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;
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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;
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468 | }
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469 |
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470 | /* Overflow detection */
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471 |
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472 | /* Calculate approximate logarithm of answer */
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473 | s = x;
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474 | s = frexpl( s, &lx);
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475 | e = (lx - 1)*n;
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476 | if ((e == 0) || (e > 64) || (e < -64)) {
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477 | s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
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478 | s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
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479 | } else {
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480 | s = LOGE2L * e;
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481 | }
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482 |
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483 | if (s > MAXLOGL)
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484 | return huge * huge; /* overflow */
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485 |
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486 | if (s < MINLOGL)
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487 | return twom10000 * twom10000; /* underflow */
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488 | /* Handle tiny denormal answer, but with less accuracy
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489 | * since roundoff error in 1.0/x will be amplified.
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490 | * The precise demarcation should be the gradual underflow threshold.
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491 | */
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492 | if (s < -MAXLOGL+2.0) {
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493 | x = 1.0/x;
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494 | sign = -sign;
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495 | }
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496 |
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497 | /* First bit of the power */
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498 | if (n & 1)
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499 | y = x;
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500 | else
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501 | y = 1.0;
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502 |
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503 | ww = x;
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504 | n >>= 1;
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505 | while (n) {
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506 | ww = ww * ww; /* arg to the 2-to-the-kth power */
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507 | if (n & 1) /* if that bit is set, then include in product */
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508 | y *= ww;
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509 | n >>= 1;
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510 | }
|
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511 |
|
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512 | if (sign < 0)
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513 | y = 1.0/y;
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514 | return y;
|
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515 | }
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516 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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517 | // TODO: broken implementation to make things compile
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518 | long double powl(long double x, long double y)
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519 | {
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520 | return pow(x, y);
|
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521 | }
|
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522 | #endif
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