mirror of git://git.musl-libc.org/musl
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]
This commit is contained in:
Szabolcs Nagy
parent
64623cd59a
commit
482ccd2f74
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@ -1,54 +1,19 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/e_acosh.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*
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*/
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/* acosh(x)
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* Method :
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* Based on
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* acosh(x) = log [ x + sqrt(x*x-1) ]
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* we have
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* acosh(x) := log(x)+ln2, if x is large; else
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* acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
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* acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
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*
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* Special cases:
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* acosh(x) is NaN with signal if x<1.
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* acosh(NaN) is NaN without signal.
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*/
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#include "libm.h"
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static const double
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ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
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/* acosh(x) = log(x + sqrt(x*x-1)) */
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double acosh(double x)
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{
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double t;
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int32_t hx;
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uint32_t lx;
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union {double f; uint64_t i;} u = {.f = x};
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unsigned e = u.i >> 52 & 0x7ff;
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EXTRACT_WORDS(hx, lx, x);
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if (hx < 0x3ff00000) { /* x < 1 */
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return (x-x)/(x-x);
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} else if (hx >= 0x41b00000) { /* x > 2**28 */
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if (hx >= 0x7ff00000) /* x is inf of NaN */
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return x+x;
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return log(x) + ln2; /* acosh(huge) = log(2x) */
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} else if ((hx-0x3ff00000 | lx) == 0) {
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return 0.0; /* acosh(1) = 0 */
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} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
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t = x*x;
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return log(2.0*x - 1.0/(x+sqrt(t-1.0)));
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} else { /* 1 < x < 2 */
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t = x-1.0;
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return log1p(t + sqrt(2.0*t+t*t));
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}
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/* x < 1 domain error is handled in the called functions */
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if (e < 0x3ff + 1)
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/* |x| < 2, up to 2ulp error in [1,1.125] */
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return log1p(x-1 + sqrt((x-1)*(x-1)+2*(x-1)));
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if (e < 0x3ff + 26)
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/* |x| < 0x1p26 */
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return log(2*x - 1/(x+sqrt(x*x-1)));
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/* |x| >= 0x1p26 or nan */
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return log(x) + 0.693147180559945309417232121458176568;
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}
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@ -1,42 +1,17 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/e_acoshf.c */
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/*
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include "libm.h"
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static const float
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ln2 = 6.9314718246e-01; /* 0x3f317218 */
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/* acosh(x) = log(x + sqrt(x*x-1)) */
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float acoshf(float x)
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{
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float t;
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int32_t hx;
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union {float f; int32_t i;} u = {.f = x};
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GET_FLOAT_WORD(hx, x);
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if (hx < 0x3f800000) { /* x < 1 */
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return (x-x)/(x-x);
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} else if (hx >= 0x4d800000) { /* x > 2**28 */
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if (hx >= 0x7f800000) /* x is inf of NaN */
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return x + x;
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return logf(x) + ln2; /* acosh(huge)=log(2x) */
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} else if (hx == 0x3f800000) {
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return 0.0f; /* acosh(1) = 0 */
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} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
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t = x*x;
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return logf(2.0f*x - 1.0f/(x+sqrtf(t-1.0f)));
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} else { /* 1 < x < 2 */
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t = x-1.0f;
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return log1pf(t + sqrtf(2.0f*t+t*t));
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}
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if (u.i < 0x3f800000+(1<<23))
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/* x < 2, invalid if x < 1 or nan */
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/* up to 2ulp error in [1,1.125] */
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return log1pf(x-1 + sqrtf((x-1)*(x-1)+2*(x-1)));
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if (u.i < 0x3f800000+(12<<23))
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/* x < 0x1p12 */
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return logf(2*x - 1/(x+sqrtf(x*x-1)));
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/* x >= 0x1p12 */
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return logf(x) + 0.693147180559945309417232121458176568f;
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}
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@ -1,28 +1,3 @@
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/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_acoshl.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* acoshl(x)
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* Method :
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* Based on
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* acoshl(x) = logl [ x + sqrtl(x*x-1) ]
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* we have
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* acoshl(x) := logl(x)+ln2, if x is large; else
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* acoshl(x) := logl(2x-1/(sqrtl(x*x-1)+x)) if x>2; else
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* acoshl(x) := log1pl(t+sqrtl(2.0*t+t*t)); where t=x-1.
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*
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* Special cases:
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* acoshl(x) is NaN with signal if x<1.
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* acoshl(NaN) is NaN without signal.
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*/
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#include "libm.h"
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#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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@ -31,29 +6,20 @@ long double acoshl(long double x)
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return acosh(x);
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}
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#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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static const long double
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ln2 = 6.931471805599453094287e-01L; /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
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/* acosh(x) = log(x + sqrt(x*x-1)) */
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long double acoshl(long double x)
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{
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long double t;
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uint32_t se,i0,i1;
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union {
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long double f;
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struct{uint64_t m; int16_t se; uint16_t pad;} i;
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} u = {.f = x};
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GET_LDOUBLE_WORDS(se, i0, i1, x);
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if (se < 0x3fff || se & 0x8000) { /* x < 1 */
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return (x-x)/(x-x);
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} else if (se >= 0x401d) { /* x > 2**30 */
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if (se >= 0x7fff) /* x is inf or NaN */
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return x+x;
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return logl(x) + ln2; /* acoshl(huge) = logl(2x) */
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} else if (((se-0x3fff)|i0|i1) == 0) {
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return 0.0; /* acosh(1) = 0 */
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} else if (se > 0x4000) { /* x > 2 */
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t = x*x;
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return logl(2.0*x - 1.0/(x + sqrtl(t - 1.0)));
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}
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/* 1 < x <= 2 */
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t = x - 1.0;
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return log1pl(t + sqrtl(2.0*t + t*t));
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if (u.i.se < 0x3fff + 1)
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/* x < 2, invalid if x < 1 or nan */
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return log1pl(x-1 + sqrtl((x-1)*(x-1)+2*(x-1)));
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if (u.i.se < 0x3fff + 32)
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/* x < 0x1p32 */
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return logl(2*x - 1/(x+sqrtl(x*x-1)));
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return logl(x) + 0.693147180559945309417232121458176568L;
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}
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#endif
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@ -1,55 +1,28 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/s_asinh.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* asinh(x)
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* Method :
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* Based on
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* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
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* we have
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* asinh(x) := x if 1+x*x=1,
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* := sign(x)*(log(x)+ln2)) for large |x|, else
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* := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
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* := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
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*/
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#include "libm.h"
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static const double
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ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
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huge= 1.00000000000000000000e+300;
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/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
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double asinh(double x)
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{
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double t,w;
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int32_t hx,ix;
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union {double f; uint64_t i;} u = {.f = x};
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unsigned e = u.i >> 52 & 0x7ff;
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unsigned s = u.i >> 63;
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GET_HIGH_WORD(hx, x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x7ff00000) /* x is inf or NaN */
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return x+x;
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if (ix < 0x3e300000) { /* |x| < 2**-28 */
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/* return x inexact except 0 */
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if (huge+x > 1.0)
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return x;
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/* |x| */
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u.i &= (uint64_t)-1/2;
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x = u.f;
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if (e >= 0x3ff + 26) {
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/* |x| >= 0x1p26 or inf or nan */
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x = log(x) + 0.693147180559945309417232121458176568;
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} else if (e >= 0x3ff + 1) {
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/* |x| >= 2 */
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x = log(2*x + 1/(sqrt(x*x+1)+x));
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} else if (e >= 0x3ff - 26) {
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/* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
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x = log1p(x + x*x/(sqrt(x*x+1)+1));
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} else {
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/* |x| < 0x1p-26, raise inexact if x != 0 */
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FORCE_EVAL(x + 0x1p1000);
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}
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if (ix > 0x41b00000) { /* |x| > 2**28 */
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w = log(fabs(x)) + ln2;
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} else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
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t = fabs(x);
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w = log(2.0*t + 1.0/(sqrt(x*x+1.0)+t));
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} else { /* 2.0 > |x| > 2**-28 */
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t = x*x;
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w =log1p(fabs(x) + t/(1.0+sqrt(1.0+t)));
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}
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if (hx > 0)
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return w;
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return -w;
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return s ? -x : x;
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}
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@ -1,48 +1,28 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/s_asinhf.c */
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/*
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
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* ====================================================
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*/
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#include "libm.h"
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static const float
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ln2 = 6.9314718246e-01, /* 0x3f317218 */
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huge= 1.0000000000e+30;
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/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
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float asinhf(float x)
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{
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float t,w;
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int32_t hx,ix;
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union {float f; uint32_t i;} u = {.f = x};
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uint32_t i = u.i & 0x7fffffff;
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unsigned s = u.i >> 31;
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GET_FLOAT_WORD(hx, x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x7f800000) /* x is inf or NaN */
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return x+x;
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if (ix < 0x31800000) { /* |x| < 2**-28 */
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/* return x inexact except 0 */
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if (huge+x > 1.0f)
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return x;
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/* |x| */
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u.i = i;
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x = u.f;
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if (i >= 0x3f800000 + (12<<23)) {
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/* |x| >= 0x1p12 or inf or nan */
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x = logf(x) + 0.693147180559945309417232121458176568f;
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} else if (i >= 0x3f800000 + (1<<23)) {
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/* |x| >= 2 */
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x = logf(2*x + 1/(sqrtf(x*x+1)+x));
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} else if (i >= 0x3f800000 - (12<<23)) {
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/* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */
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x = log1pf(x + x*x/(sqrtf(x*x+1)+1));
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} else {
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/* |x| < 0x1p-12, raise inexact if x!=0 */
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FORCE_EVAL(x + 0x1p120f);
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}
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if (ix > 0x4d800000) { /* |x| > 2**28 */
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w = logf(fabsf(x)) + ln2;
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} else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
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t = fabsf(x);
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w = logf(2.0f*t + 1.0f/(sqrtf(x*x+1.0f)+t));
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} else { /* 2.0 > |x| > 2**-28 */
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t = x*x;
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w =log1pf(fabsf(x) + t/(1.0f+sqrtf(1.0f+t)));
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}
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if (hx > 0)
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return w;
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return -w;
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return s ? -x : x;
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}
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|
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@ -1,25 +1,3 @@
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/* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_asinhl.c */
|
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/*
|
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* ====================================================
|
||||
* 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.
|
||||
* ====================================================
|
||||
*/
|
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/* asinhl(x)
|
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* Method :
|
||||
* Based on
|
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* asinhl(x) = signl(x) * logl [ |x| + sqrtl(x*x+1) ]
|
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* we have
|
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* asinhl(x) := x if 1+x*x=1,
|
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* := signl(x)*(logl(x)+ln2)) for large |x|, else
|
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* := signl(x)*logl(2|x|+1/(|x|+sqrtl(x*x+1))) if|x|>2, else
|
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* := signl(x)*log1pl(|x| + x^2/(1 + sqrtl(1+x^2)))
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*/
|
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|
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#include "libm.h"
|
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|
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#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
|
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|
|
@ -28,35 +6,33 @@ long double asinhl(long double x)
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return asinh(x);
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}
|
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#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
|
||||
static const long double
|
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ln2 = 6.931471805599453094287e-01L, /* 0x3FFE, 0xB17217F7, 0xD1CF79AC */
|
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huge = 1.000000000000000000e+4900L;
|
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|
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/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
|
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long double asinhl(long double x)
|
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{
|
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long double t,w;
|
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int32_t hx,ix;
|
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union {
|
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long double f;
|
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struct{uint64_t m; uint16_t se; uint16_t pad;} i;
|
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} u = {.f = x};
|
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unsigned e = u.i.se & 0x7fff;
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unsigned s = u.i.se >> 15;
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|
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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;
|
||||
/* |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 (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)));
|
||||
}
|
||||
if (hx & 0x8000)
|
||||
return -w;
|
||||
return w;
|
||||
return s ? -x : x;
|
||||
}
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -1,58 +1,21 @@
|
|||
/* 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;
|
||||
|
||||
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;
|
||||
/* |x| */
|
||||
u.i &= (uint64_t)-1/2;
|
||||
x = u.f;
|
||||
|
||||
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;
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,42 +1,20 @@
|
|||
/* 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;
|
||||
|
||||
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;
|
||||
/* |x| */
|
||||
u.i &= 0x7fffffff;
|
||||
x = u.f;
|
||||
|
||||
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;
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,31 +1,3 @@
|
|||
/* 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;
|
||||
|
||||
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;
|
||||
/* |x| */
|
||||
u.i.se = e;
|
||||
x = u.f;
|
||||
|
||||
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
|
||||
|
|
|
|||
Loading…
Reference in New Issue