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526 lines
12 KiB
C
526 lines
12 KiB
C
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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/* powl.c
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*
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* Power function, long double precision
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, z, powl();
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*
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* z = powl( x, y );
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*
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*
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* DESCRIPTION:
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*
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* Computes x raised to the yth power. Analytically,
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*
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* x**y = exp( y log(x) ).
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*
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* Following Cody and Waite, this program uses a lookup table
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* of 2**-i/32 and pseudo extended precision arithmetic to
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* obtain several extra bits of accuracy in both the logarithm
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* and the exponential.
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*
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*
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* ACCURACY:
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*
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* The relative error of pow(x,y) can be estimated
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* by y dl ln(2), where dl is the absolute error of
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* the internally computed base 2 logarithm. At the ends
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* of the approximation interval the logarithm equal 1/32
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* and its relative error is about 1 lsb = 1.1e-19. Hence
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* the predicted relative error in the result is 2.3e-21 y .
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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*
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* IEEE +-1000 40000 2.8e-18 3.7e-19
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* .001 < x < 1000, with log(x) uniformly distributed.
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* -1000 < y < 1000, y uniformly distributed.
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*
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* IEEE 0,8700 60000 6.5e-18 1.0e-18
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* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
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*
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*
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* ERROR MESSAGES:
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*
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* message condition value returned
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* pow overflow x**y > MAXNUM INFINITY
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* pow underflow x**y < 1/MAXNUM 0.0
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* pow domain x<0 and y noninteger 0.0
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*
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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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long double powl(long double x, long double y)
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{
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return pow(x, y);
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}
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#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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/* Table size */
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#define NXT 32
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/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
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* on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
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*/
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static const long double P[] = {
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8.3319510773868690346226E-4L,
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4.9000050881978028599627E-1L,
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1.7500123722550302671919E0L,
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1.4000100839971580279335E0L,
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};
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static const long double Q[] = {
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/* 1.0000000000000000000000E0L,*/
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5.2500282295834889175431E0L,
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8.4000598057587009834666E0L,
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4.2000302519914740834728E0L,
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};
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/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
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* If i is even, A[i] + B[i/2] gives additional accuracy.
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*/
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static const long double A[33] = {
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1.0000000000000000000000E0L,
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9.7857206208770013448287E-1L,
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9.5760328069857364691013E-1L,
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9.3708381705514995065011E-1L,
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9.1700404320467123175367E-1L,
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8.9735453750155359320742E-1L,
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8.7812608018664974155474E-1L,
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8.5930964906123895780165E-1L,
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8.4089641525371454301892E-1L,
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8.2287773907698242225554E-1L,
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8.0524516597462715409607E-1L,
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7.8799042255394324325455E-1L,
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7.7110541270397041179298E-1L,
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7.5458221379671136985669E-1L,
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7.3841307296974965571198E-1L,
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7.2259040348852331001267E-1L,
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7.0710678118654752438189E-1L,
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6.9195494098191597746178E-1L,
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6.7712777346844636413344E-1L,
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6.6261832157987064729696E-1L,
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6.4841977732550483296079E-1L,
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6.3452547859586661129850E-1L,
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6.2092890603674202431705E-1L,
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6.0762367999023443907803E-1L,
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5.9460355750136053334378E-1L,
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5.8186242938878875689693E-1L,
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5.6939431737834582684856E-1L,
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5.5719337129794626814472E-1L,
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5.4525386633262882960438E-1L,
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5.3357020033841180906486E-1L,
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5.2213689121370692017331E-1L,
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5.1094857432705833910408E-1L,
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5.0000000000000000000000E-1L,
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};
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static const long double B[17] = {
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0.0000000000000000000000E0L,
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2.6176170809902549338711E-20L,
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-1.0126791927256478897086E-20L,
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1.3438228172316276937655E-21L,
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1.2207982955417546912101E-20L,
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-6.3084814358060867200133E-21L,
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1.3164426894366316434230E-20L,
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-1.8527916071632873716786E-20L,
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1.8950325588932570796551E-20L,
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1.5564775779538780478155E-20L,
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6.0859793637556860974380E-21L,
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-2.0208749253662532228949E-20L,
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1.4966292219224761844552E-20L,
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3.3540909728056476875639E-21L,
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-8.6987564101742849540743E-22L,
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-1.2327176863327626135542E-20L,
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0.0000000000000000000000E0L,
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};
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/* 2^x = 1 + x P(x),
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* on the interval -1/32 <= x <= 0
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*/
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static const long double R[] = {
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1.5089970579127659901157E-5L,
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1.5402715328927013076125E-4L,
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1.3333556028915671091390E-3L,
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9.6181291046036762031786E-3L,
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5.5504108664798463044015E-2L,
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2.4022650695910062854352E-1L,
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6.9314718055994530931447E-1L,
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};
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#define MEXP (NXT*16384.0L)
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/* The following if denormal numbers are supported, else -MEXP: */
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#define MNEXP (-NXT*(16384.0L+64.0L))
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/* log2(e) - 1 */
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#define LOG2EA 0.44269504088896340735992L
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#define F W
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#define Fa Wa
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#define Fb Wb
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#define G W
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#define Ga Wa
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#define Gb u
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#define H W
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#define Ha Wb
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#define Hb Wb
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static const long double MAXLOGL = 1.1356523406294143949492E4L;
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static const long double MINLOGL = -1.13994985314888605586758E4L;
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static const long double LOGE2L = 6.9314718055994530941723E-1L;
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static const long double huge = 0x1p10000L;
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/* XXX Prevent gcc from erroneously constant folding this. */
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static const volatile long double twom10000 = 0x1p-10000L;
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static long double reducl(long double);
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static long double powil(long double, int);
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long double __polevll(long double x, const long double *P, int n);
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long double __p1evll(long double x, const long double *P, int n);
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long double powl(long double x, long double y)
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{
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/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
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int i, nflg, iyflg, yoddint;
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long e;
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volatile long double z=0;
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long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
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/* make sure no invalid exception is raised by nan comparison */
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if (isnan(x)) {
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if (!isnan(y) && y == 0.0)
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return 1.0;
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return x;
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}
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if (isnan(y)) {
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if (x == 1.0)
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return 1.0;
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return y;
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}
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if (x == 1.0)
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return 1.0; /* 1**y = 1, even if y is nan */
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if (x == -1.0 && !isfinite(y))
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return 1.0; /* -1**inf = 1 */
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if (y == 0.0)
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return 1.0; /* x**0 = 1, even if x is nan */
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if (y == 1.0)
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return x;
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if (y >= LDBL_MAX) {
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if (x > 1.0 || x < -1.0)
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return INFINITY;
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if (x != 0.0)
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return 0.0;
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}
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if (y <= -LDBL_MAX) {
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if (x > 1.0 || x < -1.0)
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return 0.0;
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if (x != 0.0 || y == -INFINITY)
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return INFINITY;
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}
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if (x >= LDBL_MAX) {
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if (y > 0.0)
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return INFINITY;
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return 0.0;
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}
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w = floorl(y);
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/* Set iyflg to 1 if y is an integer. */
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iyflg = 0;
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if (w == y)
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iyflg = 1;
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/* Test for odd integer y. */
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yoddint = 0;
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if (iyflg) {
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ya = fabsl(y);
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ya = floorl(0.5 * ya);
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yb = 0.5 * fabsl(w);
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if( ya != yb )
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yoddint = 1;
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}
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if (x <= -LDBL_MAX) {
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if (y > 0.0) {
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if (yoddint)
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return -INFINITY;
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return INFINITY;
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}
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if (y < 0.0) {
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if (yoddint)
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return -0.0;
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return 0.0;
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}
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}
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nflg = 0; /* (x<0)**(odd int) */
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if (x <= 0.0) {
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if (x == 0.0) {
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if (y < 0.0) {
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if (signbit(x) && yoddint)
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/* (-0.0)**(-odd int) = -inf, divbyzero */
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return -1.0/0.0;
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/* (+-0.0)**(negative) = inf, divbyzero */
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return 1.0/0.0;
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}
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if (signbit(x) && yoddint)
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return -0.0;
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return 0.0;
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}
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if (iyflg == 0)
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return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
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/* (x<0)**(integer) */
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if (yoddint)
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nflg = 1; /* negate result */
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x = -x;
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}
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/* (+integer)**(integer) */
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if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
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w = powil(x, (int)y);
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return nflg ? -w : w;
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}
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/* separate significand from exponent */
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x = frexpl(x, &i);
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e = i;
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/* find significand in antilog table A[] */
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i = 1;
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if (x <= A[17])
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i = 17;
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if (x <= A[i+8])
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i += 8;
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if (x <= A[i+4])
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i += 4;
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if (x <= A[i+2])
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i += 2;
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if (x >= A[1])
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i = -1;
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i += 1;
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/* Find (x - A[i])/A[i]
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* in order to compute log(x/A[i]):
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*
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* log(x) = log( a x/a ) = log(a) + log(x/a)
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*
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* log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
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*/
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x -= A[i];
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x -= B[i/2];
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x /= A[i];
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/* rational approximation for log(1+v):
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*
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* log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
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*/
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z = x*x;
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w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
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w = w - 0.5*z;
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/* Convert to base 2 logarithm:
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* multiply by log2(e) = 1 + LOG2EA
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*/
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z = LOG2EA * w;
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z += w;
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z += LOG2EA * x;
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z += x;
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/* Compute exponent term of the base 2 logarithm. */
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w = -i;
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w /= NXT;
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w += e;
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/* Now base 2 log of x is w + z. */
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/* Multiply base 2 log by y, in extended precision. */
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/* separate y into large part ya
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* and small part yb less than 1/NXT
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*/
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ya = reducl(y);
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yb = y - ya;
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/* (w+z)(ya+yb)
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* = w*ya + w*yb + z*y
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*/
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F = z * y + w * yb;
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Fa = reducl(F);
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Fb = F - Fa;
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G = Fa + w * ya;
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Ga = reducl(G);
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Gb = G - Ga;
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H = Fb + Gb;
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Ha = reducl(H);
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w = (Ga + Ha) * NXT;
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/* Test the power of 2 for overflow */
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if (w > MEXP)
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return huge * huge; /* overflow */
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if (w < MNEXP)
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return twom10000 * twom10000; /* underflow */
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e = w;
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Hb = H - Ha;
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if (Hb > 0.0) {
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e += 1;
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Hb -= 1.0/NXT; /*0.0625L;*/
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}
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/* Now the product y * log2(x) = Hb + e/NXT.
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*
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* Compute base 2 exponential of Hb,
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* where -0.0625 <= Hb <= 0.
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*/
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z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
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/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
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* Find lookup table entry for the fractional power of 2.
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*/
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if (e < 0)
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i = 0;
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else
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i = 1;
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i = e/NXT + i;
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e = NXT*i - e;
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w = A[e];
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z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
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z = z + w;
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z = scalbnl(z, i); /* multiply by integer power of 2 */
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if (nflg)
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z = -z;
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return z;
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}
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/* Find a multiple of 1/NXT that is within 1/NXT of x. */
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static long double reducl(long double x)
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{
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long double t;
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t = x * NXT;
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t = floorl(t);
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t = t / NXT;
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return t;
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}
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/*
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* Positive real raised to integer power, long double precision
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, powil();
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* int n;
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*
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* y = powil( x, n );
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*
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*
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* DESCRIPTION:
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*
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* Returns argument x>0 raised to the nth power.
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* The routine efficiently decomposes n as a sum of powers of
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* two. The desired power is a product of two-to-the-kth
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* powers of x. Thus to compute the 32767 power of x requires
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* 28 multiplications instead of 32767 multiplications.
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic x domain n domain # trials peak rms
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* IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
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* IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
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* IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
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*
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* Returns MAXNUM on overflow, zero on underflow.
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*/
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static long double powil(long double x, int nn)
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{
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long double ww, y;
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long double s;
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int n, e, sign, lx;
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if (nn == 0)
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return 1.0;
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if (nn < 0) {
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sign = -1;
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n = -nn;
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} else {
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sign = 1;
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n = nn;
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}
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/* Overflow detection */
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/* Calculate approximate logarithm of answer */
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s = x;
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s = frexpl( s, &lx);
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e = (lx - 1)*n;
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if ((e == 0) || (e > 64) || (e < -64)) {
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s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
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s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
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} else {
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s = LOGE2L * e;
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}
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if (s > MAXLOGL)
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return huge * huge; /* overflow */
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if (s < MINLOGL)
|
|
return twom10000 * twom10000; /* underflow */
|
|
/* Handle tiny denormal answer, but with less accuracy
|
|
* since roundoff error in 1.0/x will be amplified.
|
|
* The precise demarcation should be the gradual underflow threshold.
|
|
*/
|
|
if (s < -MAXLOGL+2.0) {
|
|
x = 1.0/x;
|
|
sign = -sign;
|
|
}
|
|
|
|
/* First bit of the power */
|
|
if (n & 1)
|
|
y = x;
|
|
else
|
|
y = 1.0;
|
|
|
|
ww = x;
|
|
n >>= 1;
|
|
while (n) {
|
|
ww = ww * ww; /* arg to the 2-to-the-kth power */
|
|
if (n & 1) /* if that bit is set, then include in product */
|
|
y *= ww;
|
|
n >>= 1;
|
|
}
|
|
|
|
if (sign < 0)
|
|
y = 1.0/y;
|
|
return y;
|
|
}
|
|
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
|
// TODO: broken implementation to make things compile
|
|
long double powl(long double x, long double y)
|
|
{
|
|
return pow(x, y);
|
|
}
|
|
#endif
|