Mercurial > cpython-lru_cache-weakref

annotate Modules/_math.c @ 80661:94de77bd0b4b 2.7

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Issue #16194: document imp.load_dynamic problems when read several extension modules from the same binary object file.
author Andrew Svetlov <andrew.svetlov@gmail.com>
date Thu, 29 Nov 2012 19:51:28 +0200
parents 377036cb13e9
children
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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 }
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59 else {
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60 return log(x)+ln2; /* acosh(huge)=log(2x) */
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61 }
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62 }
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63 else if (x == 1.) {
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64 return 0.0; /* acosh(1) = 0 */
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65 }
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66 else if (x > 2.) { /* 2 < x < 2**28 */
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67 double t = x*x;
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68 return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
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69 }
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70 else { /* 1 < x <= 2 */
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71 double t = x - 1.0;
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72 return m_log1p(t + sqrt(2.0*t + t*t));
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73 }
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74 }
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75
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76
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77 /* asinh(x)
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78 * Method :
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79 * Based on
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80 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
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81 * we have
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82 * asinh(x) := x if 1+x*x=1,
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83 * := sign(x)*(log(x)+ln2)) for large |x|, else
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84 * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
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85 * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
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86 */
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87
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88 double
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89 _Py_asinh(double x)
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90 {
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91 double w;
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92 double absx = fabs(x);
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93
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94 if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
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95 return x+x;
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96 }
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97 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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98 return x; /* return x inexact except 0 */
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99 }
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100 if (absx > two_pow_p28) { /* |x| > 2**28 */
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101 w = log(absx)+ln2;
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102 }
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103 else if (absx > 2.0) { /* 2 < |x| < 2**28 */
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104 w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
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105 }
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106 else { /* 2**-28 <= |x| < 2= */
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107 double t = x*x;
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108 w = m_log1p(absx + t / (1.0 + sqrt(1.0 + t)));
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109 }
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110 return copysign(w, x);
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111
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112 }
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113
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114 /* atanh(x)
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115 * Method :
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116 * 1.Reduced x to positive by atanh(-x) = -atanh(x)
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117 * 2.For x>=0.5
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118 * 1 2x x
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119 * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * -------)
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120 * 2 1 - x 1 - x
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121 *
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122 * For x<0.5
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123 * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
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124 *
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125 * Special cases:
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126 * atanh(x) is NaN if |x| >= 1 with signal;
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127 * atanh(NaN) is that NaN with no signal;
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128 *
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129 */
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130
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131 double
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132 _Py_atanh(double x)
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133 {
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134 double absx;
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135 double t;
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136
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137 if (Py_IS_NAN(x)) {
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138 return x+x;
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139 }
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140 absx = fabs(x);
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141 if (absx >= 1.) { /* |x| >= 1 */
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142 errno = EDOM;
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143 #ifdef Py_NAN
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144 return Py_NAN;
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145 #else
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146 return x/zero;
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147 #endif
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148 }
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149 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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150 return x;
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151 }
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152 if (absx < 0.5) { /* |x| < 0.5 */
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153 t = absx+absx;
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154 t = 0.5 * m_log1p(t + t*absx / (1.0 - absx));
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155 }
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156 else { /* 0.5 <= |x| <= 1.0 */
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157 t = 0.5 * m_log1p((absx + absx) / (1.0 - absx));
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158 }
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159 return copysign(t, x);
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160 }
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161
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162 /* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed
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163 to avoid the significant loss of precision that arises from direct
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164 evaluation of the expression exp(x) - 1, for x near 0. */
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165
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166 double
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167 _Py_expm1(double x)
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168 {
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169 /* For abs(x) >= log(2), it's safe to evaluate exp(x) - 1 directly; this
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170 also works fine for infinities and nans.
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171
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172 For smaller x, we can use a method due to Kahan that achieves close to
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173 full accuracy.
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174 */
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175
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176 if (fabs(x) < 0.7) {
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177 double u;
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178 u = exp(x);
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179 if (u == 1.0)
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180 return x;
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181 else
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182 return (u - 1.0) * x / log(u);
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183 }
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184 else
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185 return exp(x) - 1.0;
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186 }
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187
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188 /* log1p(x) = log(1+x). The log1p function is designed to avoid the
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189 significant loss of precision that arises from direct evaluation when x is
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190 small. */
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191
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192 #ifdef HAVE_LOG1P
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193
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194 double
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195 _Py_log1p(double x)
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196 {
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197 /* Some platforms supply a log1p function but don't respect the sign of
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198 zero: log1p(-0.0) gives 0.0 instead of the correct result of -0.0.
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199
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200 To save fiddling with configure tests and platform checks, we handle the
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201 special case of zero input directly on all platforms.
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202 */
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203 if (x == 0.0) {
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204 return x;
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205 }
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206 else {
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207 return log1p(x);
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208 }
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209 }
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210
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211 #else
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212
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213 double
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214 _Py_log1p(double x)
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215 {
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216 /* For x small, we use the following approach. Let y be the nearest float
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217 to 1+x, then
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218
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219 1+x = y * (1 - (y-1-x)/y)
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220
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221 so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny, the
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222 second term is well approximated by (y-1-x)/y. If abs(x) >=
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223 DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
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224 then y-1-x will be exactly representable, and is computed exactly by
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225 (y-1)-x.
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226
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227 If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
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228 round-to-nearest then this method is slightly dangerous: 1+x could be
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229 rounded up to 1+DBL_EPSILON instead of down to 1, and in that case
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230 y-1-x will not be exactly representable any more and the result can be
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231 off by many ulps. But this is easily fixed: for a floating-point
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232 number |x| < DBL_EPSILON/2., the closest floating-point number to
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233 log(1+x) is exactly x.
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234 */
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235
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236 double y;
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237 if (fabs(x) < DBL_EPSILON/2.) {
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238 return x;
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239 }
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240 else if (-0.5 <= x && x <= 1.) {
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241 /* WARNING: it's possible than an overeager compiler
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242 will incorrectly optimize the following two lines
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243 to the equivalent of "return log(1.+x)". If this
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244 happens, then results from log1p will be inaccurate
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245 for small x. */
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246 y = 1.+x;
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247 return log(y)-((y-1.)-x)/y;
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248 }
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249 else {
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250 /* NaNs and infinities should end up here */
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251 return log(1.+x);
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252 }
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253 }
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254
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255 #endif /* ifdef HAVE_LOG1P */