax = maxcontext.emax = MPD_MAX_EMAX + 100;
varcontext.emin = maxcontext.emin = MPD_MIN_EMIN - 100;
maxcontext.prec = MPD_MAX_PREC + 100;
maxprec = ctx->prec;
maxprec += 2;
initprec = MPD_RDIGITS-3;
i = recpr_schedule_prec(klist, maxprec, initprec);
for (; i >= 0; i--) {
/* Loop invariant: z->digits <= klist[i]+7 */
/* Let s := z**2, exact result */
_mpd_qmul_exact(&s, z, z, &maxcontext, status);
varcontext.prec = 2*klist[i] + 5;
if (v->digits > varcontext.prec) {
/* Let t := v, truncated to n >= 2*k+5 fraction digits */
mpd_qshiftr(&t, v, v->digits-varcontext.prec, status);
t.exp = -varcontext.prec;
/* Let t := trunc(v)*s, truncated to n >= 2*k+1 fraction digits */
mpd_qmul(&t, &t, &s, &varcontext, status);
}
else { /* v->digits <= 2*k+5 */
/* Let t := v*s, truncated to n >= 2*k+1 fraction digits */
mpd_qmul(&t, v, &s, &varcontext, status);
}
/* Let s := 2*z, exact result */
_mpd_qmul_exact(&s, z, &two, &maxcontext, status);
/* s.digits < t.digits <= 2*k+5, |adjexp(s)-adjexp(t)| <= 1,
* so the subtraction generates at most 2*k+6 <= klist[i+1]+7
* digits. The loop invariant is preserved. */
_mpd_qsub_exact(z, &s, &t, &maxcontext, status);
}
if (!mpd_isspecial(z)) {
z->exp -= adj;
mpd_set_flags(z, sign);
}
mpd_del(&s);
mpd_del(&t);
mpd_qfinalize(z, ctx, status);
}
/*
* Internal function for large numbers:
*
* q, r = divmod(coeff(a), coeff(b))
*
* Strategy: Multiply the dividend by the reciprocal of the divisor. The
* inexact result is fixed by a small loop, using at most one iteration.
*
* ACL2 proofs:
* ------------
* 1) q is a natural number. (ndivmod-quotient-natp)
* 2) r is a natural number. (ndivmod-remainder-natp)
* 3) a = q * b + r (ndivmod-q*b+r==a)
* 4) r < b (ndivmod-remainder-<-b)
*/
static void
_mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
uint32_t *status)
{
mpd_context_t workctx;
mpd_t *qq = q, *rr = r;
mpd_t aa, bb;
int k;
_mpd_copy_shared(&aa, a);
_mpd_copy_shared(&bb, b);
mpd_set_positive(&aa);
mpd_set_positive(&bb);
aa.exp = 0;
bb.exp = 0;
if (q == a || q == b) {
if ((qq = mpd_qnew()) == NULL) {
*status |= MPD_Malloc_error;
goto nanresult;
}
}
if (r == a || r == b) {
if ((rr = mpd_qnew()) == NULL) {
*status |= MPD_Malloc_error;
goto nanresult;
}
}
mpd_maxcontext(&workctx);
/* Let prec := adigits - bdigits + 4 */
workctx.prec = a->digits - b->digits + 1 + 3;
if (a->digits > MPD_MAX_PREC || workctx.prec > MPD_MAX_PREC) {
*status |= MPD_Division_impossible;
goto nanresult;
}
/* Let x := _mpd_qreciprocal(b, prec)
* Then x is bounded by:
* 1) 1/b - 10**(-prec - bdigits) < x < 1/b + 10**(-prec - bdigits)
* 2) 1/b - 10**(-adigits - 4) < x < 1/b + 10**(-adigits - 4)
*/
_mpd_qreciprocal(rr, &bb, &workctx, &workctx.status);
/* Get an estimate for the quotient. Let q := a * x
* Then q is bounded by:
* 3) a/b - 10**-4 < q < a/b + 10**-4
*/
_mpd_qmul(qq, &aa, rr, &workctx, &workctx.status);
/* Truncate q to an integer:
* 4) a/b - 2 < trunc(q) < a/b + 1
*/
mpd_qtrunc(qq, qq, &workctx, &workctx.status);
workctx.prec = aa.digits + 3;
workctx.emax = MPD_MAX_EMAX + 3;
workctx.emin = MPD_MIN_EMIN - 3;
/* Multiply the estimate for q by b:
* 5) a - 2 * b < trunc(q) * b < a + b
*/
_mpd_qmul(rr, &bb, qq, &workctx, &workctx.status);
/* Get the estimate for r such that a = q * b + r. */
_mpd_qsub_exact(rr, &aa, rr, &workctx, &workctx.status);
/* Fix the result. At this point -b < r < 2*b, so the correction loop
takes at most one iteration. */
for (k = 0;; k++) {
if (mpd_isspecial(qq) || mpd_isspecial(rr)) {
*status |= (workctx.status&MPD_Errors);
goto nanresult;
}
if (k > 2) { /* Allow two iterations despite the proof. */
mpd_err_warn("libmpdec: internal error in " /* GCOV_NOT_REACHED */
"_mpd_base_ndivmod: please report"); /* GCOV_NOT_REACHED */
*status |= MPD_Invalid_operation; /* GCOV_NOT_REACHED */
goto nanresult; /* GCOV_NOT_REACHED */
}
/* r < 0 */
else if (_mpd_cmp(&zero, rr) == 1) {
_mpd_qadd_exact(rr, rr, &bb, &workctx, &workctx.status);
_mpd_qadd_exact(qq, qq, &minus_one, &workctx, &workctx.status);
}
/* 0 <= r < b */
else if (_mpd_cmp(rr, &bb) == -1) {
break;
}
/* r >= b */
else {
_mpd_qsub_exact(rr, rr, &bb, &workctx, &workctx.status);
_mpd_qadd_exact(qq, qq, &one, &workctx, &workctx.status);
}
}
if (qq != q) {
if (!mpd_qcopy(q, qq, status)) {
goto nanresult; /* GCOV_UNLIKELY */
}
mpd_del(qq);
}
if (rr != r) {
if (!mpd_qcopy(r, rr, status)) {
goto nanresult; /* GCOV_UNLIKELY */
}
mpd_del(rr);
}
*status |= (workctx.status&MPD_Errors);
return;
nanresult:
if (qq && qq != q) mpd_del(qq);
if (rr && rr != r) mpd_del(rr);
mpd_setspecial(q, MPD_POS, MPD_NAN);
mpd_setspecial(r, MPD_POS, MPD_NAN);
}
/* LIBMPDEC_ONLY */
/*
* Schedule the optimal precision increase for the Newton iteration.
* v := input operand
* z_0 := initial approximation
* initprec := natural number such that abs(sqrt(v) - z_0) < 10**-initprec
* maxprec := target precision
*
* For convenience the output klist contains the elements in reverse order:
* klist := [k_n-1, ..., k_0], where
* 1) k_0 <= initprec and
* 2) abs(sqrt(v) - result) < 10**(-2*k_n-1 + 2) <= 10**-maxprec.
*/
static inline int
invroot_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2],
mpd_ssize_t maxprec, mpd_ssize_t initprec)
{
mpd_ssize_t k;
int i;
assert(maxprec >= 3 && initprec >= 3);
if (maxprec <= initprec) return -1;
i = 0; k = maxprec;
do {
k = (k+3) / 2;
klist[i++] = k;
} while (k > initprec);
return i-1;
}
/*
* Initial approximation for the inverse square root function.
* Input:
* v := rational number, with 1 <= v < 100
* vhat := floor(v * 10**6)
* Output:
* z := approximation to 1/sqrt(v), such that abs(z - 1/sqrt(v)) < 10**-3.
*/
static inline void
_invroot_init_approx(mpd_t *z, mpd_uint_t vhat)
{
mpd_uint_t lo = 1000;
mpd_uint_t hi = 10000;
mpd_uint_t a, sq;
assert(lo*lo <= vhat && vhat < (hi+1)*(hi+1));
for(;;) {
a = (lo + hi) / 2;
sq = a * a;
if (vhat >= sq) {
if (vhat < sq + 2*a + 1) {
break;
}
lo = a + 1;
}
else {
hi = a - 1;
}
}
/*
* After the binary search we have:
* 1) a**2 <= floor(v * 10**6) < (a + 1)**2
* This implies:
* 2) a**2 <= v * 10**6 < (a + 1)**2
* 3) a <= sqrt(v) * 10**3 < a + 1
* Since 10**3 <= a:
* 4) 0 <= 10**prec/a - 1/sqrt(v) < 10**-prec
* We have:
* 5) 10**3/a - 10**-3 < floor(10**9/a) * 10**-6 <= 10**3/a
* Merging 4) and 5):
* 6) abs(floor(10**9/a) * 10**-6 - 1/sqrt(v)) < 10**-3
*/
mpd_minalloc(z);
mpd_clear_flags(z);
z->data[0] = 1000000000UL / a;
z->len = 1;
z->exp = -6;
mpd_setdigits(z);
}
/*
* Set 'result' to 1/sqrt(a).
* Relative error: abs(result - 1/sqrt(a)) < 10**-prec * 1/sqrt(a)
*/
static void
_mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
uint32_t workstatus = 0;
mpd_context_t varcontext, maxcontext;
mpd_t *z = result; /* current approximation */
mpd_t *v; /* a, normalized to a number between 1 and 100 */
MPD_NEW_SHARED(vtmp, a); /* by default v will share data with a */
MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */
MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */
MPD_NEW_CONST(one_half,0,-1,1,1,1,5);
MPD_NEW_CONST(three,0,0,1,1,1,3);
mpd_ssize_t klist[MPD_MAX_PREC_LOG2];
mpd_ssize_t ideal_exp, shift;
mpd_ssize_t adj, tz;
mpd_ssize_t maxprec, fracdigits;
mpd_uint_t vhat, dummy;
int i, n;
ideal_exp = -(a->exp - (a->exp & 1)) / 2;
v = &vtmp;
if (result == a) {
if ((v = mpd_qncopy(a)) == NULL) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
}
/* normalize a to 1 <= v < 100 */
if ((v->digits+v->exp) & 1) {
fracdigits = v->digits - 1;
v->exp = -fracdigits;
n = (v->digits > 7) ? 7 : (int)v->digits;
/* Let vhat := floor(v * 10**(2*initprec)) */
_mpd_get_msdigits(&dummy, &vhat, v, n);
if (n < 7) {
vhat *= mpd_pow10[7-n];
}
}
else {
fracdigits = v->digits - 2;
v->exp = -fracdigits;
n = (v->digits > 8) ? 8 : (int)v->digits;
/* Let vhat := floor(v * 10**(2*initprec)) */
_mpd_get_msdigits(&dummy, &vhat, v, n);
if (n < 8) {
vhat *= mpd_pow10[8-n];
}
}
adj = (a->exp-v->exp) / 2;
/* initial approximation */
_invroot_init_approx(z, vhat);
mpd_maxcontext(&maxcontext);
mpd_maxcontext(&varcontext);
varcontext.round = MPD_ROUND_TRUNC;
maxprec = ctx->prec + 1;
/* initprec == 3 */
i = invroot_schedule_prec(klist, maxprec, 3);
for (; i >= 0; i--) {
varcontext.prec = 2*klist[i]+2;
mpd_qmul(&s, z, z, &maxcontext, &workstatus);
if (v->digits > varcontext.prec) {
shift = v->digits - varcontext.prec;
mpd_qshiftr(&t, v, shift, &workstatus);
t.exp += shift;
mpd_qmul(&t, &t, &s, &varcontext, &workstatus);
}
else {
mpd_qmul(&t, v, &s, &varcontext, &workstatus);
}
mpd_qsub(&t, &three, &t, &maxcontext, &workstatus);
mpd_qmul(z, z, &t, &varcontext, &workstatus);
mpd_qmul(z, z, &one_half, &maxcontext, &workstatus);
}
z->exp -= adj;
tz = mpd_trail_zeros(result);
shift = ideal_exp - result->exp;
shift = (tz > shift) ? shift : tz;
if (shift > 0) {
mpd_qshiftr_inplace(result, shift);
result->exp += shift;
}
mpd_del(&s);
mpd_del(&t);
if (v != &vtmp) mpd_del(v);
*status |= (workstatus&MPD_Errors);
*status |= (MPD_Rounded|MPD_Inexact);
}
void
mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t workctx;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
/* positive infinity */
_settriple(result, MPD_POS, 0, mpd_etiny(ctx));
*status |= MPD_Clamped;
return;
}
if (mpd_iszero(a)) {
mpd_setspecial(result, mpd_sign(a), MPD_INF);
*status |= MPD_Division_by_zero;
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
workctx = *ctx;
workctx.prec += 2;
workctx.round = MPD_ROUND_HALF_EVEN;
_mpd_qinvroot(result, a, &workctx, status);
mpd_qfinalize(result, ctx, status);
}
/* END LIBMPDEC_ONLY */
/* Algorithm from decimal.py */
void
mpd_qsqrt(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
uint32_t *status)
{
mpd_context_t maxcontext;
MPD_NEW_STATIC(c,0,0,0,0);
MPD_NEW_STATIC(q,0,0,0,0);
MPD_NEW_STATIC(r,0,0,0,0);
MPD_NEW_CONST(two,0,0,1,1,1,2);
mpd_ssize_t prec, ideal_exp;
mpd_ssize_t l, shift;
int exact = 0;
ideal_exp = (a->exp - (a->exp & 1)) / 2;
if (mpd_isspecial(a)) {
if (mpd_qcheck_nan(result, a, ctx, status)) {
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
mpd_setspecial(result, MPD_POS, MPD_INF);
return;
}
if (mpd_iszero(a)) {
_settriple(result, mpd_sign(a), 0, ideal_exp);
mpd_qfinalize(result, ctx, status);
return;
}
if (mpd_isnegative(a)) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
mpd_maxcontext(&maxcontext);
prec = ctx->prec + 1;
if (!mpd_qcopy(&c, a, status)) {
goto malloc_error;
}
c.exp = 0;
if (a->exp & 1) {
if (!mpd_qshiftl(&c, &c, 1, status)) {
goto malloc_error;
}
l = (a->digits >> 1) + 1;
}
else {
l = (a->digits + 1) >> 1;
}
shift = prec - l;
if (shift >= 0) {
if (!mpd_qshiftl(&c, &c, 2*shift, status)) {
goto malloc_error;
}
exact = 1;
}
else {
exact = !mpd_qshiftr_inplace(&c, -2*shift);
}
ideal_exp -= shift;
/* find result = floor(sqrt(c)) using Newton's method */
if (!mpd_qshiftl(result, &one, prec, status)) {
goto malloc_error;
}
while (1) {
_mpd_qdivmod(&q, &r, &c, result, &maxcontext, &maxcontext.status);
if (mpd_isspecial(result) || mpd_isspecial(&q)) {
mpd_seterror(result, maxcontext.status&MPD_Errors, status);
goto out;
}
if (_mpd_cmp(result, &q) <= 0) {
break;
}
_mpd_qadd_exact(result, result, &q, &maxcontext, &maxcontext.status);
if (mpd_isspecial(result)) {
mpd_seterror(result, maxcontext.status&MPD_Errors, status);
goto out;
}
_mpd_qdivmod(result, &r, result, &two, &maxcontext, &maxcontext.status);
}
if (exact) {
_mpd_qmul_exact(&r, result, result, &maxcontext, &maxcontext.status);
if (mpd_isspecial(&r)) {
mpd_seterror(result, maxcontext.status&MPD_Errors, status);
goto out;
}
exact = (_mpd_cmp(&r, &c) == 0);
}
if (exact) {
if (shift >= 0) {
mpd_qshiftr_inplace(result, shift);
}
else {
if (!mpd_qshiftl(result, result, -shift, status)) {
goto malloc_error;
}
}
ideal_exp += shift;
}
else {
int lsd = (int)mpd_lsd(result->data[0]);
if (lsd == 0 || lsd == 5) {
result->data[0] += 1;
}
}
result->exp = ideal_exp;
out:
mpd_del(&c);
mpd_del(&q);
mpd_del(&r);
maxcontext = *ctx;
maxcontext.round = MPD_ROUND_HALF_EVEN;
mpd_qfinalize(result, &maxcontext, status);
return;
malloc_error:
mpd_seterror(result, MPD_Malloc_error, status);
goto out;
}
/******************************************************************************/
/* Base conversions */
/******************************************************************************/
/* Space needed to represent an integer mpd_t in base 'base'. */
size_t
mpd_sizeinbase(const mpd_t *a, uint32_t base)
{
double x;
size_t digits;
assert(mpd_isinteger(a));
assert(base >= 2);
if (mpd_iszero(a)) {
return 1;
}
digits = a->digits+a->exp;
assert(digits > 0);
#ifdef CONFIG_64
/* ceil(2711437152599294 / log10(2)) + 4 == 2**53 */
if (digits > 2711437152599294ULL) {
return SIZE_MAX;
}
#endif
x = (double)digits / log10(base);
return (x > SIZE_MAX-1) ? SIZE_MAX : (size_t)x + 1;
}
/* Space needed to import a base 'base' integer of length 'srclen'. */
static mpd_ssize_t
_mpd_importsize(size_t srclen, uint32_t base)
{
double x;
assert(srclen > 0);
assert(base >= 2);
#if SIZE_MAX == UINT64_MAX
if (srclen > (1ULL<<53)) {
return MPD_SSIZE_MAX;
}
#endif
x = (double)srclen * (log10(base)/MPD_RDIGITS);
return (x >= MPD_MAXIMPORT) ? MPD_SSIZE_MAX : (mpd_ssize_t)x + 1;
}
static uint8_t
mpd_resize_u16(uint16_t **w, size_t nmemb)
{
uint8_t err = 0;
*w = mpd_realloc(*w, nmemb, sizeof **w, &err);
return !err;
}
static uint8_t
mpd_resize_u32(uint32_t **w, size_t nmemb)
{
uint8_t err = 0;
*w = mpd_realloc(*w, nmemb, sizeof **w, &err);
return !err;
}
static size_t
_baseconv_to_u16(uint16_t **w, size_t wlen, mpd_uint_t wbase,
mpd_uint_t *u, mpd_ssize_t ulen)
{
size_t n = 0;
assert(wlen > 0 && ulen > 0);
assert(wbase <= (1U<<16));
do {
if (n >= wlen) {
if (!mpd_resize_u16(w, n+1)) {
return SIZE_MAX;
}
wlen = n+1;
}
(*w)[n++] = (uint16_t)_mpd_shortdiv(u, u, ulen, wbase);
/* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */
ulen = _mpd_real_size(u, ulen);
} while (u[ulen-1] != 0);
return n;
}
static size_t
_coeff_from_u16(mpd_t *w, mpd_ssize_t wlen,
const mpd_uint_t *u, size_t ulen, uint32_t ubase,
uint32_t *status)
{
mpd_ssize_t n = 0;
mpd_uint_t carry;
assert(wlen > 0 && ulen > 0);
assert(ubase <= (1U<<16));
w->data[n++] = u[--ulen];
while (--ulen != SIZE_MAX) {
carry = _mpd_shortmul_c(w->data, w->data, n, ubase);
if (carry) {
if (n >= wlen) {
if (!mpd_qresize(w, n+1, status)) {
return SIZE_MAX;
}
wlen = n+1;
}
w->data[n++] = carry;
}
carry = _mpd_shortadd(w->data, n, u[ulen]);
if (carry) {
if (n >= wlen) {
if (!mpd_qresize(w, n+1, status)) {
return SIZE_MAX;
}
wlen = n+1;
}
w->data[n++] = carry;
}
}
return n;
}
/* target base wbase < source base ubase */
static size_t
_baseconv_to_smaller(uint32_t **w, size_t wlen, uint32_t wbase,
mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase)
{
size_t n = 0;
assert(wlen > 0 && ulen > 0);
assert(wbase < ubase);
do {
if (n >= wlen) {
if (!mpd_resize_u32(w, n+1)) {
return SIZE_MAX;
}
wlen = n+1;
}
(*w)[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase);
/* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */
ulen = _mpd_real_size(u, ulen);
} while (u[ulen-1] != 0);
return n;
}
#ifdef CONFIG_32
/* target base 'wbase' == source base 'ubase' */
static size_t
_copy_equal_base(uint32_t **w, size_t wlen,
const uint32_t *u, size_t ulen)
{
if (wlen < ulen) {
if (!mpd_resize_u32(w, ulen)) {
return SIZE_MAX;
}
}
memcpy(*w, u, ulen * (sizeof **w));
return ulen;
}
/* target base 'wbase' > source base 'ubase' */
static size_t
_baseconv_to_larger(uint32_t **w, size_t wlen, mpd_uint_t wbase,
const mpd_uint_t *u, size_t ulen, mpd_uint_t ubase)
{
size_t n = 0;
mpd_uint_t carry;
assert(wlen > 0 && ulen > 0);
assert(ubase < wbase);
(*w)[n++] = u[--ulen];
while (--ulen != SIZE_MAX) {
carry = _mpd_shortmul_b(*w, *w, n, ubase, wbase);
if (carry) {
if (n >= wlen) {
if (!mpd_resize_u32(w, n+1)) {
return SIZE_MAX;
}
wlen = n+1;
}
(*w)[n++] = carry;
}
carry = _mpd_shortadd_b(*w, n, u[ulen], wbase);
if (carry) {
if (n >= wlen) {
if (!mpd_resize_u32(w, n+1)) {
return SIZE_MAX;
}
wlen = n+1;
}
(*w)[n++] = carry;
}
}
return n;
}
/* target base wbase < source base ubase */
static size_t
_coeff_from_larger_base(mpd_t *w, size_t wlen, mpd_uint_t wbase,
mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase,
uint32_t *status)
{
size_t n = 0;
assert(wlen > 0 && ulen > 0);
assert(wbase < ubase);
do {
if (n >= wlen) {
if (!mpd_qresize(w, n+1, status)) {
return SIZE_MAX;
}
wlen = n+1;
}
w->data[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase);
/* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */
ulen = _mpd_real_size(u, ulen);
} while (u[ulen-1] != 0);
return n;
}
#endif
/* target base 'wbase' > source base 'ubase' */
static size_t
_coeff_from_smaller_base(mpd_t *w, mpd_ssize_t wlen, mpd_uint_t wbase,
const uint32_t *u, size_t ulen, mpd_uint_t ubase,
uint32_t *status)
{
mpd_ssize_t n = 0;
mpd_uint_t carry;
assert(wlen > 0 && ulen > 0);
assert(wbase > ubase);
w->data[n++] = u[--ulen];
while (--ulen != SIZE_MAX) {
carry = _mpd_shortmul_b(w->data, w->data, n, ubase, wbase);
if (carry) {
if (n >= wlen) {
if (!mpd_qresize(w, n+1, status)) {
return SIZE_MAX;
}
wlen = n+1;
}
w->data[n++] = carry;
}
carry = _mpd_shortadd_b(w->data, n, u[ulen], wbase);
if (carry) {
if (n >= wlen) {
if (!mpd_qresize(w, n+1, status)) {
return SIZE_MAX;
}
wlen = n+1;
}
w->data[n++] = carry;
}
}
return n;
}
/*
* Convert an integer mpd_t to a multiprecision integer with base <= 2**16.
* The least significant word of the result is (*rdata)[0].
*
* If rdata is NULL, space is allocated by the function and rlen is irrelevant.
* In case of an error any allocated storage is freed and rdata is set back to
* NULL.
*
* If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation
* functions and rlen MUST be correct. If necessary, the function will resize
* rdata. In case of an error the caller must free rdata.
*
* Return value: In case of success, the exact length of rdata, SIZE_MAX
* otherwise.
*/
size_t
mpd_qexport_u16(uint16_t **rdata, size_t rlen, uint32_t rbase,
const mpd_t *src, uint32_t *status)
{
MPD_NEW_STATIC(tsrc,0,0,0,0);
int alloc = 0; /* rdata == NULL */
size_t n;
assert(rbase <= (1U<<16));
if (mpd_isspecial(src) || !_mpd_isint(src)) {
*status |= MPD_Invalid_operation;
return SIZE_MAX;
}
if (*rdata == NULL) {
rlen = mpd_sizeinbase(src, rbase);
if (rlen == SIZE_MAX) {
*status |= MPD_Invalid_operation;
return SIZE_MAX;
}
*rdata = mpd_alloc(rlen, sizeof **rdata);
if (*rdata == NULL) {
goto malloc_error;
}
alloc = 1;
}
if (mpd_iszero(src)) {
**rdata = 0;
return 1;
}
if (src->exp >= 0) {
if (!mpd_qshiftl(&tsrc, src, src->exp, status)) {
goto malloc_error;
}
}
else {
if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) {
goto malloc_error;
}
}
n = _baseconv_to_u16(rdata, rlen, rbase, tsrc.data, tsrc.len);
if (n == SIZE_MAX) {
goto malloc_error;
}
out:
mpd_del(&tsrc);
return n;
malloc_error:
if (alloc) {
mpd_free(*rdata);
*rdata = NULL;
}
n = SIZE_MAX;
*status |= MPD_Malloc_error;
goto out;
}
/*
* Convert an integer mpd_t to a multiprecision integer with base<=UINT32_MAX.
* The least significant word of the result is (*rdata)[0].
*
* If rdata is NULL, space is allocated by the function and rlen is irrelevant.
* In case of an error any allocated storage is freed and rdata is set back to
* NULL.
*
* If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation
* functions and rlen MUST be correct. If necessary, the function will resize
* rdata. In case of an error the caller must free rdata.
*
* Return value: In case of success, the exact length of rdata, SIZE_MAX
* otherwise.
*/
size_t
mpd_qexport_u32(uint32_t **rdata, size_t rlen, uint32_t rbase,
const mpd_t *src, uint32_t *status)
{
MPD_NEW_STATIC(tsrc,0,0,0,0);
int alloc = 0; /* rdata == NULL */
size_t n;
if (mpd_isspecial(src) || !_mpd_isint(src)) {
*status |= MPD_Invalid_operation;
return SIZE_MAX;
}
if (*rdata == NULL) {
rlen = mpd_sizeinbase(src, rbase);
if (rlen == SIZE_MAX) {
*status |= MPD_Invalid_operation;
return SIZE_MAX;
}
*rdata = mpd_alloc(rlen, sizeof **rdata);
if (*rdata == NULL) {
goto malloc_error;
}
alloc = 1;
}
if (mpd_iszero(src)) {
**rdata = 0;
return 1;
}
if (src->exp >= 0) {
if (!mpd_qshiftl(&tsrc, src, src->exp, status)) {
goto malloc_error;
}
}
else {
if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) {
goto malloc_error;
}
}
#ifdef CONFIG_64
n = _baseconv_to_smaller(rdata, rlen, rbase,
tsrc.data, tsrc.len, MPD_RADIX);
#else
if (rbase == MPD_RADIX) {
n = _copy_equal_base(rdata, rlen, tsrc.data, tsrc.len);
}
else if (rbase < MPD_RADIX) {
n = _baseconv_to_smaller(rdata, rlen, rbase,
tsrc.data, tsrc.len, MPD_RADIX);
}
else {
n = _baseconv_to_larger(rdata, rlen, rbase,
tsrc.data, tsrc.len, MPD_RADIX);
}
#endif
if (n == SIZE_MAX) {
goto malloc_error;
}
out:
mpd_del(&tsrc);
return n;
malloc_error:
if (alloc) {
mpd_free(*rdata);
*rdata = NULL;
}
n = SIZE_MAX;
*status |= MPD_Malloc_error;
goto out;
}
/*
* Converts a multiprecision integer with base <= UINT16_MAX+1 to an mpd_t.
* The least significant word of the source is srcdata[0].
*/
void
mpd_qimport_u16(mpd_t *result,
const uint16_t *srcdata, size_t srclen,
uint8_t srcsign, uint32_t srcbase,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_uint_t *usrc; /* uint16_t src copied to an mpd_uint_t array */
mpd_ssize_t rlen; /* length of the result */
size_t n;
assert(srclen > 0);
assert(srcbase <= (1U<<16));
rlen = _mpd_importsize(srclen, srcbase);
if (rlen == MPD_SSIZE_MAX) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc);
if (usrc == NULL) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
for (n = 0; n < srclen; n++) {
usrc[n] = srcdata[n];
}
if (!mpd_qresize(result, rlen, status)) {
goto finish;
}
n = _coeff_from_u16(result, rlen, usrc, srclen, srcbase, status);
if (n == SIZE_MAX) {
goto finish;
}
mpd_set_flags(result, srcsign);
result->exp = 0;
result->len = n;
mpd_setdigits(result);
mpd_qresize(result, result->len, status);
mpd_qfinalize(result, ctx, status);
finish:
mpd_free(usrc);
}
/*
* Converts a multiprecision integer with base <= UINT32_MAX to an mpd_t.
* The least significant word of the source is srcdata[0].
*/
void
mpd_qimport_u32(mpd_t *result,
const uint32_t *srcdata, size_t srclen,
uint8_t srcsign, uint32_t srcbase,
const mpd_context_t *ctx, uint32_t *status)
{
mpd_ssize_t rlen; /* length of the result */
size_t n;
assert(srclen > 0);
rlen = _mpd_importsize(srclen, srcbase);
if (rlen == MPD_SSIZE_MAX) {
mpd_seterror(result, MPD_Invalid_operation, status);
return;
}
if (!mpd_qresize(result, rlen, status)) {
return;
}
#ifdef CONFIG_64
n = _coeff_from_smaller_base(result, rlen, MPD_RADIX,
srcdata, srclen, srcbase,
status);
#else
if (srcbase == MPD_RADIX) {
if (!mpd_qresize(result, srclen, status)) {
return;
}
memcpy(result->data, srcdata, srclen * (sizeof *srcdata));
n = srclen;
}
else if (srcbase < MPD_RADIX) {
n = _coeff_from_smaller_base(result, rlen, MPD_RADIX,
srcdata, srclen, srcbase,
status);
}
else {
mpd_uint_t *usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc);
if (usrc == NULL) {
mpd_seterror(result, MPD_Malloc_error, status);
return;
}
for (n = 0; n < srclen; n++) {
usrc[n] = srcdata[n];
}
n = _coeff_from_larger_base(result, rlen, MPD_RADIX,
usrc, (mpd_ssize_t)srclen, srcbase,
status);
mpd_free(usrc);
}
#endif
if (n == SIZE_MAX) {
return;
}
mpd_set_flags(result, srcsign);
result->exp = 0;
result->len = n;
mpd_setdigits(result);
mpd_qresize(result, result->len, status);
mpd_qfinalize(result, ctx, status);
}
