Mercurial > cpython-lru_cache-weakref

annotate Modules/_math.c @ 58765:2ba7677bfa95 legacy-trunk

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testCopysign was defined twice in test_math; combine the definitions
author Mark Dickinson <dickinsm@gmail.com>
date Sat, 06 Feb 2010 23:11:25 +0000
parents 1a25b420ec00
children fd25030223b0
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1 /* Definitions of some C99 math library functions, for those platforms
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2 that don't implement these functions already. */
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3
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4 #include "Python.h"
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5 #include <float.h>
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6 #include "_math.h"
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7
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8 /* The following copyright notice applies to the original
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9 implementations of acosh, asinh and atanh. */
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10
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11 /*
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12 * ====================================================
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13 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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14 *
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15 * Developed at SunPro, a Sun Microsystems, Inc. business.
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16 * Permission to use, copy, modify, and distribute this
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17 * software is freely granted, provided that this notice
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18 * is preserved.
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19 * ====================================================
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20 */
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21
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22 static const double ln2 = 6.93147180559945286227E-01;
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23 static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */
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24 static const double two_pow_p28 = 268435456.0; /* 2**28 */
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25 static const double zero = 0.0;
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26
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27 /* acosh(x)
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28 * Method :
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29 * Based on
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30 * acosh(x) = log [ x + sqrt(x*x-1) ]
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31 * we have
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32 * acosh(x) := log(x)+ln2, if x is large; else
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33 * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
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34 * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
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35 *
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36 * Special cases:
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37 * acosh(x) is NaN with signal if x<1.
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38 * acosh(NaN) is NaN without signal.
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39 */
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40
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41 double
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42 _Py_acosh(double x)
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43 {
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44 if (Py_IS_NAN(x)) {
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45 return x+x;
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46 }
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47 if (x < 1.) { /* x < 1; return a signaling NaN */
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48 errno = EDOM;
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49 #ifdef Py_NAN
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50 return Py_NAN;
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51 #else
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52 return (x-x)/(x-x);
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53 #endif
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54 }
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55 else if (x >= two_pow_p28) { /* x > 2**28 */
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56 if (Py_IS_INFINITY(x)) {
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57 return x+x;
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58 } else {
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59 return log(x)+ln2; /* acosh(huge)=log(2x) */
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60 }
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61 }
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62 else if (x == 1.) {
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63 return 0.0; /* acosh(1) = 0 */
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64 }
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65 else if (x > 2.) { /* 2 < x < 2**28 */
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66 double t = x*x;
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67 return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
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68 }
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69 else { /* 1 < x <= 2 */
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70 double t = x - 1.0;
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71 return m_log1p(t + sqrt(2.0*t + t*t));
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72 }
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73 }
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74
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75
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76 /* asinh(x)
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77 * Method :
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78 * Based on
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79 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
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80 * we have
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81 * asinh(x) := x if 1+x*x=1,
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82 * := sign(x)*(log(x)+ln2)) for large |x|, else
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83 * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
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84 * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
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85 */
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86
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87 double
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88 _Py_asinh(double x)
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89 {
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90 double w;
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91 double absx = fabs(x);
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92
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93 if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
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94 return x+x;
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95 }
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96 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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97 return x; /* return x inexact except 0 */
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98 }
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99 if (absx > two_pow_p28) { /* |x| > 2**28 */
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100 w = log(absx)+ln2;
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101 }
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102 else if (absx > 2.0) { /* 2 < |x| < 2**28 */
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103 w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
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104 }
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105 else { /* 2**-28 <= |x| < 2= */
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106 double t = x*x;
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107 w = m_log1p(absx + t / (1.0 + sqrt(1.0 + t)));
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108 }
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109 return copysign(w, x);
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110
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111 }
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112
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113 /* atanh(x)
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114 * Method :
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115 * 1.Reduced x to positive by atanh(-x) = -atanh(x)
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116 * 2.For x>=0.5
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117 * 1 2x x
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118 * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
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119 * 2 1 - x 1 - x
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120 *
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121 * For x<0.5
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122 * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
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123 *
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124 * Special cases:
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125 * atanh(x) is NaN if |x| >= 1 with signal;
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126 * atanh(NaN) is that NaN with no signal;
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127 *
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128 */
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129
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130 double
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131 _Py_atanh(double x)
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132 {
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133 double absx;
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134 double t;
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135
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136 if (Py_IS_NAN(x)) {
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137 return x+x;
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138 }
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139 absx = fabs(x);
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140 if (absx >= 1.) { /* |x| >= 1 */
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141 errno = EDOM;
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142 #ifdef Py_NAN
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143 return Py_NAN;
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144 #else
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145 return x/zero;
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146 #endif
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147 }
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148 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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149 return x;
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150 }
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151 if (absx < 0.5) { /* |x| < 0.5 */
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152 t = absx+absx;
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153 t = 0.5 * m_log1p(t + t*absx / (1.0 - absx));
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154 }
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155 else { /* 0.5 <= |x| <= 1.0 */
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156 t = 0.5 * m_log1p((absx + absx) / (1.0 - absx));
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157 }
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158 return copysign(t, x);
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159 }
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160
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161 /* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed
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162 to avoid the significant loss of precision that arises from direct
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163 evaluation of the expression exp(x) - 1, for x near 0. */
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164
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165 double
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166 _Py_expm1(double x)
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167 {
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168 /* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this
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169 also works fine for infinities and nans.
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170
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171 For smaller x, we can use a method due to Kahan that achieves close to
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172 full accuracy.
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173 */
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174
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175 if (fabs(x) < 0.7) {
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176 double u;
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177 u = exp(x);
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178 if (u == 1.0)
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179 return x;
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180 else
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181 return (u - 1.0) * x / log(u);
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182 }
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183 else
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184 return exp(x) - 1.0;
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185 }
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186
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187 /* log1p(x) = log(1+x). The log1p function is designed to avoid the
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188 significant loss of precision that arises from direct evaluation when x is
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189 small. */
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190
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191 double
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192 _Py_log1p(double x)
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193 {
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194 /* For x small, we use the following approach. Let y be the nearest float
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195 to 1+x, then
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196
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197 1+x = y * (1 - (y-1-x)/y)
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198
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199 so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny, the
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200 second term is well approximated by (y-1-x)/y. If abs(x) >=
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201 DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
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202 then y-1-x will be exactly representable, and is computed exactly by
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203 (y-1)-x.
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204
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205 If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
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206 round-to-nearest then this method is slightly dangerous: 1+x could be
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207 rounded up to 1+DBL_EPSILON instead of down to 1, and in that case
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208 y-1-x will not be exactly representable any more and the result can be
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209 off by many ulps. But this is easily fixed: for a floating-point
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210 number |x| < DBL_EPSILON/2., the closest floating-point number to
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211 log(1+x) is exactly x.
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212 */
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213
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214 double y;
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215 if (fabs(x) < DBL_EPSILON/2.) {
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216 return x;
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217 } else if (-0.5 <= x && x <= 1.) {
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218 /* WARNING: it's possible than an overeager compiler
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219 will incorrectly optimize the following two lines
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220 to the equivalent of "return log(1.+x)". If this
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221 happens, then results from log1p will be inaccurate
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222 for small x. */
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223 y = 1.+x;
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224 return log(y)-((y-1.)-x)/y;
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225 } else {
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226 /* NaNs and infinities should end up here */
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227 return log(1.+x);
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228 }
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229 }