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math: tanh.c cleanup similar to sinh, cosh

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comments are kept in the double version of the function

compared to fdlibm/freebsd we partition the domain into one
more part and select different threshold points:
now the [log(5/3)/2,log(3)/2] and [log(3)/2,inf] domains
should have <1.5ulp error
(so only the last bit may be wrong, assuming good exp, expm1)

(note that log(3)/2 and log(5/3)/2 are the points where tanh
changes resolution: tanh(log(3)/2)=0.5, tanh(log(5/3)/2)=0.25)

for some x < log(5/3)/2 (~=0.2554) the error can be >1.5ulp
but it should be <2ulp
(the freebsd code had some >2ulp errors in [0.255,1])

even with the extra logic the new code produces smaller
object files
This commit is contained in:
Szabolcs Nagy
2012-12-16 19:52:42 +01:00
parent f143458223
commit e42a977fe5
3 changed files with 83 additions and 173 deletions
Download Patch File Download Diff File Expand all files Collapse all files

94
src/math/tanh.c
View File

@ -1,73 +1,41 @@
/* origin: FreeBSD /usr/src/lib/msun/src/s_tanh.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* Tanh(x)
* Return the Hyperbolic Tangent of x
*
* Method :
* x -x
* e - e
* 0. tanh(x) is defined to be -----------
* x -x
* e + e
* 1. reduce x to non-negative by tanh(-x) = -tanh(x).
* 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0
* -t
* 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x)
* t + 2
* 2
* 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x)
* t + 2
* 22 <= x <= INF : tanh(x) := 1.
*
* Special cases:
* tanh(NaN) is NaN;
* only tanh(0)=0 is exact for finite argument.
*/
#include "libm.h" #include "libm.h"
static const double tiny = 1.0e-300, huge = 1.0e300; /* tanh(x) = (exp(x) - exp(-x))/(exp(x) + exp(-x))
* = (exp(2*x) - 1)/(exp(2*x) - 1 + 2)
* = (1 - exp(-2*x))/(exp(-2*x) - 1 + 2)
*/
double tanh(double x) double tanh(double x)
{ {
double t,z; union {double f; uint64_t i;} u = {.f = x};
int32_t jx,ix; uint32_t w;
int sign;
double t;
GET_HIGH_WORD(jx, x); /* x = |x| */
ix = jx & 0x7fffffff; sign = u.i >> 63;
u.i &= (uint64_t)-1/2;
x = u.f;
w = u.i >> 32;
/* x is INF or NaN */ if (w > 0x3fe193ea) {
if (ix >= 0x7ff00000) { /* |x| > log(3)/2 ~= 0.5493 or nan */
if (jx >= 0) if (w > 0x40340000) {
return 1.0f/x + 1.0f; /* tanh(+-inf)=+-1 */ /* |x| > 20 or nan */
else /* note: this branch avoids raising overflow */
return 1.0f/x - 1.0f; /* tanh(NaN) = NaN */ /* raise inexact if x!=+-inf and handle nan */
} t = 1 + 0/(x + 0x1p-120f);
if (ix < 0x40360000) { /* |x| < 22 */
if (ix < 0x3e300000) { /* |x| < 2**-28 */
/* tanh(tiny) = tiny with inexact */
if (huge+x > 1.0f)
return x;
}
if (ix >= 0x3ff00000) { /* |x| >= 1 */
t = expm1(2.0f*fabs(x));
z = 1.0f - 2.0f/(t+2.0f);
} else { } else {
t = expm1(-2.0f*fabs(x)); t = expm1(2*x);
z= -t/(t+2.0f); t = 1 - 2/(t+2);
} }
} else { /* |x| >= 22, return +-1 */ } else if (w > 0x3fd058ae) {
z = 1.0f - tiny; /* raise inexact */ /* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1(2*x);
t = t/(t+2);
} else {
/* |x| is small, up to 2ulp error in [0.1,0.2554] */
t = expm1(-2*x);
t = -t/(t+2);
} }
return jx >= 0 ? z : -z; return sign ? -t : t;
} }

70
src/math/tanhf.c
View File

@ -1,55 +1,35 @@
/* origin: FreeBSD /usr/src/lib/msun/src/s_tanhf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include "libm.h" #include "libm.h"
static const float
tiny = 1.0e-30,
huge = 1.0e30;
float tanhf(float x) float tanhf(float x)
{ {
float t,z; union {float f; uint32_t i;} u = {.f = x};
int32_t jx,ix; uint32_t w;
int sign;
float t;
GET_FLOAT_WORD(jx, x); /* x = |x| */
ix = jx & 0x7fffffff; sign = u.i >> 31;
u.i &= 0x7fffffff;
x = u.f;
w = u.i;
/* x is INF or NaN */ if (w > 0x3f0c9f54) {
if(ix >= 0x7f800000) { /* |x| > log(3)/2 ~= 0.5493 or nan */
if (jx >= 0) if (w > 0x41200000) {
return 1.0f/x + 1.0f; /* tanh(+-inf)=+-1 */ /* |x| > 10 */
else t = 1 + 0/(x + 0x1p-120f);
return 1.0f/x - 1.0f; /* tanh(NaN) = NaN */
}
if (ix < 0x41100000) { /* |x| < 9 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
/* tanh(tiny) = tiny with inexact */
if (huge+x > 1.0f)
return x;
}
if (ix >= 0x3f800000) { /* |x|>=1 */
t = expm1f(2.0f*fabsf(x));
z = 1.0f - 2.0f/(t+2.0f);
} else { } else {
t = expm1f(-2.0f*fabsf(x)); t = expm1f(2*x);
z = -t/(t+2.0f); t = 1 - 2/(t+2);
} }
} else { /* |x| >= 9, return +-1 */ } else if (w > 0x3e82c578) {
z = 1.0f - tiny; /* raise inexact */ /* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1f(2*x);
t = t/(t+2);
} else {
/* |x| is small */
t = expm1f(-2*x);
t = -t/(t+2);
} }
return jx >= 0 ? z : -z; return sign ? -t : t;
} }

92
src/math/tanhl.c
View File

@ -1,38 +1,3 @@
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_tanhl.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* tanhl(x)
* Return the Hyperbolic Tangent of x
*
* Method :
* x -x
* e - e
* 0. tanhl(x) is defined to be -----------
* x -x
* e + e
* 1. reduce x to non-negative by tanhl(-x) = -tanhl(x).
* 2. 0 <= x <= 2**-55 : tanhl(x) := x*(one+x)
* -t
* 2**-55 < x <= 1 : tanhl(x) := -----; t = expm1l(-2x)
* t + 2
* 2
* 1 <= x <= 23.0 : tanhl(x) := 1- ----- ; t=expm1l(2x)
* t + 2
* 23.0 < x <= INF : tanhl(x) := 1.
*
* Special cases:
* tanhl(NaN) is NaN;
* only tanhl(0)=0 is exact for finite argument.
*/
#include "libm.h" #include "libm.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
@ -41,43 +6,40 @@ long double tanhl(long double x)
return tanh(x); return tanh(x);
} }
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
static const long double tiny = 1.0e-4900L;
long double tanhl(long double x) long double tanhl(long double x)
{ {
long double t,z; union {
int32_t se; long double f;
uint32_t jj0,jj1,ix; struct{uint64_t m; uint16_t se; uint16_t pad;} i;
} u = {.f = x};
unsigned ex = u.i.se & 0x7fff;
unsigned sign = u.i.se & 0x8000;
uint32_t w;
long double t;
/* High word of |x|. */ /* x = |x| */
GET_LDOUBLE_WORDS(se, jj0, jj1, x); u.i.se = ex;
ix = se & 0x7fff; x = u.f;
w = u.i.m >> 32;
/* x is INF or NaN */ if (ex > 0x3ffe || (ex == 0x3ffe && w > 0x8c9f53d5)) {
if (ix == 0x7fff) { /* |x| > log(3)/2 ~= 0.5493 or nan */
/* for NaN it's not important which branch: tanhl(NaN) = NaN */ if (ex >= 0x3fff+5) {
if (se & 0x8000) /* |x| >= 32 */
return 1.0/x-1.0; /* tanhl(-inf)= -1; */ t = 1 + 0/(x + 0x1p-120f);
return 1.0/x+1.0; /* tanhl(+inf)= +1 */
}
/* |x| < 23 */
if (ix < 0x4003 || (ix == 0x4003 && jj0 < 0xb8000000u)) {
if ((ix|jj0|jj1) == 0) /* x == +- 0 */
return x;
if (ix < 0x3fc8) /* |x| < 2**-55 */
return x*(1.0+tiny); /* tanh(small) = small */
if (ix >= 0x3fff) { /* |x| >= 1 */
t = expm1l(2.0*fabsl(x));
z = 1.0 - 2.0/(t+2.0);
} else { } else {
t = expm1l(-2.0*fabsl(x)); t = expm1l(2*x);
z = -t/(t+2.0); t = 1 - 2/(t+2);
} }
/* |x| > 23, return +-1 */ } else if (ex > 0x3ffd || (ex == 0x3ffd && w > 0x82c577d4)) {
/* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1l(2*x);
t = t/(t+2);
} else { } else {
z = 1.0 - tiny; /* raise inexact flag */ /* |x| is small */
t = expm1l(-2*x);
t = -t/(t+2);
} }
return se & 0x8000 ? -z : z; return sign ? -t : t;
} }
#endif #endif