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The cexp() and {,c}{cos,sin}h functions all need to be able to compute

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exp(x) scaled down by some factor, and the challenge is doing this
accurately when exp(x) would overflow.  This change replaces all of
the tricks we've been using with common __ldexp_exp() and
__ldexp_cexp() routines that handle all the scaling.

bde plans to improve on this further by moving the guts of exp() into
k_exp.c and handling the scaling in a more direct manner.  But the
current approach is simple and adequate for now.
This commit is contained in:
David Schultz 2011-10-21 06:27:56 +00:00
parent f2ea2b9d27
commit 12188b77a2
Notes: svn2git 2020-12-20 02:59:44 +00:00 
svn path=/head/; revision=226597
6 changed files with 210 additions and 33 deletions
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2
lib/msun/Makefile
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@ -49,7 +49,7 @@ COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \
e_pow.c e_powf.c e_rem_pio2.c \
e_rem_pio2f.c e_remainder.c e_remainderf.c e_scalb.c e_scalbf.c \
e_sinh.c e_sinhf.c e_sqrt.c e_sqrtf.c fenv.c \
k_cos.c k_cosf.c k_rem_pio2.c k_sin.c k_sinf.c \
k_cos.c k_cosf.c k_exp.c k_expf.c k_rem_pio2.c k_sin.c k_sinf.c \
k_tan.c k_tanf.c \
s_asinh.c s_asinhf.c s_atan.c s_atanf.c s_carg.c s_cargf.c s_cargl.c \
s_cbrt.c s_cbrtf.c s_ceil.c s_ceilf.c \

108
lib/msun/src/k_exp.c Normal file
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@ -0,0 +1,108 @@
/*-
* Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
* 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 <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <complex.h>
#include "math.h"
#include "math_private.h"
static const uint32_t k = 1799; /* constant for reduction */
static const double kln2 = 1246.97177782734161156; /* k * ln2 */
/*
* Compute exp(x), scaled to avoid spurious overflow. An exponent is
* returned separately in 'expt'.
*
* Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
* Output: 2**1023 <= y < 2**1024
*/
static double
__frexp_exp(double x, int *expt)
{
double exp_x;
uint32_t hx;
/*
* We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
* minimize |exp(kln2) - 2**k|. We also scale the exponent of
* exp_x to MAX_EXP so that the result can be multiplied by
* a tiny number without losing accuracy due to denormalization.
*/
exp_x = exp(x - kln2);
GET_HIGH_WORD(hx, exp_x);
*expt = (hx >> 20) - (0x3ff + 1023) + k;
SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
return (exp_x);
}
/*
* __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
* They are intended for large arguments (real part >= ln(DBL_MAX))
* where care is needed to avoid overflow.
*
* The present implementation is narrowly tailored for our hyperbolic and
* exponential functions. We assume expt is small (0 or -1), and the caller
* has filtered out very large x, for which overflow would be inevitable.
*/
double
__ldexp_exp(double x, int expt)
{
double exp_x, scale;
int ex_expt;
exp_x = __frexp_exp(x, &ex_expt);
expt += ex_expt;
INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
return (exp_x * scale);
}
double complex
__ldexp_cexp(double complex z, int expt)
{
double x, y, exp_x, scale1, scale2;
int ex_expt, half_expt;
x = creal(z);
y = cimag(z);
exp_x = __frexp_exp(x, &ex_expt);
expt += ex_expt;
/*
* Arrange so that scale1 * scale2 == 2**expt. We use this to
* compensate for scalbn being horrendously slow.
*/
half_expt = expt / 2;
INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
half_expt = expt - half_expt;
INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
return (cpack(cos(y) * exp_x * scale1 * scale2,
sin(y) * exp_x * scale1 * scale2));
}

87
lib/msun/src/k_expf.c Normal file
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@ -0,0 +1,87 @@
/*-
* Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
* 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 <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <complex.h>
#include "math.h"
#include "math_private.h"
static const uint32_t k = 235; /* constant for reduction */
static const float kln2 = 162.88958740F; /* k * ln2 */
/*
* See k_exp.c for details.
*
* Input: ln(FLT_MAX) <= x < ln(2 * FLT_MAX / FLT_MIN_DENORM) ~= 192.7
* Output: 2**127 <= y < 2**128
*/
static float
__frexp_expf(float x, int *expt)
{
double exp_x;
uint32_t hx;
exp_x = expf(x - kln2);
GET_FLOAT_WORD(hx, exp_x);
*expt = (hx >> 23) - (0x7f + 127) + k;
SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 127) << 23));
return (exp_x);
}
float
__ldexp_expf(float x, int expt)
{
float exp_x, scale;
int ex_expt;
exp_x = __frexp_expf(x, &ex_expt);
expt += ex_expt;
SET_FLOAT_WORD(scale, (0x7f + expt) << 23);
return (exp_x * scale);
}
float complex
__ldexp_cexpf(float complex z, int expt)
{
float x, y, exp_x, scale1, scale2;
int ex_expt, half_expt;
x = crealf(z);
y = cimagf(z);
exp_x = __frexp_expf(x, &ex_expt);
expt += ex_expt;
half_expt = expt / 2;
SET_FLOAT_WORD(scale1, (0x7f + half_expt) << 23);
half_expt = expt - half_expt;
SET_FLOAT_WORD(scale2, (0x7f + half_expt) << 23);
return (cpackf(cosf(y) * exp_x * scale1 * scale2,
sinf(y) * exp_x * scale1 * scale2));
}

8
lib/msun/src/math_private.h
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@ -397,6 +397,10 @@ int __ieee754_rem_pio2(double,double*);
double __kernel_sin(double,double,int);
double __kernel_cos(double,double);
double __kernel_tan(double,double,int);
double __ldexp_exp(double,int);
#ifdef _COMPLEX_H
double complex __ldexp_cexp(double complex,int);
#endif
/* float precision kernel functions */
#ifdef INLINE_REM_PIO2F
@ -415,6 +419,10 @@ float __kernel_cosdf(double);
__inline
#endif
float __kernel_tandf(double,int);
float __ldexp_expf(float,int);
#ifdef _COMPLEX_H
float complex __ldexp_cexpf(float complex,int);
#endif
/* long double precision kernel functions */
long double __kernel_sinl(long double, long double, int);

19
lib/msun/src/s_cexp.c
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@ -34,18 +34,13 @@ __FBSDID("$FreeBSD$");
static const uint32_t
exp_ovfl = 0x40862e42, /* high bits of MAX_EXP * ln2 ~= 710 */
cexp_ovfl = 0x4096b8e4, /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
k = 1799; /* constant for reduction */
static const double
kln2 = 1246.97177782734161156; /* k * ln2 */
cexp_ovfl = 0x4096b8e4; /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
double complex
cexp(double complex z)
{
double x, y, exp_x;
uint32_t hx, hy, lx, ly;
int scale;
x = creal(z);
y = cimag(z);
@ -77,17 +72,9 @@ cexp(double complex z)
if (hx >= exp_ovfl && hx <= cexp_ovfl) {
/*
* x is between 709.7 and 1454.3, so we must scale to avoid
* overflow in exp(x). We use exp(x) = exp(x - kln2) * 2**k,
* carefully chosen to minimize |exp(kln2) - 2**k|. We also
* scale the exponent of exp(x) to MANT_DIG to avoid loss of
* accuracy due to underflow if sin(y) is tiny.
* overflow in exp(x).
*/
exp_x = exp(x - kln2);
GET_HIGH_WORD(hx, exp_x);
SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 52) << 20));
scale = (hx >> 20) - (0x3ff + 52) + k;
return (cpack(scalbn(cos(y) * exp_x, scale),
scalbn(sin(y) * exp_x, scale)));
return (__ldexp_cexp(z, 0));
} else {
/*
* Cases covered here:

19
lib/msun/src/s_cexpf.c
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@ -34,18 +34,13 @@ __FBSDID("$FreeBSD$");
static const uint32_t
exp_ovfl = 0x42b17218, /* MAX_EXP * ln2 ~= 88.722839355 */
cexp_ovfl = 0x43400074, /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
k = 235; /* constant for reduction */
static const float
kln2 = 162.88958740f; /* k * ln2 */
cexp_ovfl = 0x43400074; /* (MAX_EXP - MIN_DENORM_EXP) * ln2 */
float complex
cexpf(float complex z)
{
float x, y, exp_x;
uint32_t hx, hy;
int scale;
x = crealf(z);
y = cimagf(z);
@ -77,17 +72,9 @@ cexpf(float complex z)
if (hx >= exp_ovfl && hx <= cexp_ovfl) {
/*
* x is between 88.7 and 192, so we must scale to avoid
* overflow in expf(x). We use exp(x) = exp(x - kln2) * 2**k,
* carefully chosen to minimize |exp(kln2) - 2**k|. We also
* scale the exponent of exp(x) to MANT_DIG to avoid loss of
* accuracy due to underflow if sin(y) is tiny.
* overflow in expf(x).
*/
exp_x = expf(x - kln2);
GET_FLOAT_WORD(hx, exp_x);
SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 23) << 23));
scale = (hx >> 23) - (0x7f + 23) + k;
return (cpackf(scalbnf(cosf(y) * exp_x, scale),
scalbnf(sinf(y) * exp_x, scale)));
return (__ldexp_cexpf(z, 0));
} else {
/*
* Cases covered here: