Mercurial > cpython-lru_cache-weakref

annotate Python/pymath.c @ 48178:cf331fef3293 legacy-trunk

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author Brett Cannon <bcannon@gmail.com>
date Mon, 04 Aug 2008 21:30:53 +0000
parents b69f443ef480
children 922b9e03b260
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1 #include "Python.h"
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2
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3 #ifndef HAVE_HYPOT
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4 double hypot(double x, double y)
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5 {
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6 double yx;
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7
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8 x = fabs(x);
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9 y = fabs(y);
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10 if (x < y) {
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11 double temp = x;
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12 x = y;
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13 y = temp;
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14 }
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15 if (x == 0.)
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16 return 0.;
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17 else {
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18 yx = y/x;
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19 return x*sqrt(1.+yx*yx);
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20 }
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21 }
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22 #endif /* HAVE_HYPOT */
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23
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24 #ifndef HAVE_COPYSIGN
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25 static double
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26 copysign(double x, double y)
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27 {
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28 /* use atan2 to distinguish -0. from 0. */
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29 if (y > 0. || (y == 0. && atan2(y, -1.) > 0.)) {
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30 return fabs(x);
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31 } else {
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32 return -fabs(x);
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33 }
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34 }
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35 #endif /* HAVE_COPYSIGN */
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36
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37 #ifndef HAVE_LOG1P
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38 double
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39 log1p(double x)
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40 {
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41 /* For x small, we use the following approach. Let y be the nearest
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42 float to 1+x, then
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43
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44 1+x = y * (1 - (y-1-x)/y)
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45
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46 so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny,
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47 the second term is well approximated by (y-1-x)/y. If abs(x) >=
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48 DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
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49 then y-1-x will be exactly representable, and is computed exactly
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50 by (y-1)-x.
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51
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52 If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
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53 round-to-nearest then this method is slightly dangerous: 1+x could
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54 be rounded up to 1+DBL_EPSILON instead of down to 1, and in that
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55 case y-1-x will not be exactly representable any more and the
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56 result can be off by many ulps. But this is easily fixed: for a
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57 floating-point number |x| < DBL_EPSILON/2., the closest
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58 floating-point number to log(1+x) is exactly x.
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59 */
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60
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61 double y;
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62 if (fabs(x) < DBL_EPSILON/2.) {
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63 return x;
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64 } else if (-0.5 <= x && x <= 1.) {
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65 /* WARNING: it's possible than an overeager compiler
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66 will incorrectly optimize the following two lines
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67 to the equivalent of "return log(1.+x)". If this
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68 happens, then results from log1p will be inaccurate
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69 for small x. */
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70 y = 1.+x;
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71 return log(y)-((y-1.)-x)/y;
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72 } else {
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73 /* NaNs and infinities should end up here */
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74 return log(1.+x);
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75 }
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76 }
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77 #endif /* HAVE_LOG1P */
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78
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79 /*
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80 * ====================================================
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81 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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82 *
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83 * Developed at SunPro, a Sun Microsystems, Inc. business.
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84 * Permission to use, copy, modify, and distribute this
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85 * software is freely granted, provided that this notice
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86 * is preserved.
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87 * ====================================================
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88 */
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89
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90 static const double ln2 = 6.93147180559945286227E-01;
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91 static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */
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92 static const double two_pow_p28 = 268435456.0; /* 2**28 */
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93 static const double zero = 0.0;
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94
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95 /* asinh(x)
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96 * Method :
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97 * Based on
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98 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
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99 * we have
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100 * asinh(x) := x if 1+x*x=1,
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101 * := sign(x)*(log(x)+ln2)) for large |x|, else
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102 * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
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103 * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
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104 */
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105
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106 #ifndef HAVE_ASINH
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107 double
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108 asinh(double x)
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109 {
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110 double w;
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111 double absx = fabs(x);
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112
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113 if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
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114 return x+x;
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115 }
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116 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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117 return x; /* return x inexact except 0 */
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118 }
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119 if (absx > two_pow_p28) { /* |x| > 2**28 */
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120 w = log(absx)+ln2;
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121 }
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122 else if (absx > 2.0) { /* 2 < |x| < 2**28 */
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123 w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
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124 }
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125 else { /* 2**-28 <= |x| < 2= */
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126 double t = x*x;
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127 w = log1p(absx + t / (1.0 + sqrt(1.0 + t)));
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128 }
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129 return copysign(w, x);
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130
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131 }
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132 #endif /* HAVE_ASINH */
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133
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134 /* acosh(x)
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135 * Method :
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136 * Based on
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137 * acosh(x) = log [ x + sqrt(x*x-1) ]
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138 * we have
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139 * acosh(x) := log(x)+ln2, if x is large; else
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140 * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
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141 * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
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142 *
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143 * Special cases:
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144 * acosh(x) is NaN with signal if x<1.
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145 * acosh(NaN) is NaN without signal.
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146 */
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147
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148 #ifndef HAVE_ACOSH
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149 double
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150 acosh(double x)
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151 {
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152 if (Py_IS_NAN(x)) {
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153 return x+x;
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154 }
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155 if (x < 1.) { /* x < 1; return a signaling NaN */
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156 errno = EDOM;
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157 #ifdef Py_NAN
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158 return Py_NAN;
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159 #else
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160 return (x-x)/(x-x);
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161 #endif
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162 }
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163 else if (x >= two_pow_p28) { /* x > 2**28 */
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164 if (Py_IS_INFINITY(x)) {
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165 return x+x;
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166 } else {
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167 return log(x)+ln2; /* acosh(huge)=log(2x) */
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168 }
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169 }
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170 else if (x == 1.) {
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171 return 0.0; /* acosh(1) = 0 */
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172 }
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173 else if (x > 2.) { /* 2 < x < 2**28 */
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174 double t = x*x;
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175 return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
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176 }
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177 else { /* 1 < x <= 2 */
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178 double t = x - 1.0;
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179 return log1p(t + sqrt(2.0*t + t*t));
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180 }
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181 }
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182 #endif /* HAVE_ACOSH */
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183
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184 /* atanh(x)
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185 * Method :
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186 * 1.Reduced x to positive by atanh(-x) = -atanh(x)
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187 * 2.For x>=0.5
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188 * 1 2x x
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189 * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
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190 * 2 1 - x 1 - x
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191 *
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192 * For x<0.5
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193 * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
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194 *
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195 * Special cases:
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196 * atanh(x) is NaN if |x| >= 1 with signal;
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197 * atanh(NaN) is that NaN with no signal;
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198 *
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199 */
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200
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201 #ifndef HAVE_ATANH
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202 double
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203 atanh(double x)
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204 {
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205 double absx;
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206 double t;
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207
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208 if (Py_IS_NAN(x)) {
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209 return x+x;
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210 }
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211 absx = fabs(x);
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212 if (absx >= 1.) { /* |x| >= 1 */
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213 errno = EDOM;
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214 #ifdef Py_NAN
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215 return Py_NAN;
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216 #else
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217 return x/zero;
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218 #endif
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219 }
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220 if (absx < two_pow_m28) { /* |x| < 2**-28 */
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221 return x;
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222 }
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223 if (absx < 0.5) { /* |x| < 0.5 */
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224 t = absx+absx;
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225 t = 0.5 * log1p(t + t*absx / (1.0 - absx));
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226 }
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227 else { /* 0.5 <= |x| <= 1.0 */
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228 t = 0.5 * log1p((absx + absx) / (1.0 - absx));
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229 }
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230 return copysign(t, x);
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231 }
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232 #endif /* HAVE_ATANH */