svn commit: r336299 - in head: include lib/msun lib/msun/ld128 lib/msun/ld80 lib/msun/man lib/msun/src
Matt Macy
mmacy at FreeBSD.org
Sun Jul 15 00:23:12 UTC 2018
Author: mmacy
Date: Sun Jul 15 00:23:10 2018
New Revision: 336299
URL: https://svnweb.freebsd.org/changeset/base/336299
Log:
msun: add ld80/ld128 powl, cpow, cpowf, cpowl from openbsd
This corresponds to the latest status (hasn't changed in 9+
years) from openbsd of ld80/ld128 powl, and source cpowf, cpow,
cpowl (the complex power functions for float complex, double
complex, and long double complex) which are required for C99
compliance and were missing from FreeBSD. Also required for
some numerical codes using complex numbered Hamiltonians.
Thanks to jhb for tracking down the issue with making
weak_reference compile on powerpc.
When asked to review, bde said "I don't like it" - but
provided no actionable feedback or superior implementations.
Discussed with: jhb
Submitted by: jmd
Differential Revision: https://reviews.freebsd.org/D15919
Added:
head/lib/msun/ld128/e_powl.c (contents, props changed)
head/lib/msun/ld80/e_powl.c (contents, props changed)
head/lib/msun/man/cpow.3 (contents, props changed)
head/lib/msun/src/polevll.c (contents, props changed)
head/lib/msun/src/s_cpow.c (contents, props changed)
head/lib/msun/src/s_cpowf.c (contents, props changed)
head/lib/msun/src/s_cpowl.c (contents, props changed)
Modified:
head/include/complex.h
head/lib/msun/Makefile
head/lib/msun/Symbol.map
head/lib/msun/man/complex.3
head/lib/msun/src/e_pow.c
head/lib/msun/src/imprecise.c
head/lib/msun/src/math_private.h
Modified: head/include/complex.h
==============================================================================
--- head/include/complex.h Sat Jul 14 23:53:51 2018 (r336298)
+++ head/include/complex.h Sun Jul 15 00:23:10 2018 (r336299)
@@ -109,6 +109,10 @@ double complex conj(double complex) __pure2;
float complex conjf(float complex) __pure2;
long double complex
conjl(long double complex) __pure2;
+float complex cpowf(float complex, float complex);
+double complex cpow(double complex, double complex);
+long double complex
+ cpowl(long double complex, long double complex);
float complex cprojf(float complex) __pure2;
double complex cproj(double complex) __pure2;
long double complex
Modified: head/lib/msun/Makefile
==============================================================================
--- head/lib/msun/Makefile Sat Jul 14 23:53:51 2018 (r336298)
+++ head/lib/msun/Makefile Sun Jul 15 00:23:10 2018 (r336299)
@@ -56,6 +56,7 @@ COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \
imprecise.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 \
+ polevll.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 s_clog.c s_clogf.c \
s_copysign.c s_copysignf.c s_cos.c s_cosf.c \
@@ -98,7 +99,7 @@ COMMON_SRCS+= s_copysignl.c s_fabsl.c s_llrintl.c s_lr
COMMON_SRCS+= catrigl.c \
e_acoshl.c e_acosl.c e_asinl.c e_atan2l.c e_atanhl.c \
e_coshl.c e_fmodl.c e_hypotl.c \
- e_lgammal.c e_lgammal_r.c \
+ e_lgammal.c e_lgammal_r.c e_powl.c \
e_remainderl.c e_sinhl.c e_sqrtl.c \
invtrig.c k_cosl.c k_sinl.c k_tanl.c \
s_asinhl.c s_atanl.c s_cbrtl.c s_ceill.c \
@@ -115,6 +116,7 @@ COMMON_SRCS+= catrig.c catrigf.c \
s_ccosh.c s_ccoshf.c s_cexp.c s_cexpf.c \
s_cimag.c s_cimagf.c s_cimagl.c \
s_conj.c s_conjf.c s_conjl.c \
+ s_cpow.c s_cpowf.c s_cpowl.c \
s_cproj.c s_cprojf.c s_creal.c s_crealf.c s_creall.c \
s_csinh.c s_csinhf.c s_ctanh.c s_ctanhf.c
@@ -134,7 +136,7 @@ INCS+= fenv.h math.h
MAN= acos.3 acosh.3 asin.3 asinh.3 atan.3 atan2.3 atanh.3 \
ceil.3 cacos.3 ccos.3 ccosh.3 cexp.3 \
- cimag.3 clog.3 copysign.3 cos.3 cosh.3 csqrt.3 erf.3 \
+ cimag.3 clog.3 copysign.3 cos.3 cosh.3 cpow.3 csqrt.3 erf.3 \
exp.3 fabs.3 fdim.3 \
feclearexcept.3 feenableexcept.3 fegetenv.3 \
fegetround.3 fenv.3 floor.3 \
@@ -172,6 +174,7 @@ MLINKS+=clog.3 clogf.3 clog.3 clogl.3
MLINKS+=copysign.3 copysignf.3 copysign.3 copysignl.3
MLINKS+=cos.3 cosf.3 cos.3 cosl.3
MLINKS+=cosh.3 coshf.3 cosh.3 coshl.3
+MLINKS+=cpow.3 cpowf.3 cpow.3 cpowl.3
MLINKS+=csqrt.3 csqrtf.3 csqrt.3 csqrtl.3
MLINKS+=erf.3 erfc.3 erf.3 erff.3 erf.3 erfcf.3 erf.3 erfl.3 erf.3 erfcl.3
MLINKS+=exp.3 expm1.3 exp.3 expm1f.3 exp.3 expm1l.3 exp.3 pow.3 exp.3 powf.3 \
Modified: head/lib/msun/Symbol.map
==============================================================================
--- head/lib/msun/Symbol.map Sat Jul 14 23:53:51 2018 (r336298)
+++ head/lib/msun/Symbol.map Sun Jul 15 00:23:10 2018 (r336299)
@@ -274,10 +274,10 @@ FBSD_1.3 {
log1pl;
log2l;
logl;
+ powl;
sinhl;
tanhl;
/* Implemented as weak aliases for imprecise versions */
- powl;
tgammal;
};
@@ -297,6 +297,9 @@ FBSD_1.5 {
clog;
clogf;
clogl;
+ cpow;
+ cpowf;
+ cpowl;
sincos;
sincosf;
sincosl;
Added: head/lib/msun/ld128/e_powl.c
==============================================================================
--- /dev/null 00:00:00 1970 (empty, because file is newly added)
+++ head/lib/msun/ld128/e_powl.c Sun Jul 15 00:23:10 2018 (r336299)
@@ -0,0 +1,443 @@
+/*-
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+
+/*
+ * Copyright (c) 2008 Stephen L. Moshier <steve at moshier.net>
+ *
+ * Permission to use, copy, modify, and distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
+ * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
+ * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
+ * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
+ */
+
+/* powl(x,y) return x**y
+ *
+ * n
+ * Method: Let x = 2 * (1+f)
+ * 1. Compute and return log2(x) in two pieces:
+ * log2(x) = w1 + w2,
+ * where w1 has 113-53 = 60 bit trailing zeros.
+ * 2. Perform y*log2(x) = n+y' by simulating muti-precision
+ * arithmetic, where |y'|<=0.5.
+ * 3. Return x**y = 2**n*exp(y'*log2)
+ *
+ * Special cases:
+ * 1. (anything) ** 0 is 1
+ * 2. (anything) ** 1 is itself
+ * 3. (anything) ** NAN is NAN
+ * 4. NAN ** (anything except 0) is NAN
+ * 5. +-(|x| > 1) ** +INF is +INF
+ * 6. +-(|x| > 1) ** -INF is +0
+ * 7. +-(|x| < 1) ** +INF is +0
+ * 8. +-(|x| < 1) ** -INF is +INF
+ * 9. +-1 ** +-INF is NAN
+ * 10. +0 ** (+anything except 0, NAN) is +0
+ * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
+ * 12. +0 ** (-anything except 0, NAN) is +INF
+ * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
+ * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
+ * 15. +INF ** (+anything except 0,NAN) is +INF
+ * 16. +INF ** (-anything except 0,NAN) is +0
+ * 17. -INF ** (anything) = -0 ** (-anything)
+ * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
+ * 19. (-anything except 0 and inf) ** (non-integer) is NAN
+ *
+ */
+
+#include <sys/cdefs.h>
+__FBSDID("$FreeBSD$");
+
+#include <float.h>
+#include <math.h>
+
+#include "math_private.h"
+
+static const long double bp[] = {
+ 1.0L,
+ 1.5L,
+};
+
+/* log_2(1.5) */
+static const long double dp_h[] = {
+ 0.0,
+ 5.8496250072115607565592654282227158546448E-1L
+};
+
+/* Low part of log_2(1.5) */
+static const long double dp_l[] = {
+ 0.0,
+ 1.0579781240112554492329533686862998106046E-16L
+};
+
+static const long double zero = 0.0L,
+ one = 1.0L,
+ two = 2.0L,
+ two113 = 1.0384593717069655257060992658440192E34L,
+ huge = 1.0e3000L,
+ tiny = 1.0e-3000L;
+
+/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
+ z = (x-1)/(x+1)
+ 1 <= x <= 1.25
+ Peak relative error 2.3e-37 */
+static const long double LN[] =
+{
+ -3.0779177200290054398792536829702930623200E1L,
+ 6.5135778082209159921251824580292116201640E1L,
+ -4.6312921812152436921591152809994014413540E1L,
+ 1.2510208195629420304615674658258363295208E1L,
+ -9.9266909031921425609179910128531667336670E-1L
+};
+static const long double LD[] =
+{
+ -5.129862866715009066465422805058933131960E1L,
+ 1.452015077564081884387441590064272782044E2L,
+ -1.524043275549860505277434040464085593165E2L,
+ 7.236063513651544224319663428634139768808E1L,
+ -1.494198912340228235853027849917095580053E1L
+ /* 1.0E0 */
+};
+
+/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
+ 0 <= x <= 0.5
+ Peak relative error 5.7e-38 */
+static const long double PN[] =
+{
+ 5.081801691915377692446852383385968225675E8L,
+ 9.360895299872484512023336636427675327355E6L,
+ 4.213701282274196030811629773097579432957E4L,
+ 5.201006511142748908655720086041570288182E1L,
+ 9.088368420359444263703202925095675982530E-3L,
+};
+static const long double PD[] =
+{
+ 3.049081015149226615468111430031590411682E9L,
+ 1.069833887183886839966085436512368982758E8L,
+ 8.259257717868875207333991924545445705394E5L,
+ 1.872583833284143212651746812884298360922E3L,
+ /* 1.0E0 */
+};
+
+static const long double
+ /* ln 2 */
+ lg2 = 6.9314718055994530941723212145817656807550E-1L,
+ lg2_h = 6.9314718055994528622676398299518041312695E-1L,
+ lg2_l = 2.3190468138462996154948554638754786504121E-17L,
+ ovt = 8.0085662595372944372e-0017L,
+ /* 2/(3*log(2)) */
+ cp = 9.6179669392597560490661645400126142495110E-1L,
+ cp_h = 9.6179669392597555432899980587535537779331E-1L,
+ cp_l = 5.0577616648125906047157785230014751039424E-17L;
+
+long double
+powl(long double x, long double y)
+{
+ long double z, ax, z_h, z_l, p_h, p_l;
+ long double yy1, t1, t2, r, s, t, u, v, w;
+ long double s2, s_h, s_l, t_h, t_l;
+ int32_t i, j, k, yisint, n;
+ u_int32_t ix, iy;
+ int32_t hx, hy;
+ ieee_quad_shape_type o, p, q;
+
+ p.value = x;
+ hx = p.parts32.mswhi;
+ ix = hx & 0x7fffffff;
+
+ q.value = y;
+ hy = q.parts32.mswhi;
+ iy = hy & 0x7fffffff;
+
+
+ /* y==zero: x**0 = 1 */
+ if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
+ return one;
+
+ /* 1.0**y = 1; -1.0**+-Inf = 1 */
+ if (x == one)
+ return one;
+ if (x == -1.0L && iy == 0x7fff0000
+ && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
+ return one;
+
+ /* +-NaN return x+y */
+ if ((ix > 0x7fff0000)
+ || ((ix == 0x7fff0000)
+ && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
+ || (iy > 0x7fff0000)
+ || ((iy == 0x7fff0000)
+ && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
+ return x + y;
+
+ /* determine if y is an odd int when x < 0
+ * yisint = 0 ... y is not an integer
+ * yisint = 1 ... y is an odd int
+ * yisint = 2 ... y is an even int
+ */
+ yisint = 0;
+ if (hx < 0)
+ {
+ if (iy >= 0x40700000) /* 2^113 */
+ yisint = 2; /* even integer y */
+ else if (iy >= 0x3fff0000) /* 1.0 */
+ {
+ if (floorl (y) == y)
+ {
+ z = 0.5 * y;
+ if (floorl (z) == z)
+ yisint = 2;
+ else
+ yisint = 1;
+ }
+ }
+ }
+
+ /* special value of y */
+ if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
+ {
+ if (iy == 0x7fff0000) /* y is +-inf */
+ {
+ if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
+ p.parts32.lswlo) == 0)
+ return y - y; /* +-1**inf is NaN */
+ else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */
+ return (hy >= 0) ? y : zero;
+ else /* (|x|<1)**-,+inf = inf,0 */
+ return (hy < 0) ? -y : zero;
+ }
+ if (iy == 0x3fff0000)
+ { /* y is +-1 */
+ if (hy < 0)
+ return one / x;
+ else
+ return x;
+ }
+ if (hy == 0x40000000)
+ return x * x; /* y is 2 */
+ if (hy == 0x3ffe0000)
+ { /* y is 0.5 */
+ if (hx >= 0) /* x >= +0 */
+ return sqrtl (x);
+ }
+ }
+
+ ax = fabsl (x);
+ /* special value of x */
+ if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
+ {
+ if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
+ {
+ z = ax; /*x is +-0,+-inf,+-1 */
+ if (hy < 0)
+ z = one / z; /* z = (1/|x|) */
+ if (hx < 0)
+ {
+ if (((ix - 0x3fff0000) | yisint) == 0)
+ {
+ z = (z - z) / (z - z); /* (-1)**non-int is NaN */
+ }
+ else if (yisint == 1)
+ z = -z; /* (x<0)**odd = -(|x|**odd) */
+ }
+ return z;
+ }
+ }
+
+ /* (x<0)**(non-int) is NaN */
+ if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
+ return (x - x) / (x - x);
+
+ /* |y| is huge.
+ 2^-16495 = 1/2 of smallest representable value.
+ If (1 - 1/131072)^y underflows, y > 1.4986e9 */
+ if (iy > 0x401d654b)
+ {
+ /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
+ if (iy > 0x407d654b)
+ {
+ if (ix <= 0x3ffeffff)
+ return (hy < 0) ? huge * huge : tiny * tiny;
+ if (ix >= 0x3fff0000)
+ return (hy > 0) ? huge * huge : tiny * tiny;
+ }
+ /* over/underflow if x is not close to one */
+ if (ix < 0x3ffeffff)
+ return (hy < 0) ? huge * huge : tiny * tiny;
+ if (ix > 0x3fff0000)
+ return (hy > 0) ? huge * huge : tiny * tiny;
+ }
+
+ n = 0;
+ /* take care subnormal number */
+ if (ix < 0x00010000)
+ {
+ ax *= two113;
+ n -= 113;
+ o.value = ax;
+ ix = o.parts32.mswhi;
+ }
+ n += ((ix) >> 16) - 0x3fff;
+ j = ix & 0x0000ffff;
+ /* determine interval */
+ ix = j | 0x3fff0000; /* normalize ix */
+ if (j <= 0x3988)
+ k = 0; /* |x|<sqrt(3/2) */
+ else if (j < 0xbb67)
+ k = 1; /* |x|<sqrt(3) */
+ else
+ {
+ k = 0;
+ n += 1;
+ ix -= 0x00010000;
+ }
+
+ o.value = ax;
+ o.parts32.mswhi = ix;
+ ax = o.value;
+
+ /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
+ u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
+ v = one / (ax + bp[k]);
+ s = u * v;
+ s_h = s;
+
+ o.value = s_h;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ s_h = o.value;
+ /* t_h=ax+bp[k] High */
+ t_h = ax + bp[k];
+ o.value = t_h;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ t_h = o.value;
+ t_l = ax - (t_h - bp[k]);
+ s_l = v * ((u - s_h * t_h) - s_h * t_l);
+ /* compute log(ax) */
+ s2 = s * s;
+ u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
+ v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
+ r = s2 * s2 * u / v;
+ r += s_l * (s_h + s);
+ s2 = s_h * s_h;
+ t_h = 3.0 + s2 + r;
+ o.value = t_h;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ t_h = o.value;
+ t_l = r - ((t_h - 3.0) - s2);
+ /* u+v = s*(1+...) */
+ u = s_h * t_h;
+ v = s_l * t_h + t_l * s;
+ /* 2/(3log2)*(s+...) */
+ p_h = u + v;
+ o.value = p_h;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ p_h = o.value;
+ p_l = v - (p_h - u);
+ z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
+ z_l = cp_l * p_h + p_l * cp + dp_l[k];
+ /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
+ t = (long double) n;
+ t1 = (((z_h + z_l) + dp_h[k]) + t);
+ o.value = t1;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ t1 = o.value;
+ t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
+
+ /* s (sign of result -ve**odd) = -1 else = 1 */
+ s = one;
+ if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
+ s = -one; /* (-ve)**(odd int) */
+
+ /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
+ yy1 = y;
+ o.value = yy1;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ yy1 = o.value;
+ p_l = (y - yy1) * t1 + y * t2;
+ p_h = yy1 * t1;
+ z = p_l + p_h;
+ o.value = z;
+ j = o.parts32.mswhi;
+ if (j >= 0x400d0000) /* z >= 16384 */
+ {
+ /* if z > 16384 */
+ if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
+ o.parts32.lswlo) != 0)
+ return s * huge * huge; /* overflow */
+ else
+ {
+ if (p_l + ovt > z - p_h)
+ return s * huge * huge; /* overflow */
+ }
+ }
+ else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */
+ {
+ /* z < -16495 */
+ if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
+ o.parts32.lswlo)
+ != 0)
+ return s * tiny * tiny; /* underflow */
+ else
+ {
+ if (p_l <= z - p_h)
+ return s * tiny * tiny; /* underflow */
+ }
+ }
+ /* compute 2**(p_h+p_l) */
+ i = j & 0x7fffffff;
+ k = (i >> 16) - 0x3fff;
+ n = 0;
+ if (i > 0x3ffe0000)
+ { /* if |z| > 0.5, set n = [z+0.5] */
+ n = floorl (z + 0.5L);
+ t = n;
+ p_h -= t;
+ }
+ t = p_l + p_h;
+ o.value = t;
+ o.parts32.lswlo = 0;
+ o.parts32.lswhi &= 0xf8000000;
+ t = o.value;
+ u = t * lg2_h;
+ v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
+ z = u + v;
+ w = v - (z - u);
+ /* exp(z) */
+ t = z * z;
+ u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
+ v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
+ t1 = z - t * u / v;
+ r = (z * t1) / (t1 - two) - (w + z * w);
+ z = one - (r - z);
+ o.value = z;
+ j = o.parts32.mswhi;
+ j += (n << 16);
+ if ((j >> 16) <= 0)
+ z = scalbnl (z, n); /* subnormal output */
+ else
+ {
+ o.parts32.mswhi = j;
+ z = o.value;
+ }
+ return s * z;
+}
Added: head/lib/msun/ld80/e_powl.c
==============================================================================
--- /dev/null 00:00:00 1970 (empty, because file is newly added)
+++ head/lib/msun/ld80/e_powl.c Sun Jul 15 00:23:10 2018 (r336299)
@@ -0,0 +1,616 @@
+/*-
+ * Copyright (c) 2008 Stephen L. Moshier <steve at moshier.net>
+ *
+ * Permission to use, copy, modify, and distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
+ * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
+ * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
+ * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
+ */
+
+/* powl.c
+ *
+ * Power function, long double precision
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, z, powl();
+ *
+ * z = powl( x, y );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Computes x raised to the yth power. Analytically,
+ *
+ * x**y = exp( y log(x) ).
+ *
+ * Following Cody and Waite, this program uses a lookup table
+ * of 2**-i/32 and pseudo extended precision arithmetic to
+ * obtain several extra bits of accuracy in both the logarithm
+ * and the exponential.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * The relative error of pow(x,y) can be estimated
+ * by y dl ln(2), where dl is the absolute error of
+ * the internally computed base 2 logarithm. At the ends
+ * of the approximation interval the logarithm equal 1/32
+ * and its relative error is about 1 lsb = 1.1e-19. Hence
+ * the predicted relative error in the result is 2.3e-21 y .
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ *
+ * IEEE +-1000 40000 2.8e-18 3.7e-19
+ * .001 < x < 1000, with log(x) uniformly distributed.
+ * -1000 < y < 1000, y uniformly distributed.
+ *
+ * IEEE 0,8700 60000 6.5e-18 1.0e-18
+ * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
+ *
+ *
+ * ERROR MESSAGES:
+ *
+ * message condition value returned
+ * pow overflow x**y > MAXNUM INFINITY
+ * pow underflow x**y < 1/MAXNUM 0.0
+ * pow domain x<0 and y noninteger 0.0
+ *
+ */
+
+#include <sys/cdefs.h>
+__FBSDID("$FreeBSD$");
+
+#include <float.h>
+#include <math.h>
+
+#include "math_private.h"
+
+/* Table size */
+#define NXT 32
+/* log2(Table size) */
+#define LNXT 5
+
+/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
+ * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
+ */
+static long double P[] = {
+ 8.3319510773868690346226E-4L,
+ 4.9000050881978028599627E-1L,
+ 1.7500123722550302671919E0L,
+ 1.4000100839971580279335E0L,
+};
+static long double Q[] = {
+/* 1.0000000000000000000000E0L,*/
+ 5.2500282295834889175431E0L,
+ 8.4000598057587009834666E0L,
+ 4.2000302519914740834728E0L,
+};
+/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
+ * If i is even, A[i] + B[i/2] gives additional accuracy.
+ */
+static long double A[33] = {
+ 1.0000000000000000000000E0L,
+ 9.7857206208770013448287E-1L,
+ 9.5760328069857364691013E-1L,
+ 9.3708381705514995065011E-1L,
+ 9.1700404320467123175367E-1L,
+ 8.9735453750155359320742E-1L,
+ 8.7812608018664974155474E-1L,
+ 8.5930964906123895780165E-1L,
+ 8.4089641525371454301892E-1L,
+ 8.2287773907698242225554E-1L,
+ 8.0524516597462715409607E-1L,
+ 7.8799042255394324325455E-1L,
+ 7.7110541270397041179298E-1L,
+ 7.5458221379671136985669E-1L,
+ 7.3841307296974965571198E-1L,
+ 7.2259040348852331001267E-1L,
+ 7.0710678118654752438189E-1L,
+ 6.9195494098191597746178E-1L,
+ 6.7712777346844636413344E-1L,
+ 6.6261832157987064729696E-1L,
+ 6.4841977732550483296079E-1L,
+ 6.3452547859586661129850E-1L,
+ 6.2092890603674202431705E-1L,
+ 6.0762367999023443907803E-1L,
+ 5.9460355750136053334378E-1L,
+ 5.8186242938878875689693E-1L,
+ 5.6939431737834582684856E-1L,
+ 5.5719337129794626814472E-1L,
+ 5.4525386633262882960438E-1L,
+ 5.3357020033841180906486E-1L,
+ 5.2213689121370692017331E-1L,
+ 5.1094857432705833910408E-1L,
+ 5.0000000000000000000000E-1L,
+};
+static long double B[17] = {
+ 0.0000000000000000000000E0L,
+ 2.6176170809902549338711E-20L,
+-1.0126791927256478897086E-20L,
+ 1.3438228172316276937655E-21L,
+ 1.2207982955417546912101E-20L,
+-6.3084814358060867200133E-21L,
+ 1.3164426894366316434230E-20L,
+-1.8527916071632873716786E-20L,
+ 1.8950325588932570796551E-20L,
+ 1.5564775779538780478155E-20L,
+ 6.0859793637556860974380E-21L,
+-2.0208749253662532228949E-20L,
+ 1.4966292219224761844552E-20L,
+ 3.3540909728056476875639E-21L,
+-8.6987564101742849540743E-22L,
+-1.2327176863327626135542E-20L,
+ 0.0000000000000000000000E0L,
+};
+
+/* 2^x = 1 + x P(x),
+ * on the interval -1/32 <= x <= 0
+ */
+static long double R[] = {
+ 1.5089970579127659901157E-5L,
+ 1.5402715328927013076125E-4L,
+ 1.3333556028915671091390E-3L,
+ 9.6181291046036762031786E-3L,
+ 5.5504108664798463044015E-2L,
+ 2.4022650695910062854352E-1L,
+ 6.9314718055994530931447E-1L,
+};
+
+#define douba(k) A[k]
+#define doubb(k) B[k]
+#define MEXP (NXT*16384.0L)
+/* The following if denormal numbers are supported, else -MEXP: */
+#define MNEXP (-NXT*(16384.0L+64.0L))
+/* log2(e) - 1 */
+#define LOG2EA 0.44269504088896340735992L
+
+#define F W
+#define Fa Wa
+#define Fb Wb
+#define G W
+#define Ga Wa
+#define Gb u
+#define H W
+#define Ha Wb
+#define Hb Wb
+
+static const long double MAXLOGL = 1.1356523406294143949492E4L;
+static const long double MINLOGL = -1.13994985314888605586758E4L;
+static const long double LOGE2L = 6.9314718055994530941723E-1L;
+static volatile long double z;
+static long double w, W, Wa, Wb, ya, yb, u;
+static const long double huge = 0x1p10000L;
+#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
+static const long double twom10000 = 0x1p-10000L;
+#else
+static volatile long double twom10000 = 0x1p-10000L;
+#endif
+
+static long double reducl( long double );
+static long double powil ( long double, int );
+
+long double
+powl(long double x, long double y)
+{
+/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
+int i, nflg, iyflg, yoddint;
+long e;
+
+if( y == 0.0L )
+ return( 1.0L );
+
+if( x == 1.0L )
+ return( 1.0L );
+
+if( isnan(x) )
+ return( x );
+if( isnan(y) )
+ return( y );
+
+if( y == 1.0L )
+ return( x );
+
+if( !isfinite(y) && x == -1.0L )
+ return( 1.0L );
+
+if( y >= LDBL_MAX )
+ {
+ if( x > 1.0L )
+ return( INFINITY );
+ if( x > 0.0L && x < 1.0L )
+ return( 0.0L );
+ if( x < -1.0L )
+ return( INFINITY );
+ if( x > -1.0L && x < 0.0L )
+ return( 0.0L );
+ }
+if( y <= -LDBL_MAX )
+ {
+ if( x > 1.0L )
+ return( 0.0L );
+ if( x > 0.0L && x < 1.0L )
+ return( INFINITY );
+ if( x < -1.0L )
+ return( 0.0L );
+ if( x > -1.0L && x < 0.0L )
+ return( INFINITY );
+ }
+if( x >= LDBL_MAX )
+ {
+ if( y > 0.0L )
+ return( INFINITY );
+ return( 0.0L );
+ }
+
+w = floorl(y);
+/* Set iyflg to 1 if y is an integer. */
+iyflg = 0;
+if( w == y )
+ iyflg = 1;
+
+/* Test for odd integer y. */
+yoddint = 0;
+if( iyflg )
+ {
+ ya = fabsl(y);
+ ya = floorl(0.5L * ya);
+ yb = 0.5L * fabsl(w);
+ if( ya != yb )
+ yoddint = 1;
+ }
+
+if( x <= -LDBL_MAX )
+ {
+ if( y > 0.0L )
+ {
+ if( yoddint )
+ return( -INFINITY );
+ return( INFINITY );
+ }
+ if( y < 0.0L )
+ {
+ if( yoddint )
+ return( -0.0L );
+ return( 0.0 );
+ }
+ }
+
+
+nflg = 0; /* flag = 1 if x<0 raised to integer power */
+if( x <= 0.0L )
+ {
+ if( x == 0.0L )
+ {
+ if( y < 0.0 )
+ {
+ if( signbit(x) && yoddint )
+ return( -INFINITY );
+ return( INFINITY );
+ }
+ if( y > 0.0 )
+ {
+ if( signbit(x) && yoddint )
+ return( -0.0L );
+ return( 0.0 );
+ }
+ if( y == 0.0L )
+ return( 1.0L ); /* 0**0 */
+ else
+ return( 0.0L ); /* 0**y */
+ }
+ else
+ {
+ if( iyflg == 0 )
+ return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
+ nflg = 1;
+ }
+ }
+
+/* Integer power of an integer. */
+
+if( iyflg )
+ {
+ i = w;
+ w = floorl(x);
+ if( (w == x) && (fabsl(y) < 32768.0) )
+ {
+ w = powil( x, (int) y );
+ return( w );
+ }
+ }
+
+
+if( nflg )
+ x = fabsl(x);
+
+/* separate significand from exponent */
+x = frexpl( x, &i );
+e = i;
+
+/* find significand in antilog table A[] */
+i = 1;
+if( x <= douba(17) )
+ i = 17;
+if( x <= douba(i+8) )
+ i += 8;
+if( x <= douba(i+4) )
+ i += 4;
+if( x <= douba(i+2) )
+ i += 2;
+if( x >= douba(1) )
+ i = -1;
+i += 1;
+
+
+/* Find (x - A[i])/A[i]
+ * in order to compute log(x/A[i]):
+ *
+ * log(x) = log( a x/a ) = log(a) + log(x/a)
+ *
+ * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
+ */
+x -= douba(i);
+x -= doubb(i/2);
+x /= douba(i);
+
+
+/* rational approximation for log(1+v):
+ *
+ * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
+ */
+z = x*x;
+w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
+w = w - ldexpl( z, -1 ); /* w - 0.5 * z */
+
+/* Convert to base 2 logarithm:
+ * multiply by log2(e) = 1 + LOG2EA
+ */
+z = LOG2EA * w;
+z += w;
+z += LOG2EA * x;
+z += x;
+
+/* Compute exponent term of the base 2 logarithm. */
+w = -i;
+w = ldexpl( w, -LNXT ); /* divide by NXT */
+w += e;
+/* Now base 2 log of x is w + z. */
+
+/* Multiply base 2 log by y, in extended precision. */
+
+/* separate y into large part ya
+ * and small part yb less than 1/NXT
+ */
+ya = reducl(y);
+yb = y - ya;
+
+/* (w+z)(ya+yb)
+ * = w*ya + w*yb + z*y
+ */
+F = z * y + w * yb;
+Fa = reducl(F);
+Fb = F - Fa;
+
+G = Fa + w * ya;
+Ga = reducl(G);
+Gb = G - Ga;
+
+H = Fb + Gb;
+Ha = reducl(H);
+w = ldexpl( Ga+Ha, LNXT );
+
+/* Test the power of 2 for overflow */
+if( w > MEXP )
+ return (huge * huge); /* overflow */
+
+if( w < MNEXP )
+ return (twom10000 * twom10000); /* underflow */
+
+e = w;
+Hb = H - Ha;
+
+if( Hb > 0.0L )
+ {
+ e += 1;
+ Hb -= (1.0L/NXT); /*0.0625L;*/
+ }
+
+/* Now the product y * log2(x) = Hb + e/NXT.
+ *
+ * Compute base 2 exponential of Hb,
+ * where -0.0625 <= Hb <= 0.
+ */
+z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */
+
+/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
+ * Find lookup table entry for the fractional power of 2.
+ */
+if( e < 0 )
+ i = 0;
+else
+ i = 1;
+i = e/NXT + i;
+e = NXT*i - e;
+w = douba( e );
+z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
+z = z + w;
+z = ldexpl( z, i ); /* multiply by integer power of 2 */
+
+if( nflg )
+ {
+/* For negative x,
+ * find out if the integer exponent
+ * is odd or even.
+ */
+ w = ldexpl( y, -1 );
+ w = floorl(w);
+ w = ldexpl( w, 1 );
+ if( w != y )
+ z = -z; /* odd exponent */
+ }
+
+return( z );
+}
+
+
+/* Find a multiple of 1/NXT that is within 1/NXT of x. */
+static long double
*** DIFF OUTPUT TRUNCATED AT 1000 LINES ***
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