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615 lines
12 KiB
C
615 lines
12 KiB
C
/* $OpenBSD: e_powl.c,v 1.5 2013/11/12 20:35:19 martynas Exp $ */
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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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*
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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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*
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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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*
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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 <float.h>
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#include <openlibm.h>
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#include "math_private.h"
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/* Table size */
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#define NXT 32
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/* log2(Table size) */
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#define LNXT 5
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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 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 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 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 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 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 douba(k) A[k]
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#define doubb(k) B[k]
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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 volatile long double z;
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static long double w, W, Wa, Wb, ya, yb, u;
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static const long double huge = 0x1p10000L;
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#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
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static const long double twom10000 = 0x1p-10000L;
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#else
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static volatile long double twom10000 = 0x1p-10000L;
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#endif
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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
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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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if( y == 0.0L )
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return( 1.0L );
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if( x == 1.0L )
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return( 1.0L );
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if( isnan(x) )
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return( x );
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if( isnan(y) )
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return( y );
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if( y == 1.0L )
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return( x );
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if( !isfinite(y) && x == -1.0L )
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return( 1.0L );
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if( y >= LDBL_MAX )
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{
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if( x > 1.0L )
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return( INFINITY );
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if( x > 0.0L && x < 1.0L )
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return( 0.0L );
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if( x < -1.0L )
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return( INFINITY );
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if( x > -1.0L && x < 0.0L )
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return( 0.0L );
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}
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if( y <= -LDBL_MAX )
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{
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if( x > 1.0L )
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return( 0.0L );
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if( x > 0.0L && x < 1.0L )
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return( INFINITY );
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if( x < -1.0L )
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return( 0.0L );
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if( x > -1.0L && x < 0.0L )
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return( INFINITY );
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}
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if( x >= LDBL_MAX )
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{
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if( y > 0.0L )
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return( INFINITY );
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return( 0.0L );
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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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{
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ya = fabsl(y);
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ya = floorl(0.5L * ya);
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yb = 0.5L * 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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{
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if( y > 0.0L )
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{
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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 < 0.0L )
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{
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if( yoddint )
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return( -0.0L );
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return( 0.0 );
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}
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}
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nflg = 0; /* flag = 1 if x<0 raised to integer power */
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if( x <= 0.0L )
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{
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if( x == 0.0L )
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{
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if( y < 0.0 )
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{
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if( signbit(x) && 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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{
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if( signbit(x) && yoddint )
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return( -0.0L );
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return( 0.0 );
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}
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if( y == 0.0L )
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return( 1.0L ); /* 0**0 */
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else
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return( 0.0L ); /* 0**y */
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}
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else
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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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nflg = 1;
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}
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}
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/* Integer power of an integer. */
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if( iyflg )
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{
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i = w;
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w = floorl(x);
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if( (w == x) && (fabsl(y) < 32768.0) )
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{
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w = powil( x, (int) y );
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return( w );
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}
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}
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if( nflg )
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x = fabsl(x);
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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 <= douba(17) )
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i = 17;
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if( x <= douba(i+8) )
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i += 8;
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if( x <= douba(i+4) )
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i += 4;
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if( x <= douba(i+2) )
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i += 2;
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if( x >= douba(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 -= douba(i);
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x -= doubb(i/2);
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x /= douba(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 - ldexpl( z, -1 ); /* 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 = ldexpl( w, -LNXT ); /* divide by 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 = ldexpl( Ga+Ha, LNXT );
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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 > 0.0L )
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{
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e += 1;
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Hb -= (1.0L/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 = douba( 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 = ldexpl( z, i ); /* multiply by integer power of 2 */
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if( nflg )
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{
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/* For negative x,
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* find out if the integer exponent
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* is odd or even.
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*/
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w = ldexpl( y, -1 );
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w = floorl(w);
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w = ldexpl( w, 1 );
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if( w != y )
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z = -z; /* odd exponent */
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}
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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
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reducl(long double x)
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{
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long double t;
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t = ldexpl( x, LNXT );
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t = floorl( t );
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t = ldexpl( t, -LNXT );
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return(t);
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}
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/* powil.c
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*
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* Real raised to integer power, long double precision
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*
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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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*
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* DESCRIPTION:
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*
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* Returns argument x 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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*
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* ACCURACY:
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*
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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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*/
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static long double
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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, asign, lx;
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if( x == 0.0L )
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{
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if( nn == 0 )
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return( 1.0L );
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else if( nn < 0 )
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return( LDBL_MAX );
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else
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return( 0.0L );
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}
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if( nn == 0 )
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return( 1.0L );
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if( x < 0.0L )
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{
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asign = -1;
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x = -x;
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}
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else
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asign = 0;
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if( nn < 0 )
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{
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sign = -1;
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n = -nn;
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}
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else
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{
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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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{
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s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
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s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
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}
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else
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{
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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 )
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return (twom10000 * twom10000); /* underflow */
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/* Handle tiny denormal answer, but with less accuracy
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* since roundoff error in 1.0/x will be amplified.
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* The precise demarcation should be the gradual underflow threshold.
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*/
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if( s < (-MAXLOGL+2.0L) )
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{
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x = 1.0L/x;
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sign = -sign;
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}
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/* First bit of the power */
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if( n & 1 )
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y = x;
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else
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{
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y = 1.0L;
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asign = 0;
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}
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ww = x;
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n >>= 1;
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while( n )
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{
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ww = ww * ww; /* arg to the 2-to-the-kth power */
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if( n & 1 ) /* if that bit is set, then include in product */
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y *= ww;
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n >>= 1;
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}
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if( asign )
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y = -y; /* odd power of negative number */
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if( sign < 0 )
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y = 1.0L/y;
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return(y);
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}
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