[398] | 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
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| 2 | /*
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| 3 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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| 4 | *
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| 5 | * Permission to use, copy, modify, and distribute this software for any
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| 6 | * purpose with or without fee is hereby granted, provided that the above
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| 7 | * copyright notice and this permission notice appear in all copies.
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| 8 | *
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| 9 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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| 10 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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| 11 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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| 12 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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| 13 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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| 14 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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| 15 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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| 16 | */
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| 17 | /* powl.c
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| 18 | *
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| 19 | * Power function, long double precision
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| 20 | *
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| 21 | *
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| 22 | * SYNOPSIS:
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| 23 | *
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| 24 | * long double x, y, z, powl();
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| 25 | *
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| 26 | * z = powl( x, y );
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| 27 | *
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| 28 | *
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| 29 | * DESCRIPTION:
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| 30 | *
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| 31 | * Computes x raised to the yth power. Analytically,
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| 32 | *
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| 33 | * x**y = exp( y log(x) ).
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| 34 | *
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| 35 | * Following Cody and Waite, this program uses a lookup table
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| 36 | * of 2**-i/32 and pseudo extended precision arithmetic to
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| 37 | * obtain several extra bits of accuracy in both the logarithm
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| 38 | * and the exponential.
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| 39 | *
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| 40 | *
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| 41 | * ACCURACY:
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| 42 | *
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| 43 | * The relative error of pow(x,y) can be estimated
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| 44 | * by y dl ln(2), where dl is the absolute error of
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| 45 | * the internally computed base 2 logarithm. At the ends
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| 46 | * of the approximation interval the logarithm equal 1/32
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| 47 | * and its relative error is about 1 lsb = 1.1e-19. Hence
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| 48 | * the predicted relative error in the result is 2.3e-21 y .
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| 49 | *
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| 50 | * Relative error:
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| 51 | * arithmetic domain # trials peak rms
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| 52 | *
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| 53 | * IEEE +-1000 40000 2.8e-18 3.7e-19
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| 54 | * .001 < x < 1000, with log(x) uniformly distributed.
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| 55 | * -1000 < y < 1000, y uniformly distributed.
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| 56 | *
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| 57 | * IEEE 0,8700 60000 6.5e-18 1.0e-18
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| 58 | * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
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| 59 | *
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| 60 | *
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| 61 | * ERROR MESSAGES:
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| 62 | *
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| 63 | * message condition value returned
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| 64 | * pow overflow x**y > MAXNUM INFINITY
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| 65 | * pow underflow x**y < 1/MAXNUM 0.0
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| 66 | * pow domain x<0 and y noninteger 0.0
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| 67 | *
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| 68 | */
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| 69 |
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| 70 | #include "libm.h"
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| 71 |
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| 72 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 73 | long double powl(long double x, long double y)
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| 74 | {
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| 75 | return pow(x, y);
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| 76 | }
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| 77 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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| 78 |
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| 79 | /* Table size */
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| 80 | #define NXT 32
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| 81 |
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| 82 | /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
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| 83 | * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
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| 84 | */
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| 85 | static const long double P[] = {
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| 86 | 8.3319510773868690346226E-4L,
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| 87 | 4.9000050881978028599627E-1L,
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| 88 | 1.7500123722550302671919E0L,
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| 89 | 1.4000100839971580279335E0L,
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| 90 | };
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| 91 | static const long double Q[] = {
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| 92 | /* 1.0000000000000000000000E0L,*/
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| 93 | 5.2500282295834889175431E0L,
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| 94 | 8.4000598057587009834666E0L,
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| 95 | 4.2000302519914740834728E0L,
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| 96 | };
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| 97 | /* A[i] = 2^(-i/32), rounded to IEEE long double precision.
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| 98 | * If i is even, A[i] + B[i/2] gives additional accuracy.
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| 99 | */
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| 100 | static const long double A[33] = {
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| 101 | 1.0000000000000000000000E0L,
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| 102 | 9.7857206208770013448287E-1L,
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| 103 | 9.5760328069857364691013E-1L,
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| 104 | 9.3708381705514995065011E-1L,
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| 105 | 9.1700404320467123175367E-1L,
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| 106 | 8.9735453750155359320742E-1L,
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| 107 | 8.7812608018664974155474E-1L,
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| 108 | 8.5930964906123895780165E-1L,
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| 109 | 8.4089641525371454301892E-1L,
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| 110 | 8.2287773907698242225554E-1L,
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| 111 | 8.0524516597462715409607E-1L,
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| 112 | 7.8799042255394324325455E-1L,
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| 113 | 7.7110541270397041179298E-1L,
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| 114 | 7.5458221379671136985669E-1L,
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| 115 | 7.3841307296974965571198E-1L,
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| 116 | 7.2259040348852331001267E-1L,
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| 117 | 7.0710678118654752438189E-1L,
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| 118 | 6.9195494098191597746178E-1L,
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| 119 | 6.7712777346844636413344E-1L,
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| 120 | 6.6261832157987064729696E-1L,
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| 121 | 6.4841977732550483296079E-1L,
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| 122 | 6.3452547859586661129850E-1L,
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| 123 | 6.2092890603674202431705E-1L,
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| 124 | 6.0762367999023443907803E-1L,
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| 125 | 5.9460355750136053334378E-1L,
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| 126 | 5.8186242938878875689693E-1L,
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| 127 | 5.6939431737834582684856E-1L,
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| 128 | 5.5719337129794626814472E-1L,
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| 129 | 5.4525386633262882960438E-1L,
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| 130 | 5.3357020033841180906486E-1L,
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| 131 | 5.2213689121370692017331E-1L,
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| 132 | 5.1094857432705833910408E-1L,
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| 133 | 5.0000000000000000000000E-1L,
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| 134 | };
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| 135 | static const long double B[17] = {
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| 136 | 0.0000000000000000000000E0L,
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| 137 | 2.6176170809902549338711E-20L,
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| 138 | -1.0126791927256478897086E-20L,
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| 139 | 1.3438228172316276937655E-21L,
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| 140 | 1.2207982955417546912101E-20L,
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| 141 | -6.3084814358060867200133E-21L,
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| 142 | 1.3164426894366316434230E-20L,
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| 143 | -1.8527916071632873716786E-20L,
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| 144 | 1.8950325588932570796551E-20L,
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| 145 | 1.5564775779538780478155E-20L,
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| 146 | 6.0859793637556860974380E-21L,
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| 147 | -2.0208749253662532228949E-20L,
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| 148 | 1.4966292219224761844552E-20L,
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| 149 | 3.3540909728056476875639E-21L,
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| 150 | -8.6987564101742849540743E-22L,
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| 151 | -1.2327176863327626135542E-20L,
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| 152 | 0.0000000000000000000000E0L,
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| 153 | };
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| 154 |
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| 155 | /* 2^x = 1 + x P(x),
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| 156 | * on the interval -1/32 <= x <= 0
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| 157 | */
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| 158 | static const long double R[] = {
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| 159 | 1.5089970579127659901157E-5L,
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| 160 | 1.5402715328927013076125E-4L,
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| 161 | 1.3333556028915671091390E-3L,
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| 162 | 9.6181291046036762031786E-3L,
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| 163 | 5.5504108664798463044015E-2L,
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| 164 | 2.4022650695910062854352E-1L,
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| 165 | 6.9314718055994530931447E-1L,
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| 166 | };
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| 167 |
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| 168 | #define MEXP (NXT*16384.0L)
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| 169 | /* The following if denormal numbers are supported, else -MEXP: */
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| 170 | #define MNEXP (-NXT*(16384.0L+64.0L))
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| 171 | /* log2(e) - 1 */
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| 172 | #define LOG2EA 0.44269504088896340735992L
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| 173 |
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| 174 | #define F W
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| 175 | #define Fa Wa
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| 176 | #define Fb Wb
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| 177 | #define G W
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| 178 | #define Ga Wa
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| 179 | #define Gb u
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| 180 | #define H W
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| 181 | #define Ha Wb
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| 182 | #define Hb Wb
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| 183 |
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| 184 | static const long double MAXLOGL = 1.1356523406294143949492E4L;
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| 185 | static const long double MINLOGL = -1.13994985314888605586758E4L;
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| 186 | static const long double LOGE2L = 6.9314718055994530941723E-1L;
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| 187 | static const long double huge = 0x1p10000L;
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| 188 | /* XXX Prevent gcc from erroneously constant folding this. */
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| 189 | static const volatile long double twom10000 = 0x1p-10000L;
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| 190 |
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| 191 | static long double reducl(long double);
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| 192 | static long double powil(long double, int);
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| 193 |
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| 194 | long double powl(long double x, long double y)
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| 195 | {
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| 196 | /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
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| 197 | int i, nflg, iyflg, yoddint;
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| 198 | long e;
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| 199 | volatile long double z=0;
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| 200 | long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
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| 201 |
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| 202 | /* make sure no invalid exception is raised by nan comparision */
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| 203 | if (isnan(x)) {
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| 204 | if (!isnan(y) && y == 0.0)
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| 205 | return 1.0;
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| 206 | return x;
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| 207 | }
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| 208 | if (isnan(y)) {
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| 209 | if (x == 1.0)
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| 210 | return 1.0;
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| 211 | return y;
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| 212 | }
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| 213 | if (x == 1.0)
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| 214 | return 1.0; /* 1**y = 1, even if y is nan */
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| 215 | if (x == -1.0 && !isfinite(y))
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| 216 | return 1.0; /* -1**inf = 1 */
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| 217 | if (y == 0.0)
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| 218 | return 1.0; /* x**0 = 1, even if x is nan */
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| 219 | if (y == 1.0)
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| 220 | return x;
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| 221 | if (y >= LDBL_MAX) {
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| 222 | if (x > 1.0 || x < -1.0)
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| 223 | return INFINITY;
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| 224 | if (x != 0.0)
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| 225 | return 0.0;
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| 226 | }
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| 227 | if (y <= -LDBL_MAX) {
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| 228 | if (x > 1.0 || x < -1.0)
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| 229 | return 0.0;
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| 230 | if (x != 0.0 || y == -INFINITY)
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| 231 | return INFINITY;
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| 232 | }
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| 233 | if (x >= LDBL_MAX) {
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| 234 | if (y > 0.0)
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| 235 | return INFINITY;
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| 236 | return 0.0;
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| 237 | }
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| 238 |
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| 239 | w = floorl(y);
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| 240 |
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| 241 | /* Set iyflg to 1 if y is an integer. */
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| 242 | iyflg = 0;
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| 243 | if (w == y)
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| 244 | iyflg = 1;
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| 245 |
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| 246 | /* Test for odd integer y. */
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| 247 | yoddint = 0;
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| 248 | if (iyflg) {
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| 249 | ya = fabsl(y);
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| 250 | ya = floorl(0.5 * ya);
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| 251 | yb = 0.5 * fabsl(w);
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| 252 | if( ya != yb )
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| 253 | yoddint = 1;
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| 254 | }
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| 255 |
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| 256 | if (x <= -LDBL_MAX) {
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| 257 | if (y > 0.0) {
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| 258 | if (yoddint)
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| 259 | return -INFINITY;
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| 260 | return INFINITY;
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| 261 | }
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| 262 | if (y < 0.0) {
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| 263 | if (yoddint)
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| 264 | return -0.0;
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| 265 | return 0.0;
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| 266 | }
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| 267 | }
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| 268 | nflg = 0; /* (x<0)**(odd int) */
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| 269 | if (x <= 0.0) {
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| 270 | if (x == 0.0) {
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| 271 | if (y < 0.0) {
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| 272 | if (signbit(x) && yoddint)
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| 273 | /* (-0.0)**(-odd int) = -inf, divbyzero */
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| 274 | return -1.0/0.0;
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| 275 | /* (+-0.0)**(negative) = inf, divbyzero */
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| 276 | return 1.0/0.0;
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| 277 | }
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| 278 | if (signbit(x) && yoddint)
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| 279 | return -0.0;
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| 280 | return 0.0;
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| 281 | }
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| 282 | if (iyflg == 0)
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| 283 | return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
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| 284 | /* (x<0)**(integer) */
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| 285 | if (yoddint)
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| 286 | nflg = 1; /* negate result */
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| 287 | x = -x;
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| 288 | }
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| 289 | /* (+integer)**(integer) */
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| 290 | if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
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| 291 | w = powil(x, (int)y);
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| 292 | return nflg ? -w : w;
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| 293 | }
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| 294 |
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| 295 | /* separate significand from exponent */
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| 296 | x = frexpl(x, &i);
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| 297 | e = i;
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| 298 |
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| 299 | /* find significand in antilog table A[] */
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| 300 | i = 1;
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| 301 | if (x <= A[17])
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| 302 | i = 17;
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| 303 | if (x <= A[i+8])
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| 304 | i += 8;
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| 305 | if (x <= A[i+4])
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| 306 | i += 4;
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| 307 | if (x <= A[i+2])
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| 308 | i += 2;
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| 309 | if (x >= A[1])
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| 310 | i = -1;
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| 311 | i += 1;
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| 312 |
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| 313 | /* Find (x - A[i])/A[i]
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| 314 | * in order to compute log(x/A[i]):
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| 315 | *
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| 316 | * log(x) = log( a x/a ) = log(a) + log(x/a)
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| 317 | *
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| 318 | * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
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| 319 | */
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| 320 | x -= A[i];
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| 321 | x -= B[i/2];
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| 322 | x /= A[i];
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| 323 |
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| 324 | /* rational approximation for log(1+v):
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| 325 | *
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| 326 | * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
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| 327 | */
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| 328 | z = x*x;
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| 329 | w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
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| 330 | w = w - 0.5*z;
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| 331 |
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| 332 | /* Convert to base 2 logarithm:
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| 333 | * multiply by log2(e) = 1 + LOG2EA
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| 334 | */
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| 335 | z = LOG2EA * w;
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| 336 | z += w;
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| 337 | z += LOG2EA * x;
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| 338 | z += x;
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| 339 |
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| 340 | /* Compute exponent term of the base 2 logarithm. */
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| 341 | w = -i;
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| 342 | w /= NXT;
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| 343 | w += e;
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| 344 | /* Now base 2 log of x is w + z. */
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| 345 |
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| 346 | /* Multiply base 2 log by y, in extended precision. */
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| 347 |
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| 348 | /* separate y into large part ya
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| 349 | * and small part yb less than 1/NXT
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| 350 | */
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| 351 | ya = reducl(y);
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| 352 | yb = y - ya;
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| 353 |
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| 354 | /* (w+z)(ya+yb)
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| 355 | * = w*ya + w*yb + z*y
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| 356 | */
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| 357 | F = z * y + w * yb;
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| 358 | Fa = reducl(F);
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| 359 | Fb = F - Fa;
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| 360 |
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| 361 | G = Fa + w * ya;
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| 362 | Ga = reducl(G);
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| 363 | Gb = G - Ga;
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| 364 |
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| 365 | H = Fb + Gb;
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| 366 | Ha = reducl(H);
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| 367 | w = (Ga + Ha) * NXT;
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| 368 |
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| 369 | /* Test the power of 2 for overflow */
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| 370 | if (w > MEXP)
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| 371 | return huge * huge; /* overflow */
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| 372 | if (w < MNEXP)
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| 373 | return twom10000 * twom10000; /* underflow */
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| 374 |
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| 375 | e = w;
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| 376 | Hb = H - Ha;
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| 377 |
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| 378 | if (Hb > 0.0) {
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| 379 | e += 1;
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| 380 | Hb -= 1.0/NXT; /*0.0625L;*/
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| 381 | }
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| 382 |
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| 383 | /* Now the product y * log2(x) = Hb + e/NXT.
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| 384 | *
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| 385 | * Compute base 2 exponential of Hb,
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| 386 | * where -0.0625 <= Hb <= 0.
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| 387 | */
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| 388 | z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
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| 389 |
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| 390 | /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
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| 391 | * Find lookup table entry for the fractional power of 2.
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| 392 | */
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| 393 | if (e < 0)
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| 394 | i = 0;
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| 395 | else
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| 396 | i = 1;
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| 397 | i = e/NXT + i;
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| 398 | e = NXT*i - e;
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| 399 | w = A[e];
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| 400 | z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
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| 401 | z = z + w;
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| 402 | z = scalbnl(z, i); /* multiply by integer power of 2 */
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| 403 |
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| 404 | if (nflg)
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| 405 | z = -z;
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| 406 | return z;
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| 407 | }
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| 408 |
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| 409 |
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| 410 | /* Find a multiple of 1/NXT that is within 1/NXT of x. */
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| 411 | static long double reducl(long double x)
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| 412 | {
|
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| 413 | long double t;
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| 414 |
|
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| 415 | t = x * NXT;
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| 416 | t = floorl(t);
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| 417 | t = t / NXT;
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| 418 | return t;
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| 419 | }
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| 420 |
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| 421 | /*
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| 422 | * Positive real raised to integer power, long double precision
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| 423 | *
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| 424 | *
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| 425 | * SYNOPSIS:
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| 426 | *
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| 427 | * long double x, y, powil();
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| 428 | * int n;
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| 429 | *
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| 430 | * y = powil( x, n );
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| 431 | *
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| 432 | *
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| 433 | * DESCRIPTION:
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| 434 | *
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| 435 | * Returns argument x>0 raised to the nth power.
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| 436 | * The routine efficiently decomposes n as a sum of powers of
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| 437 | * two. The desired power is a product of two-to-the-kth
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| 438 | * powers of x. Thus to compute the 32767 power of x requires
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| 439 | * 28 multiplications instead of 32767 multiplications.
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| 440 | *
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| 441 | *
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| 442 | * ACCURACY:
|
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| 443 | *
|
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| 444 | * Relative error:
|
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| 445 | * arithmetic x domain n domain # trials peak rms
|
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| 446 | * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
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| 447 | * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
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| 448 | * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
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| 449 | *
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| 450 | * Returns MAXNUM on overflow, zero on underflow.
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| 451 | */
|
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| 452 |
|
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| 453 | static long double powil(long double x, int nn)
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| 454 | {
|
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| 455 | long double ww, y;
|
---|
| 456 | long double s;
|
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| 457 | int n, e, sign, lx;
|
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| 458 |
|
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| 459 | if (nn == 0)
|
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| 460 | return 1.0;
|
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| 461 |
|
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| 462 | if (nn < 0) {
|
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| 463 | sign = -1;
|
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| 464 | n = -nn;
|
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| 465 | } else {
|
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| 466 | sign = 1;
|
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| 467 | n = nn;
|
---|
| 468 | }
|
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| 469 |
|
---|
| 470 | /* Overflow detection */
|
---|
| 471 |
|
---|
| 472 | /* Calculate approximate logarithm of answer */
|
---|
| 473 | s = x;
|
---|
| 474 | s = frexpl( s, &lx);
|
---|
| 475 | e = (lx - 1)*n;
|
---|
| 476 | if ((e == 0) || (e > 64) || (e < -64)) {
|
---|
| 477 | s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
|
---|
| 478 | s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
|
---|
| 479 | } else {
|
---|
| 480 | s = LOGE2L * e;
|
---|
| 481 | }
|
---|
| 482 |
|
---|
| 483 | if (s > MAXLOGL)
|
---|
| 484 | return huge * huge; /* overflow */
|
---|
| 485 |
|
---|
| 486 | if (s < MINLOGL)
|
---|
| 487 | return twom10000 * twom10000; /* underflow */
|
---|
| 488 | /* Handle tiny denormal answer, but with less accuracy
|
---|
| 489 | * since roundoff error in 1.0/x will be amplified.
|
---|
| 490 | * The precise demarcation should be the gradual underflow threshold.
|
---|
| 491 | */
|
---|
| 492 | if (s < -MAXLOGL+2.0) {
|
---|
| 493 | x = 1.0/x;
|
---|
| 494 | sign = -sign;
|
---|
| 495 | }
|
---|
| 496 |
|
---|
| 497 | /* First bit of the power */
|
---|
| 498 | if (n & 1)
|
---|
| 499 | y = x;
|
---|
| 500 | else
|
---|
| 501 | y = 1.0;
|
---|
| 502 |
|
---|
| 503 | ww = x;
|
---|
| 504 | n >>= 1;
|
---|
| 505 | while (n) {
|
---|
| 506 | ww = ww * ww; /* arg to the 2-to-the-kth power */
|
---|
| 507 | if (n & 1) /* if that bit is set, then include in product */
|
---|
| 508 | y *= ww;
|
---|
| 509 | n >>= 1;
|
---|
| 510 | }
|
---|
| 511 |
|
---|
| 512 | if (sign < 0)
|
---|
| 513 | y = 1.0/y;
|
---|
| 514 | return y;
|
---|
| 515 | }
|
---|
| 516 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
---|
| 517 | // TODO: broken implementation to make things compile
|
---|
| 518 | long double powl(long double x, long double y)
|
---|
| 519 | {
|
---|
| 520 | return pow(x, y);
|
---|
| 521 | }
|
---|
| 522 | #endif
|
---|