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16616
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1 /* Exponential base 2 function. |
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2 Copyright (C) 2012 Free Software Foundation, Inc. |
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3 |
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4 This program is free software: you can redistribute it and/or modify |
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5 it under the terms of the GNU General Public License as published by |
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6 the Free Software Foundation; either version 3 of the License, or |
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7 (at your option) any later version. |
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8 |
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9 This program is distributed in the hope that it will be useful, |
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10 but WITHOUT ANY WARRANTY; without even the implied warranty of |
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11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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12 GNU General Public License for more details. |
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13 |
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14 You should have received a copy of the GNU General Public License |
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15 along with this program. If not, see <http://www.gnu.org/licenses/>. */ |
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16 |
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17 #include <config.h> |
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18 |
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19 /* Specification. */ |
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20 #include <math.h> |
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21 |
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22 #include <float.h> |
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23 |
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24 /* Best possible approximation of log(2) as a 'double'. */ |
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25 #define LOG2 0.693147180559945309417232121458176568075 |
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26 |
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27 /* Best possible approximation of 1/log(2) as a 'double'. */ |
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28 #define LOG2_INVERSE 1.44269504088896340735992468100189213743 |
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29 |
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30 /* Best possible approximation of log(2)/256 as a 'double'. */ |
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31 #define LOG2_BY_256 0.00270760617406228636491106297444600221904 |
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32 |
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33 /* Best possible approximation of 256/log(2) as a 'double'. */ |
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34 #define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181 |
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35 |
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36 double |
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37 exp2 (double x) |
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38 { |
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39 /* exp2(x) = exp(x*log(2)). |
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40 If we would compute it like this, there would be rounding errors for |
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41 integer or near-integer values of x. To avoid these, we inline the |
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42 algorithm for exp(), and the multiplication with log(2) cancels a |
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43 division by log(2). */ |
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44 |
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45 if (isnand (x)) |
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46 return x; |
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47 |
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48 if (x > (double) DBL_MAX_EXP) |
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49 /* x > DBL_MAX_EXP |
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50 hence exp2(x) > 2^DBL_MAX_EXP, overflows to Infinity. */ |
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51 return HUGE_VAL; |
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52 |
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53 if (x < (double) (DBL_MIN_EXP - 1 - DBL_MANT_DIG)) |
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54 /* x < (DBL_MIN_EXP - 1 - DBL_MANT_DIG) |
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55 hence exp2(x) < 2^(DBL_MIN_EXP-1-DBL_MANT_DIG), |
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56 underflows to zero. */ |
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57 return 0.0; |
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58 |
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59 /* Decompose x into |
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60 x = n + m/256 + y/log(2) |
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61 where |
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62 n is an integer, |
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63 m is an integer, -128 <= m <= 128, |
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64 y is a number, |y| <= log(2)/512 + epsilon = 0.00135... |
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65 Then |
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66 exp2(x) = 2^n * exp(m * log(2)/256) * exp(y) |
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67 The first factor is an ldexpl() call. |
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68 The second factor is a table lookup. |
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69 The third factor is computed |
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70 - either as sinh(y) + cosh(y) |
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71 where sinh(y) is computed through the power series: |
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72 sinh(y) = y + y^3/3! + y^5/5! + ... |
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73 and cosh(y) is computed as hypot(1, sinh(y)), |
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74 - or as exp(2*z) = (1 + tanh(z)) / (1 - tanh(z)) |
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75 where z = y/2 |
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76 and tanh(z) is computed through its power series: |
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77 tanh(z) = z |
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78 - 1/3 * z^3 |
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79 + 2/15 * z^5 |
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80 - 17/315 * z^7 |
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81 + 62/2835 * z^9 |
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82 - 1382/155925 * z^11 |
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83 + 21844/6081075 * z^13 |
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84 - 929569/638512875 * z^15 |
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85 + ... |
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86 Since |z| <= log(2)/1024 < 0.0007, the relative error of the z^7 term |
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87 is < 0.0007^6 < 2^-60 <= 2^-DBL_MANT_DIG, therefore we can truncate |
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88 the series after the z^5 term. */ |
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89 |
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90 { |
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91 double nm = round (x * 256.0); /* = 256 * n + m */ |
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92 double z = (x * 256.0 - nm) * (LOG2_BY_256 * 0.5); |
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93 |
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94 /* Coefficients of the power series for tanh(z). */ |
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95 #define TANH_COEFF_1 1.0 |
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96 #define TANH_COEFF_3 -0.333333333333333333333333333333333333334 |
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97 #define TANH_COEFF_5 0.133333333333333333333333333333333333334 |
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98 #define TANH_COEFF_7 -0.053968253968253968253968253968253968254 |
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99 #define TANH_COEFF_9 0.0218694885361552028218694885361552028218 |
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100 #define TANH_COEFF_11 -0.00886323552990219656886323552990219656886 |
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101 #define TANH_COEFF_13 0.00359212803657248101692546136990581435026 |
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102 #define TANH_COEFF_15 -0.00145583438705131826824948518070211191904 |
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103 |
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104 double z2 = z * z; |
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105 double tanh_z = |
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106 ((TANH_COEFF_5 |
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107 * z2 + TANH_COEFF_3) |
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108 * z2 + TANH_COEFF_1) |
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109 * z; |
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110 |
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111 double exp_y = (1.0 + tanh_z) / (1.0 - tanh_z); |
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112 |
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113 int n = (int) round (nm * (1.0 / 256.0)); |
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114 int m = (int) nm - 256 * n; |
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115 |
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116 /* exp_table[i] = exp((i - 128) * log(2)/256). |
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117 Computed in GNU clisp through |
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118 (setf (long-float-digits) 128) |
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119 (setq a 0L0) |
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120 (setf (long-float-digits) 256) |
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121 (dotimes (i 257) |
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122 (format t " ~D,~%" |
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123 (float (exp (* (/ (- i 128) 256) (log 2L0))) a))) */ |
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124 static const double exp_table[257] = |
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125 { |
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126 0.707106781186547524400844362104849039284, |
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127 0.709023942160207598920563322257676190836, |
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128 0.710946301084582779904674297352120049962, |
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129 0.71287387205274715340350157671438300618, |
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130 0.714806669195985005617532889137569953044, |
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131 0.71674470668389442125974978427737336719, |
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132 0.71868799872449116280161304224785251353, |
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133 0.720636559564312831364255957304947586072, |
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134 0.72259040348852331001850312073583545284, |
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135 0.724549544821017490259402705487111270714, |
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136 0.726513997924526282423036245842287293786, |
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137 0.728483777200721910815451524818606761737, |
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138 0.730458897090323494325651445155310766577, |
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139 0.732439372073202913296664682112279175616, |
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140 0.734425216668490963430822513132890712652, |
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141 0.736416445434683797507470506133110286942, |
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142 0.738413072969749655693453740187024961962, |
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143 0.740415113911235885228829945155951253966, |
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144 0.742422582936376250272386395864403155277, |
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145 0.744435494762198532693663597314273242753, |
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146 0.746453864145632424600321765743336770838, |
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147 0.748477705883617713391824861712720862423, |
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148 0.750507034813212760132561481529764324813, |
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149 0.752541865811703272039672277899716132493, |
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150 0.75458221379671136988300977551659676571, |
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151 0.756628093726304951096818488157633113612, |
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152 0.75867952059910734940489114658718937343, |
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153 0.760736509454407291763130627098242426467, |
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154 0.762799075372269153425626844758470477304, |
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155 0.76486723347364351194254345936342587308, |
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156 0.766940998920478000900300751753859329456, |
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157 0.769020386915828464216738479594307884331, |
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158 0.771105412703970411806145931045367420652, |
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159 0.773196091570510777431255778146135325272, |
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160 0.77529243884249997956151370535341912283, |
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161 0.777394469888544286059157168801667390437, |
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162 0.779502200118918483516864044737428940745, |
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163 0.781615644985678852072965367573877941354, |
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164 0.783734819982776446532455855478222575498, |
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165 0.78585974064617068462428149076570281356, |
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166 0.787990422553943243227635080090952504452, |
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167 0.790126881326412263402248482007960521995, |
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168 0.79226913262624686505993407346567890838, |
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169 0.794417192158581972116898048814333564685, |
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170 0.796571075671133448968624321559534367934, |
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171 0.798730798954313549131410147104316569576, |
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172 0.800896377841346676896923120795476813684, |
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173 0.803067828208385462848443946517563571584, |
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174 0.805245165974627154089760333678700291728, |
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175 0.807428407102430320039984581575729114268, |
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176 0.809617567597431874649880866726368203972, |
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177 0.81181266350866441589760797777344082227, |
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178 0.814013710928673883424109261007007338614, |
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179 0.816220725993637535170713864466769240053, |
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180 0.818433724883482243883852017078007231025, |
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181 0.82065272382200311435413206848451310067, |
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182 0.822877739076982422259378362362911222833, |
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183 0.825108786960308875483586738272485101678, |
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184 0.827345883828097198786118571797909120834, |
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185 0.829589046080808042697824787210781231927, |
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186 0.831838290163368217523168228488195222638, |
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187 0.834093632565291253329796170708536192903, |
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188 0.836355089820798286809404612069230711295, |
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189 0.83862267850893927589613232455870870518, |
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190 0.84089641525371454303112547623321489504, |
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191 0.84317631672419664796432298771385230143, |
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192 0.84546239963465259098692866759361830709, |
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193 0.84775468074466634749045860363936420312, |
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194 0.850053176859261734750681286748751167545, |
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195 0.852357904829025611837203530384718316326, |
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196 0.854668881550231413551897437515331498025, |
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197 0.856986123964963019301812477839166009452, |
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198 0.859309649061238957814672188228156252257, |
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199 0.861639473873136948607517116872358729753, |
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200 0.863975615480918781121524414614366207052, |
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201 0.866318091011155532438509953514163469652, |
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202 0.868666917636853124497101040936083380124, |
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203 0.871022112577578221729056715595464682243, |
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204 0.873383693099584470038708278290226842228, |
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205 0.875751676515939078050995142767930296012, |
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206 0.878126080186649741556080309687656610647, |
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207 0.880506921518791912081045787323636256171, |
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208 0.882894217966636410521691124969260937028, |
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209 0.885287987031777386769987907431242017412, |
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210 0.88768824626326062627527960009966160388, |
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211 0.89009501325771220447985955243623523504, |
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212 0.892508305659467490072110281986409916153, |
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213 0.8949281411607004980029443898876582985, |
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214 0.897354537501553593213851621063890907178, |
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215 0.899787512470267546027427696662514569756, |
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216 0.902227083903311940153838631655504844215, |
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217 0.904673269685515934269259325789226871994, |
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218 0.907126087750199378124917300181170171233, |
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219 0.909585556079304284147971563828178746372, |
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220 0.91205169270352665549806275316460097744, |
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221 0.914524515702448671545983912696158354092, |
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222 0.91700404320467123174354159479414442804, |
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223 0.919490293387946858856304371174663918816, |
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224 0.921983284479312962533570386670938449637, |
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225 0.92448303475522546419252726694739603678, |
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226 0.92698956254169278419622653516884831976, |
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227 0.929502886214410192307650717745572682403, |
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228 0.932023024198894522404814545597236289343, |
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229 0.934549994970619252444512104439799143264, |
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230 0.93708381705514995066499947497722326722, |
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231 0.93962450902828008902058735120448448827, |
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232 0.942172089516167224843810351983745154882, |
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233 0.944726577195469551733539267378681531548, |
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234 0.947287990793482820670109326713462307376, |
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235 0.949856349088277632361251759806996099924, |
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236 0.952431670908837101825337466217860725517, |
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237 0.955013975135194896221170529572799135168, |
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238 0.957603280698573646936305635147915443924, |
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239 0.960199606581523736948607188887070611744, |
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240 0.962802971818062464478519115091191368377, |
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241 0.965413395493813583952272948264534783197, |
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242 0.968030896746147225299027952283345762418, |
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243 0.970655494764320192607710617437589705184, |
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244 0.973287208789616643172102023321302921373, |
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245 0.97592605811548914795551023340047499377, |
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246 0.978572062087700134509161125813435745597, |
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247 0.981225240104463713381244885057070325016, |
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248 0.983885611616587889056366801238014683926, |
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249 0.98655319612761715646797006813220671315, |
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250 0.989228013193975484129124959065583667775, |
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251 0.99191008242510968492991311132615581644, |
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252 0.994599423483633175652477686222166314457, |
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253 0.997296056085470126257659913847922601123, |
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254 1.0, |
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255 1.00271127505020248543074558845036204047, |
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256 1.0054299011128028213513839559347998147, |
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257 1.008155898118417515783094890817201039276, |
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258 1.01088928605170046002040979056186052439, |
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259 1.013630084951489438840258929063939929597, |
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260 1.01637831491095303794049311378629406276, |
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261 1.0191339960777379496848780958207928794, |
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262 1.02189714865411667823448013478329943978, |
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263 1.02466779289713564514828907627081492763, |
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264 1.0274459491187636965388611939222137815, |
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265 1.030231637686041012871707902453904567093, |
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266 1.033024879021228422500108283970460918086, |
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267 1.035825693601957120029983209018081371844, |
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268 1.03863410196137879061243669795463973258, |
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269 1.04145012468831614126454607901189312648, |
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270 1.044273782427413840321966478739929008784, |
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271 1.04710509587928986612990725022711224056, |
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272 1.04994408580068726608203812651590790906, |
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273 1.05279077300462632711989120298074630319, |
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274 1.05564517836055715880834132515293865216, |
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275 1.058507322794512690105772109683716645074, |
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276 1.061377227289262080950567678003883726294, |
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277 1.06425491288446454978861125700158022068, |
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278 1.06714040067682361816952112099280916261, |
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279 1.0700337118202417735424119367576235685, |
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280 1.072934867525975551385035450873827585343, |
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281 1.075843889062791037803228648476057074063, |
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282 1.07876079775711979374068003743848295849, |
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283 1.081685614993215201942115594422531125643, |
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284 1.08461836221330923781610517190661434161, |
|
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285 1.087559060917769665346797830944039707867, |
|
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286 1.09050773266525765920701065576070797899, |
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287 1.09346439907288585422822014625044716208, |
|
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288 1.096429081816376823386138295859248481766, |
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289 1.09940180263022198546369696823882990404, |
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290 1.10238258330784094355641420942564685751, |
|
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291 1.10537144570174125558827469625695031104, |
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292 1.108368411723678638009423649426619850137, |
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293 1.111373503344817603850149254228916637444, |
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294 1.1143867425958925363088129569196030678, |
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295 1.11740815156736919905457996308578026665, |
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296 1.12043775240960668442900387986631301277, |
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297 1.123475567333019800733729739775321431954, |
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298 1.12652161860824189979479864378703477763, |
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299 1.129575928566288145997264988840249825907, |
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300 1.13263851959871922798707372367762308438, |
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301 1.13570941415780551424039033067611701343, |
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302 1.13878863475669165370383028384151125472, |
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303 1.14187620396956162271229760828788093894, |
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304 1.14497214443180421939441388822291589579, |
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305 1.14807647884017900677879966269734268003, |
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306 1.15118922995298270581775963520198253612, |
|
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307 1.154310420590216039548221528724806960684, |
|
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308 1.157440073633751029613085766293796821106, |
|
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309 1.16057821202749874636945947257609098625, |
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310 1.16372485877757751381357359909218531234, |
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311 1.166880036952481570555516298414089287834, |
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312 1.170043769683250188080259035792738573, |
|
|
313 1.17321608016363724753480435451324538889, |
|
|
314 1.176396991650281276284645728483848641054, |
|
|
315 1.17958652746287594548610056676944051898, |
|
|
316 1.182784710984341029924457204693850757966, |
|
|
317 1.18599156566099383137126564953421556374, |
|
|
318 1.18920711500272106671749997056047591529, |
|
|
319 1.19243138258315122214272755814543101148, |
|
|
320 1.195664392039827374583837049865451975705, |
|
|
321 1.19890616707438048177030255797630020695, |
|
|
322 1.202156731452703142096396957497765876003, |
|
|
323 1.205416109005123825604211432558411335666, |
|
|
324 1.208684323626581577354792255889216998484, |
|
|
325 1.21196139927680119446816891773249304545, |
|
|
326 1.215247359980468878116520251338798457624, |
|
|
327 1.218542229827408361758207148117394510724, |
|
|
328 1.221846032972757516903891841911570785836, |
|
|
329 1.225158793637145437709464594384845353707, |
|
|
330 1.22848053610687000569400895779278184036, |
|
|
331 1.2318112847340759358845566532127948166, |
|
|
332 1.235151063936933305692912507415415760294, |
|
|
333 1.238499898199816567833368865859612431545, |
|
|
334 1.24185781207348404859367746872659560551, |
|
|
335 1.24522483017525793277520496748615267417, |
|
|
336 1.24860097718920473662176609730249554519, |
|
|
337 1.25198627786631627006020603178920359732, |
|
|
338 1.255380757024691089579390657442301194595, |
|
|
339 1.25878443954971644307786044181516261876, |
|
|
340 1.26219735039425070801401025851841645967, |
|
|
341 1.265619514578806324196273999873453036296, |
|
|
342 1.26905095719173322255441908103233800472, |
|
|
343 1.27249170338940275123669204418460217677, |
|
|
344 1.27594177839639210038120243475928938891, |
|
|
345 1.27940120750566922691358797002785254596, |
|
|
346 1.28287001607877828072666978102151405111, |
|
|
347 1.286348229546025533601482208069738348355, |
|
|
348 1.28983587340666581223274729549155218968, |
|
|
349 1.293332973229089436725559789048704304684, |
|
|
350 1.296839554651009665933754117792451159835, |
|
|
351 1.30035564337965065101414056707091779129, |
|
|
352 1.30388126519193589857452364895199736833, |
|
|
353 1.30741644593467724479715157747196172848, |
|
|
354 1.310961211524764341922991786330755849366, |
|
|
355 1.314515587949354658485983613383997794965, |
|
|
356 1.318079601266063994690185647066116617664, |
|
|
357 1.32165327760315751432651181233060922616, |
|
|
358 1.32523664315974129462953709549872167411, |
|
|
359 1.32882972420595439547865089632866510792, |
|
|
360 1.33243254708316144935164337949073577407, |
|
|
361 1.33604513820414577344262790437186975929, |
|
|
362 1.33966752405330300536003066972435257602, |
|
|
363 1.34329973118683526382421714618163087542, |
|
|
364 1.346941786232945835788173713229537282075, |
|
|
365 1.35059371589203439140852219606013396004, |
|
|
366 1.35425554693689272829801474014070280434, |
|
|
367 1.357927306212901046494536695671766697446, |
|
|
368 1.36160902063822475558553593883194147464, |
|
|
369 1.36530071720401181543069836033754285543, |
|
|
370 1.36900242297459061192960113298219283217, |
|
|
371 1.37271416508766836928499785714471721579, |
|
|
372 1.37643597075453010021632280551868696026, |
|
|
373 1.380167867260238095581945274358283464697, |
|
|
374 1.383909881963831954872659527265192818, |
|
|
375 1.387662042298529159042861017950775988896, |
|
|
376 1.39142437577192618714983552956624344668, |
|
|
377 1.395196909966200178275574599249220994716, |
|
|
378 1.398979672538311140209528136715194969206, |
|
|
379 1.40277269122020470637471352433337881711, |
|
|
380 1.40657599381901544248361973255451684411, |
|
|
381 1.410389608217270704414375128268675481145, |
|
|
382 1.41421356237309504880168872420969807857 |
|
|
383 }; |
|
|
384 |
|
|
385 return ldexp (exp_table[128 + m] * exp_y, n); |
|
|
386 } |
|
|
387 } |
