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提交 482ccd2f 编写于 作者: S Szabolcs Nagy
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math: rewrite inverse hyperbolic functions to be simpler/smaller 

modifications:
* avoid unsigned->signed integer conversion
* do not handle special cases when they work correctly anyway
* more strict threshold values (0x1p26 instead of 0x1p28 etc)
* smaller code, cleaner branching logic
* same precision as the old code:
    acosh(x) has up to 2ulp error in [1,1.125]
    asinh(x) has up to 1.6ulp error in [0.125,0.5], [-0.5,-0.125]
    atanh(x) has up to 1.7ulp error in [0.125,0.5], [-0.5,-0.125]
上级 64623cd5
隐藏空白更改
内联 并排
Showing with 149 addition and 406 deletion
src/math/acosh.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/e_acosh.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* acosh(x)
* Method :
* Based on
* acosh(x) = log [ x + sqrt(x*x-1) ]
* we have
* acosh(x) := log(x)+ln2, if x is large; else
* acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
* acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
*
* Special cases:
* acosh(x) is NaN with signal if x<1.
* acosh(NaN) is NaN without signal.
*/
#include "libm.h"
static const double
ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
/* acosh(x) = log(x + sqrt(x*x-1)) */
double acosh(double x)
{
double t;
int32_t hx;
uint32_t lx;
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
/* x < 1 domain error is handled in the called functions */
EXTRACT_WORDS(hx, lx, x);
if (hx < 0x3ff00000) { /* x < 1 */
return (x-x)/(x-x);
} else if (hx >= 0x41b00000) { /* x > 2**28 */
if (hx >= 0x7ff00000) /* x is inf of NaN */
return x+x;
return log(x) + ln2; /* acosh(huge) = log(2x) */
} else if ((hx-0x3ff00000 | lx) == 0) {
return 0.0; /* acosh(1) = 0 */
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
t = x*x;
return log(2.0*x - 1.0/(x+sqrt(t-1.0)));
} else { /* 1 < x < 2 */
t = x-1.0;
return log1p(t + sqrt(2.0*t+t*t));
}
if (e < 0x3ff + 1)
/* |x| < 2, up to 2ulp error in [1,1.125] */
return log1p(x-1 + sqrt((x-1)*(x-1)+2*(x-1)));
if (e < 0x3ff + 26)
/* |x| < 0x1p26 */
return log(2*x - 1/(x+sqrt(x*x-1)));
/* |x| >= 0x1p26 or nan */
return log(x) + 0.693147180559945309417232121458176568;
}
src/math/acoshf.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/e_acoshf.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"
static const float
ln2 = 6.9314718246e-01; /* 0x3f317218 */
/* acosh(x) = log(x + sqrt(x*x-1)) */
float acoshf(float x)
{
float t;
int32_t hx;
union {float f; int32_t i;} u = {.f = x};
GET_FLOAT_WORD(hx, x);
if (hx < 0x3f800000) { /* x < 1 */
return (x-x)/(x-x);
} else if (hx >= 0x4d800000) { /* x > 2**28 */
if (hx >= 0x7f800000) /* x is inf of NaN */
return x + x;
return logf(x) + ln2; /* acosh(huge)=log(2x) */
} else if (hx == 0x3f800000) {
return 0.0f; /* acosh(1) = 0 */
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
t = x*x;
return logf(2.0f*x - 1.0f/(x+sqrtf(t-1.0f)));
} else { /* 1 < x < 2 */
t = x-1.0f;
return log1pf(t + sqrtf(2.0f*t+t*t));
}
if (u.i < 0x3f800000+(1<<23))
/* x < 2, invalid if x < 1 or nan */
/* up to 2ulp error in [1,1.125] */
return log1pf(x-1 + sqrtf((x-1)*(x-1)+2*(x-1)));
if (u.i < 0x3f800000+(12<<23))
/* x < 0x1p12 */
return logf(2*x - 1/(x+sqrtf(x*x-1)));
/* x >= 0x1p12 */
return logf(x) + 0.693147180559945309417232121458176568f;
}
src/math/acoshl.c
浏览文件 @ 482ccd2f
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_acoshl.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.
* ====================================================
*/
/* acoshl(x)
* Method :
* Based on
* acoshl(x) = logl [ x + sqrtl(x*x-1) ]
* we have
* acoshl(x) := logl(x)+ln2, if x is large; else
* acoshl(x) := logl(2x-1/(sqrtl(x*x-1)+x)) if x>2; else
* acoshl(x) := log1pl(t+sqrtl(2.0*t+t*t)); where t=x-1.
*
* Special cases:
* acoshl(x) is NaN with signal if x<1.
* acoshl(NaN) is NaN without signal.
*/
#include "libm.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
......@@ -31,29 +6,20 @@ long double acoshl(long double x)
return acosh(x);
}
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
static const long double
ln2 = 6.931471805599453094287e-01L; /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
/* acosh(x) = log(x + sqrt(x*x-1)) */
long double acoshl(long double x)
{
long double t;
uint32_t se,i0,i1;
union {
long double f;
struct{uint64_t m; int16_t se; uint16_t pad;} i;
} u = {.f = x};
GET_LDOUBLE_WORDS(se, i0, i1, x);
if (se < 0x3fff || se & 0x8000) { /* x < 1 */
return (x-x)/(x-x);
} else if (se >= 0x401d) { /* x > 2**30 */
if (se >= 0x7fff) /* x is inf or NaN */
return x+x;
return logl(x) + ln2; /* acoshl(huge) = logl(2x) */
} else if (((se-0x3fff)|i0|i1) == 0) {
return 0.0; /* acosh(1) = 0 */
} else if (se > 0x4000) { /* x > 2 */
t = x*x;
return logl(2.0*x - 1.0/(x + sqrtl(t - 1.0)));
}
/* 1 < x <= 2 */
t = x - 1.0;
return log1pl(t + sqrtl(2.0*t + t*t));
if (u.i.se < 0x3fff + 1)
/* x < 2, invalid if x < 1 or nan */
return log1pl(x-1 + sqrtl((x-1)*(x-1)+2*(x-1)));
if (u.i.se < 0x3fff + 32)
/* x < 0x1p32 */
return logl(2*x - 1/(x+sqrtl(x*x-1)));
return logl(x) + 0.693147180559945309417232121458176568L;
}
#endif
src/math/asinh.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/s_asinh.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.
* ====================================================
*/
/* asinh(x)
* Method :
* Based on
* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
* we have
* asinh(x) := x if 1+x*x=1,
* := sign(x)*(log(x)+ln2)) for large |x|, else
* := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
* := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
*/
#include "libm.h"
static const double
ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
huge= 1.00000000000000000000e+300;
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
double asinh(double x)
{
double t,w;
int32_t hx,ix;
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x7ff00000) /* x is inf or NaN */
return x+x;
if (ix < 0x3e300000) { /* |x| < 2**-28 */
/* return x inexact except 0 */
if (huge+x > 1.0)
return x;
}
if (ix > 0x41b00000) { /* |x| > 2**28 */
w = log(fabs(x)) + ln2;
} else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
t = fabs(x);
w = log(2.0*t + 1.0/(sqrt(x*x+1.0)+t));
} else { /* 2.0 > |x| > 2**-28 */
t = x*x;
w =log1p(fabs(x) + t/(1.0+sqrt(1.0+t)));
/* |x| */
u.i &= (uint64_t)-1/2;
x = u.f;
if (e >= 0x3ff + 26) {
/* |x| >= 0x1p26 or inf or nan */
x = log(x) + 0.693147180559945309417232121458176568;
} else if (e >= 0x3ff + 1) {
/* |x| >= 2 */
x = log(2*x + 1/(sqrt(x*x+1)+x));
} else if (e >= 0x3ff - 26) {
/* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
x = log1p(x + x*x/(sqrt(x*x+1)+1));
} else {
/* |x| < 0x1p-26, raise inexact if x != 0 */
FORCE_EVAL(x + 0x1p1000);
}
if (hx > 0)
return w;
return -w;
return s ? -x : x;
}
src/math/asinhf.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/s_asinhf.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"
static const float
ln2 = 6.9314718246e-01, /* 0x3f317218 */
huge= 1.0000000000e+30;
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
float asinhf(float x)
{
float t,w;
int32_t hx,ix;
union {float f; uint32_t i;} u = {.f = x};
uint32_t i = u.i & 0x7fffffff;
unsigned s = u.i >> 31;
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x7f800000) /* x is inf or NaN */
return x+x;
if (ix < 0x31800000) { /* |x| < 2**-28 */
/* return x inexact except 0 */
if (huge+x > 1.0f)
return x;
}
if (ix > 0x4d800000) { /* |x| > 2**28 */
w = logf(fabsf(x)) + ln2;
} else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
t = fabsf(x);
w = logf(2.0f*t + 1.0f/(sqrtf(x*x+1.0f)+t));
} else { /* 2.0 > |x| > 2**-28 */
t = x*x;
w =log1pf(fabsf(x) + t/(1.0f+sqrtf(1.0f+t)));
/* |x| */
u.i = i;
x = u.f;
if (i >= 0x3f800000 + (12<<23)) {
/* |x| >= 0x1p12 or inf or nan */
x = logf(x) + 0.693147180559945309417232121458176568f;
} else if (i >= 0x3f800000 + (1<<23)) {
/* |x| >= 2 */
x = logf(2*x + 1/(sqrtf(x*x+1)+x));
} else if (i >= 0x3f800000 - (12<<23)) {
/* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */
x = log1pf(x + x*x/(sqrtf(x*x+1)+1));
} else {
/* |x| < 0x1p-12, raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
}
if (hx > 0)
return w;
return -w;
return s ? -x : x;
}
src/math/asinhl.c
浏览文件 @ 482ccd2f
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_asinhl.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.
* ====================================================
*/
/* asinhl(x)
* Method :
* Based on
* asinhl(x) = signl(x) * logl [ |x| + sqrtl(x*x+1) ]
* we have
* asinhl(x) := x if 1+x*x=1,
* := signl(x)*(logl(x)+ln2)) for large |x|, else
* := signl(x)*logl(2|x|+1/(|x|+sqrtl(x*x+1))) if|x|>2, else
* := signl(x)*log1pl(|x| + x^2/(1 + sqrtl(1+x^2)))
*/
#include "libm.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
......@@ -28,35 +6,33 @@ long double asinhl(long double x)
return asinh(x);
}
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
static const long double
ln2 = 6.931471805599453094287e-01L, /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
huge = 1.000000000000000000e+4900L;
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
long double asinhl(long double x)
{
long double t,w;
int32_t hx,ix;
union {
long double f;
struct{uint64_t m; uint16_t se; uint16_t pad;} i;
} u = {.f = x};
unsigned e = u.i.se & 0x7fff;
unsigned s = u.i.se >> 15;
GET_LDOUBLE_EXP(hx, x);
ix = hx & 0x7fff;
if (ix == 0x7fff)
return x + x; /* x is inf or NaN */
if (ix < 0x3fde) { /* |x| < 2**-34 */
/* return x, raise inexact if x != 0 */
if (huge+x > 1.0)
return x;
}
if (ix > 0x4020) { /* |x| > 2**34 */
w = logl(fabsl(x)) + ln2;
} else if (ix > 0x4000) { /* 2**34 > |x| > 2.0 */
t = fabsl(x);
w = logl(2.0*t + 1.0/(sqrtl(x*x + 1.0) + t));
} else { /* 2.0 > |x| > 2**-28 */
t = x*x;
w =log1pl(fabsl(x) + t/(1.0 + sqrtl(1.0 + t)));
/* |x| */
u.i.se = e;
x = u.f;
if (e >= 0x3fff + 32) {
/* |x| >= 0x1p32 or inf or nan */
x = logl(x) + 0.693147180559945309417232121458176568L;
} else if (e >= 0x3fff + 1) {
/* |x| >= 2 */
x = logl(2*x + 1/(sqrtl(x*x+1)+x));
} else if (e >= 0x3fff - 32) {
/* |x| >= 0x1p-32 */
x = log1pl(x + x*x/(sqrtl(x*x+1)+1));
} else {
/* |x| < 0x1p-32, raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p1000);
}
if (hx & 0x8000)
return -w;
return w;
return s ? -x : x;
}
#endif
src/math/atanh.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/e_atanh.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* atanh(x)
* Method :
* 1.Reduced x to positive by atanh(-x) = -atanh(x)
* 2.For x>=0.5
* 1 2x x
* atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
* 2 1 - x 1 - x
*
* For x<0.5
* atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
*
* Special cases:
* atanh(x) is NaN if |x| > 1 with signal;
* atanh(NaN) is that NaN with no signal;
* atanh(+-1) is +-INF with signal.
*
*/
#include "libm.h"
static const double huge = 1e300;
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
double atanh(double x)
{
double t;
int32_t hx,ix;
uint32_t lx;
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
/* |x| */
u.i &= (uint64_t)-1/2;
x = u.f;
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7fffffff;
if ((ix | ((lx|-lx)>>31)) > 0x3ff00000) /* |x| > 1 */
return (x-x)/(x-x);
if (ix == 0x3ff00000)
return x/0.0;
if (ix < 0x3e300000 && (huge+x) > 0.0) /* x < 2**-28 */
return x;
SET_HIGH_WORD(x, ix);
if (ix < 0x3fe00000) { /* x < 0.5 */
t = x+x;
t = 0.5*log1p(t + t*x/(1.0-x));
} else
t = 0.5*log1p((x+x)/(1.0-x));
if (hx >= 0)
return t;
return -t;
if (e < 0x3ff - 1) {
/* |x| < 0.5, up to 1.7ulp error */
x = 0.5*log1p(2*x + 2*x*x/(1-x));
} else {
x = 0.5*log1p(2*x/(1-x));
}
return s ? -x : x;
}
src/math/atanhf.c
浏览文件 @ 482ccd2f
/* origin: FreeBSD /usr/src/lib/msun/src/e_atanhf.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"
static const float huge = 1e30;
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
float atanhf(float x)
{
float t;
int32_t hx,ix;
union {float f; uint32_t i;} u = {.f = x};
unsigned s = u.i >> 31;
/* |x| */
u.i &= 0x7fffffff;
x = u.f;
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix > 0x3f800000) /* |x| > 1 */
return (x-x)/(x-x);
if (ix == 0x3f800000)
return x/0.0f;
if (ix < 0x31800000 && huge+x > 0.0f) /* x < 2**-28 */
return x;
SET_FLOAT_WORD(x, ix);
if (ix < 0x3f000000) { /* x < 0.5 */
t = x+x;
t = 0.5f*log1pf(t + t*x/(1.0f-x));
} else
t = 0.5f*log1pf((x+x)/(1.0f-x));
if (hx >= 0)
return t;
return -t;
if (u.i < 0x3f800000 - (1<<23)) {
/* |x| < 0.5, up to 1.7ulp error */
x = 0.5f*log1pf(2*x + 2*x*x/(1-x));
} else {
x = 0.5f*log1pf(2*x/(1-x));
}
return s ? -x : x;
}
src/math/atanhl.c
浏览文件 @ 482ccd2f
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_atanh.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.
* ====================================================
*/
/* atanhl(x)
* Method :
* 1.Reduced x to positive by atanh(-x) = -atanh(x)
* 2.For x>=0.5
* 1 2x x
* atanhl(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
* 2 1 - x 1 - x
*
* For x<0.5
* atanhl(x) = 0.5*log1pl(2x+2x*x/(1-x))
*
* Special cases:
* atanhl(x) is NaN if |x| > 1 with signal;
* atanhl(NaN) is that NaN with no signal;
* atanhl(+-1) is +-INF with signal.
*/
#include "libm.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
......@@ -34,31 +6,26 @@ long double atanhl(long double x)
return atanh(x);
}
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
static const long double huge = 1e4900L;
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
long double atanhl(long double x)
{
long double t;
int32_t ix;
uint32_t se,i0,i1;
union {
long double f;
struct{uint64_t m; uint16_t se; uint16_t pad;} i;
} u = {.f = x};
unsigned e = u.i.se & 0x7fff;
unsigned s = u.i.se >> 15;
/* |x| */
u.i.se = e;
x = u.f;
GET_LDOUBLE_WORDS(se, i0, i1, x);
ix = se & 0x7fff;
if ((ix+((((i0&0x7fffffff)|i1)|(-((i0&0x7fffffff)|i1)))>>31)) > 0x3fff)
/* |x| > 1 */
return (x-x)/(x-x);
if (ix == 0x3fff)
return x/0.0;
if (ix < 0x3fe3 && huge+x > 0.0) /* x < 2**-28 */
return x;
SET_LDOUBLE_EXP(x, ix);
if (ix < 0x3ffe) { /* x < 0.5 */
t = x + x;
t = 0.5*log1pl(t + t*x/(1.0 - x));
} else
t = 0.5*log1pl((x + x)/(1.0 - x));
if (se <= 0x7fff)
return t;
return -t;
if (e < 0x3fff - 1) {
/* |x| < 0.5, up to 1.7ulp error */
x = 0.5*log1pl(2*x + 2*x*x/(1-x));
} else {
x = 0.5*log1pl(2*x/(1-x));
}
return s ? -x : x;
}
#endif
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