767 lines
17 KiB
C
767 lines
17 KiB
C
/* 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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/*
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Cephes Math Library Release 2.7: May, 1998
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Copyright 1984, 1991, 1998 by Stephen L. Moshier
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*/
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/*
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Modified for mingw
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2002-07-22 Danny Smith <dannysmith@users.sourceforge.net>
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*/
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#ifdef __MINGW32__
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#include "cephes_mconf.h"
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#else
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#include "mconf.h"
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static char fname[] = {"powl"};
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#endif
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#ifndef _SET_ERRNO
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#define _SET_ERRNO(x)
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#endif
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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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#ifdef UNK
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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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#ifdef DENORMAL
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#define MNEXP (-NXT*(16384.0L+64.0L))
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#else
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#define MNEXP (-NXT*16384.0L)
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#endif
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/* log2(e) - 1 */
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#define LOG2EA 0.44269504088896340735992L
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#endif
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#ifdef IBMPC
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static const unsigned short P[] = {
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0xb804,0xa8b7,0xc6f4,0xda6a,0x3ff4, XPD
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0x7de9,0xcf02,0x58c0,0xfae1,0x3ffd, XPD
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0x405a,0x3722,0x67c9,0xe000,0x3fff, XPD
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0xcd99,0x6b43,0x87ca,0xb333,0x3fff, XPD
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};
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static const unsigned short Q[] = {
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/* 0x0000,0x0000,0x0000,0x8000,0x3fff, */
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0x6307,0xa469,0x3b33,0xa800,0x4001, XPD
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0xfec2,0x62d7,0xa51c,0x8666,0x4002, XPD
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0xda32,0xd072,0xa5d7,0x8666,0x4001, XPD
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};
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static const unsigned short A[] = {
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0x0000,0x0000,0x0000,0x8000,0x3fff, XPD
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0x033a,0x722a,0xb2db,0xfa83,0x3ffe, XPD
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0xcc2c,0x2486,0x7d15,0xf525,0x3ffe, XPD
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0xf5cb,0xdcda,0xb99b,0xefe4,0x3ffe, XPD
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0x392f,0xdd24,0xc6e7,0xeac0,0x3ffe, XPD
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0x48a8,0x7c83,0x06e7,0xe5b9,0x3ffe, XPD
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0xe111,0x2a94,0xdeec,0xe0cc,0x3ffe, XPD
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0x3755,0xdaf2,0xb797,0xdbfb,0x3ffe, XPD
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0x6af4,0xd69d,0xfcca,0xd744,0x3ffe, XPD
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0xe45a,0xf12a,0x1d91,0xd2a8,0x3ffe, XPD
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0x80e4,0x1f84,0x8c15,0xce24,0x3ffe, XPD
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0x27a3,0x6e2f,0xbd86,0xc9b9,0x3ffe, XPD
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0xdadd,0x5506,0x2a11,0xc567,0x3ffe, XPD
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0x9456,0x6670,0x4cca,0xc12c,0x3ffe, XPD
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0x36bf,0x580c,0xa39f,0xbd08,0x3ffe, XPD
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0x9ee9,0x62fb,0xaf47,0xb8fb,0x3ffe, XPD
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0x6484,0xf9de,0xf333,0xb504,0x3ffe, XPD
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0x2590,0xd2ac,0xf581,0xb123,0x3ffe, XPD
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0x4ac6,0x42a1,0x3eea,0xad58,0x3ffe, XPD
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0x0ef8,0xea7c,0x5ab4,0xa9a1,0x3ffe, XPD
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0x38ea,0xb151,0xd6a9,0xa5fe,0x3ffe, XPD
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0x6819,0x0c49,0x4303,0xa270,0x3ffe, XPD
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0x11ae,0x91a1,0x3260,0x9ef5,0x3ffe, XPD
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0x5539,0xd54e,0x39b9,0x9b8d,0x3ffe, XPD
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0xa96f,0x8db8,0xf051,0x9837,0x3ffe, XPD
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0x0961,0xfef7,0xefa8,0x94f4,0x3ffe, XPD
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0xc336,0xab11,0xd373,0x91c3,0x3ffe, XPD
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0x53c0,0x45cd,0x398b,0x8ea4,0x3ffe, XPD
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0xd6e7,0xea8b,0xc1e3,0x8b95,0x3ffe, XPD
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0x8527,0x92da,0x0e80,0x8898,0x3ffe, XPD
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0x7b15,0xcc48,0xc367,0x85aa,0x3ffe, XPD
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0xa1d7,0xac2b,0x8698,0x82cd,0x3ffe, XPD
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0x0000,0x0000,0x0000,0x8000,0x3ffe, XPD
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};
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static const unsigned short B[] = {
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0x0000,0x0000,0x0000,0x0000,0x0000, XPD
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0x1f87,0xdb30,0x18f5,0xf73a,0x3fbd, XPD
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0xac15,0x3e46,0x2932,0xbf4a,0xbfbc, XPD
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0x7944,0xba66,0xa091,0xcb12,0x3fb9, XPD
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0xff78,0x40b4,0x2ee6,0xe69a,0x3fbc, XPD
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0xc895,0x5069,0xe383,0xee53,0xbfbb, XPD
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0x7cde,0x9376,0x4325,0xf8ab,0x3fbc, XPD
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0xa10c,0x25e0,0xc093,0xaefd,0xbfbd, XPD
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0x7d3e,0xea95,0x1366,0xb2fb,0x3fbd, XPD
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0x5d89,0xeb34,0x5191,0x9301,0x3fbd, XPD
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0x80d9,0xb883,0xfb10,0xe5eb,0x3fbb, XPD
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0x045d,0x288c,0xc1ec,0xbedd,0xbfbd, XPD
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0xeded,0x5c85,0x4630,0x8d5a,0x3fbd, XPD
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0x9d82,0xe5ac,0x8e0a,0xfd6d,0x3fba, XPD
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0x6dfd,0xeb58,0xaf14,0x8373,0xbfb9, XPD
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0xf938,0x7aac,0x91cf,0xe8da,0xbfbc, XPD
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0x0000,0x0000,0x0000,0x0000,0x0000, XPD
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};
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static const unsigned short R[] = {
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0xa69b,0x530e,0xee1d,0xfd2a,0x3fee, XPD
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0xc746,0x8e7e,0x5960,0xa182,0x3ff2, XPD
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0x63b6,0xadda,0xfd6a,0xaec3,0x3ff5, XPD
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0xc104,0xfd99,0x5b7c,0x9d95,0x3ff8, XPD
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0xe05e,0x249d,0x46b8,0xe358,0x3ffa, XPD
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0x5d1d,0x162c,0xeffc,0xf5fd,0x3ffc, XPD
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0x79aa,0xd1cf,0x17f7,0xb172,0x3ffe, XPD
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};
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/* 10 byte sizes versus 12 byte */
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#define douba(k) (*(long double *)(&A[(sizeof( long double )/2)*(k)]))
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#define doubb(k) (*(long double *)(&B[(sizeof( long double )/2)*(k)]))
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#define MEXP (NXT*16384.0L)
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#ifdef DENORMAL
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#define MNEXP (-NXT*(16384.0L+64.0L))
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#else
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#define MNEXP (-NXT*16384.0L)
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#endif
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static const unsigned short L[] = {0xc2ef,0x705f,0xeca5,0xe2a8,0x3ffd, XPD};
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#define LOG2EA (*(long double *)(&L[0]))
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#endif
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#ifdef MIEEE
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static long P[] = {
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0x3ff40000,0xda6ac6f4,0xa8b7b804,
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0x3ffd0000,0xfae158c0,0xcf027de9,
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0x3fff0000,0xe00067c9,0x3722405a,
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0x3fff0000,0xb33387ca,0x6b43cd99,
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};
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static long Q[] = {
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/* 0x3fff0000,0x80000000,0x00000000, */
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0x40010000,0xa8003b33,0xa4696307,
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0x40020000,0x8666a51c,0x62d7fec2,
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0x40010000,0x8666a5d7,0xd072da32,
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};
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static long A[] = {
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0x3fff0000,0x80000000,0x00000000,
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0x3ffe0000,0xfa83b2db,0x722a033a,
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0x3ffe0000,0xf5257d15,0x2486cc2c,
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0x3ffe0000,0xefe4b99b,0xdcdaf5cb,
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0x3ffe0000,0xeac0c6e7,0xdd24392f,
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0x3ffe0000,0xe5b906e7,0x7c8348a8,
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0x3ffe0000,0xe0ccdeec,0x2a94e111,
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0x3ffe0000,0xdbfbb797,0xdaf23755,
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0x3ffe0000,0xd744fcca,0xd69d6af4,
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0x3ffe0000,0xd2a81d91,0xf12ae45a,
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0x3ffe0000,0xce248c15,0x1f8480e4,
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0x3ffe0000,0xc9b9bd86,0x6e2f27a3,
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0x3ffe0000,0xc5672a11,0x5506dadd,
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0x3ffe0000,0xc12c4cca,0x66709456,
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0x3ffe0000,0xbd08a39f,0x580c36bf,
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0x3ffe0000,0xb8fbaf47,0x62fb9ee9,
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0x3ffe0000,0xb504f333,0xf9de6484,
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0x3ffe0000,0xb123f581,0xd2ac2590,
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0x3ffe0000,0xad583eea,0x42a14ac6,
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0x3ffe0000,0xa9a15ab4,0xea7c0ef8,
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0x3ffe0000,0xa5fed6a9,0xb15138ea,
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0x3ffe0000,0xa2704303,0x0c496819,
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0x3ffe0000,0x9ef53260,0x91a111ae,
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0x3ffe0000,0x9b8d39b9,0xd54e5539,
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0x3ffe0000,0x9837f051,0x8db8a96f,
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0x3ffe0000,0x94f4efa8,0xfef70961,
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0x3ffe0000,0x91c3d373,0xab11c336,
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0x3ffe0000,0x8ea4398b,0x45cd53c0,
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0x3ffe0000,0x8b95c1e3,0xea8bd6e7,
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0x3ffe0000,0x88980e80,0x92da8527,
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0x3ffe0000,0x85aac367,0xcc487b15,
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0x3ffe0000,0x82cd8698,0xac2ba1d7,
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0x3ffe0000,0x80000000,0x00000000,
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};
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static long B[51] = {
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0x00000000,0x00000000,0x00000000,
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0x3fbd0000,0xf73a18f5,0xdb301f87,
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0xbfbc0000,0xbf4a2932,0x3e46ac15,
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0x3fb90000,0xcb12a091,0xba667944,
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0x3fbc0000,0xe69a2ee6,0x40b4ff78,
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0xbfbb0000,0xee53e383,0x5069c895,
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0x3fbc0000,0xf8ab4325,0x93767cde,
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0xbfbd0000,0xaefdc093,0x25e0a10c,
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0x3fbd0000,0xb2fb1366,0xea957d3e,
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0x3fbd0000,0x93015191,0xeb345d89,
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0x3fbb0000,0xe5ebfb10,0xb88380d9,
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0xbfbd0000,0xbeddc1ec,0x288c045d,
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0x3fbd0000,0x8d5a4630,0x5c85eded,
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0x3fba0000,0xfd6d8e0a,0xe5ac9d82,
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0xbfb90000,0x8373af14,0xeb586dfd,
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0xbfbc0000,0xe8da91cf,0x7aacf938,
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0x00000000,0x00000000,0x00000000,
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};
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static long R[] = {
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0x3fee0000,0xfd2aee1d,0x530ea69b,
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0x3ff20000,0xa1825960,0x8e7ec746,
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0x3ff50000,0xaec3fd6a,0xadda63b6,
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0x3ff80000,0x9d955b7c,0xfd99c104,
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0x3ffa0000,0xe35846b8,0x249de05e,
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0x3ffc0000,0xf5fdeffc,0x162c5d1d,
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0x3ffe0000,0xb17217f7,0xd1cf79aa,
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};
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#define douba(k) (*(long double *)&A[3*(k)])
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#define doubb(k) (*(long double *)&B[3*(k)])
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#define MEXP (NXT*16384.0L)
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#ifdef DENORMAL
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#define MNEXP (-NXT*(16384.0L+64.0L))
|
||
#else
|
||
#define MNEXP (-NXT*16382.0L)
|
||
#endif
|
||
static long L[3] = {0x3ffd0000,0xe2a8eca5,0x705fc2ef};
|
||
#define LOG2EA (*(long double *)(&L[0]))
|
||
#endif
|
||
|
||
|
||
#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
|
||
|
||
#ifndef __MINGW32__
|
||
extern long double MAXNUML;
|
||
#endif
|
||
|
||
static VOLATILE long double z;
|
||
static long double w, W, Wa, Wb, ya, yb, u;
|
||
|
||
#ifdef __MINGW32__
|
||
static __inline__ long double reducl( long double );
|
||
extern long double __powil ( long double, int );
|
||
extern long double powl ( long double x, long double y);
|
||
#else
|
||
#ifdef ANSIPROT
|
||
extern long double floorl ( long double );
|
||
extern long double fabsl ( long double );
|
||
extern long double frexpl ( long double, int * );
|
||
extern long double ldexpl ( long double, int );
|
||
extern long double polevll ( long double, void *, int );
|
||
extern long double p1evll ( long double, void *, int );
|
||
extern long double __powil ( long double, int );
|
||
extern int isnanl ( long double );
|
||
extern int isfinitel ( long double );
|
||
static long double reducl( long double );
|
||
extern int signbitl ( long double );
|
||
#else
|
||
long double floorl(), fabsl(), frexpl(), ldexpl();
|
||
long double polevll(), p1evll(), __powil();
|
||
static long double reducl();
|
||
int isnanl(), isfinitel(), signbitl();
|
||
#endif
|
||
|
||
#ifdef INFINITIES
|
||
extern long double INFINITYL;
|
||
#else
|
||
#define INFINITYL MAXNUML
|
||
#endif
|
||
|
||
#ifdef NANS
|
||
extern long double NANL;
|
||
#endif
|
||
#ifdef MINUSZERO
|
||
extern long double NEGZEROL;
|
||
#endif
|
||
|
||
#endif /* __MINGW32__ */
|
||
|
||
long double powl( x, y )
|
||
long double x, 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 );
|
||
|
||
#ifdef NANS
|
||
if( isnanl(x) )
|
||
{
|
||
_SET_ERRNO (EDOM);
|
||
return( x );
|
||
}
|
||
if( isnanl(y) )
|
||
{
|
||
_SET_ERRNO (EDOM);
|
||
return( y );
|
||
}
|
||
#endif
|
||
|
||
if( y == 1.0L )
|
||
return( x );
|
||
|
||
if( isinfl(y) && (x == -1.0L || x == 1.0L) )
|
||
return( y );
|
||
|
||
if( x == 1.0L )
|
||
return( 1.0L );
|
||
|
||
if( y >= MAXNUML )
|
||
{
|
||
_SET_ERRNO (ERANGE);
|
||
#ifdef INFINITIES
|
||
if( x > 1.0L )
|
||
return( INFINITYL );
|
||
#else
|
||
if( x > 1.0L )
|
||
return( MAXNUML );
|
||
#endif
|
||
if( x > 0.0L && x < 1.0L )
|
||
return( 0.0L );
|
||
#ifdef INFINITIES
|
||
if( x < -1.0L )
|
||
return( INFINITYL );
|
||
#else
|
||
if( x < -1.0L )
|
||
return( MAXNUML );
|
||
#endif
|
||
if( x > -1.0L && x < 0.0L )
|
||
return( 0.0L );
|
||
}
|
||
if( y <= -MAXNUML )
|
||
{
|
||
_SET_ERRNO (ERANGE);
|
||
if( x > 1.0L )
|
||
return( 0.0L );
|
||
#ifdef INFINITIES
|
||
if( x > 0.0L && x < 1.0L )
|
||
return( INFINITYL );
|
||
#else
|
||
if( x > 0.0L && x < 1.0L )
|
||
return( MAXNUML );
|
||
#endif
|
||
if( x < -1.0L )
|
||
return( 0.0L );
|
||
#ifdef INFINITIES
|
||
if( x > -1.0L && x < 0.0L )
|
||
return( INFINITYL );
|
||
#else
|
||
if( x > -1.0L && x < 0.0L )
|
||
return( MAXNUML );
|
||
#endif
|
||
}
|
||
if( x >= MAXNUML )
|
||
{
|
||
#if INFINITIES
|
||
if( y > 0.0L )
|
||
return( INFINITYL );
|
||
#else
|
||
if( y > 0.0L )
|
||
return( MAXNUML );
|
||
#endif
|
||
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 <= -MAXNUML )
|
||
{
|
||
if( y > 0.0L )
|
||
{
|
||
#ifdef INFINITIES
|
||
if( yoddint )
|
||
return( -INFINITYL );
|
||
return( INFINITYL );
|
||
#else
|
||
if( yoddint )
|
||
return( -MAXNUML );
|
||
return( MAXNUML );
|
||
#endif
|
||
}
|
||
if( y < 0.0L )
|
||
{
|
||
#ifdef MINUSZERO
|
||
if( yoddint )
|
||
return( NEGZEROL );
|
||
#endif
|
||
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 )
|
||
{
|
||
#ifdef MINUSZERO
|
||
if( signbitl(x) && yoddint )
|
||
return( -INFINITYL );
|
||
#endif
|
||
#ifdef INFINITIES
|
||
return( INFINITYL );
|
||
#else
|
||
return( MAXNUML );
|
||
#endif
|
||
}
|
||
if( y > 0.0 )
|
||
{
|
||
#ifdef MINUSZERO
|
||
if( signbitl(x) && yoddint )
|
||
return( NEGZEROL );
|
||
#endif
|
||
return( 0.0 );
|
||
}
|
||
if( y == 0.0L )
|
||
return( 1.0L ); /* 0**0 */
|
||
else
|
||
return( 0.0L ); /* 0**y */
|
||
}
|
||
else
|
||
{
|
||
if( iyflg == 0 )
|
||
{ /* noninteger power of negative number */
|
||
mtherr( fname, DOMAIN );
|
||
_SET_ERRNO (EDOM);
|
||
#ifdef NANS
|
||
return(NANL);
|
||
#else
|
||
return(0.0L);
|
||
#endif
|
||
}
|
||
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 )
|
||
{
|
||
/* printf( "w = %.4Le ", w ); */
|
||
_SET_ERRNO (ERANGE);
|
||
mtherr( fname, OVERFLOW );
|
||
return( MAXNUML );
|
||
}
|
||
|
||
if( w < MNEXP )
|
||
{
|
||
/* printf( "w = %.4Le ", w ); */
|
||
_SET_ERRNO (ERANGE);
|
||
mtherr( fname, UNDERFLOW );
|
||
return( 0.0L );
|
||
}
|
||
|
||
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 __inline__ long double reducl(x)
|
||
long double x;
|
||
{
|
||
long double t;
|
||
|
||
t = ldexpl( x, LNXT );
|
||
t = floorl( t );
|
||
t = ldexpl( t, -LNXT );
|
||
return(t);
|
||
}
|
||
|