Mercurial > cpython-lru_cache-weakref

annotate Modules/cmathmodule.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 cd8d3c2d3bce
children 91374660355a
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1 /* Complex math module */
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2
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3 /* much code borrowed from mathmodule.c */
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4
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5 #include "Python.h"
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6 #include "_math.h"
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7 /* we need DBL_MAX, DBL_MIN, DBL_EPSILON, DBL_MANT_DIG and FLT_RADIX from
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8 float.h. We assume that FLT_RADIX is either 2 or 16. */
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9 #include <float.h>
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10
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11 #if (FLT_RADIX != 2 && FLT_RADIX != 16)
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12 #error "Modules/cmathmodule.c expects FLT_RADIX to be 2 or 16"
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13 #endif
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14
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15 #ifndef M_LN2
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16 #define M_LN2 (0.6931471805599453094) /* natural log of 2 */
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17 #endif
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18
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19 #ifndef M_LN10
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20 #define M_LN10 (2.302585092994045684) /* natural log of 10 */
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21 #endif
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22
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23 /*
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24 CM_LARGE_DOUBLE is used to avoid spurious overflow in the sqrt, log,
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25 inverse trig and inverse hyperbolic trig functions. Its log is used in the
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26 evaluation of exp, cos, cosh, sin, sinh, tan, and tanh to avoid unnecessary
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27 overflow.
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28 */
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29
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30 #define CM_LARGE_DOUBLE (DBL_MAX/4.)
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31 #define CM_SQRT_LARGE_DOUBLE (sqrt(CM_LARGE_DOUBLE))
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32 #define CM_LOG_LARGE_DOUBLE (log(CM_LARGE_DOUBLE))
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33 #define CM_SQRT_DBL_MIN (sqrt(DBL_MIN))
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34
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35 /*
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36 CM_SCALE_UP is an odd integer chosen such that multiplication by
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37 2**CM_SCALE_UP is sufficient to turn a subnormal into a normal.
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38 CM_SCALE_DOWN is (-(CM_SCALE_UP+1)/2). These scalings are used to compute
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39 square roots accurately when the real and imaginary parts of the argument
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40 are subnormal.
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41 */
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42
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43 #if FLT_RADIX==2
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44 #define CM_SCALE_UP (2*(DBL_MANT_DIG/2) + 1)
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45 #elif FLT_RADIX==16
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46 #define CM_SCALE_UP (4*DBL_MANT_DIG+1)
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47 #endif
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48 #define CM_SCALE_DOWN (-(CM_SCALE_UP+1)/2)
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49
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50 /* forward declarations */
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51 static Py_complex c_asinh(Py_complex);
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52 static Py_complex c_atanh(Py_complex);
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53 static Py_complex c_cosh(Py_complex);
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54 static Py_complex c_sinh(Py_complex);
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55 static Py_complex c_sqrt(Py_complex);
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56 static Py_complex c_tanh(Py_complex);
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57 static PyObject * math_error(void);
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58
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59 /* Code to deal with special values (infinities, NaNs, etc.). */
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60
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61 /* special_type takes a double and returns an integer code indicating
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62 the type of the double as follows:
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63 */
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64
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65 enum special_types {
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66 ST_NINF, /* 0, negative infinity */
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67 ST_NEG, /* 1, negative finite number (nonzero) */
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68 ST_NZERO, /* 2, -0. */
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69 ST_PZERO, /* 3, +0. */
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70 ST_POS, /* 4, positive finite number (nonzero) */
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71 ST_PINF, /* 5, positive infinity */
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72 ST_NAN /* 6, Not a Number */
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73 };
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74
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75 static enum special_types
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76 special_type(double d)
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77 {
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78 if (Py_IS_FINITE(d)) {
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79 if (d != 0) {
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80 if (copysign(1., d) == 1.)
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81 return ST_POS;
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82 else
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83 return ST_NEG;
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84 }
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85 else {
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86 if (copysign(1., d) == 1.)
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87 return ST_PZERO;
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88 else
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89 return ST_NZERO;
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90 }
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91 }
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92 if (Py_IS_NAN(d))
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93 return ST_NAN;
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94 if (copysign(1., d) == 1.)
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95 return ST_PINF;
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96 else
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97 return ST_NINF;
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98 }
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99
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100 #define SPECIAL_VALUE(z, table) \
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101 if (!Py_IS_FINITE((z).real) || !Py_IS_FINITE((z).imag)) { \
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102 errno = 0; \
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103 return table[special_type((z).real)] \
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104 [special_type((z).imag)]; \
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105 }
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106
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107 #define P Py_MATH_PI
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108 #define P14 0.25*Py_MATH_PI
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109 #define P12 0.5*Py_MATH_PI
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110 #define P34 0.75*Py_MATH_PI
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111 #define INF Py_HUGE_VAL
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112 #define N Py_NAN
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113 #define U -9.5426319407711027e33 /* unlikely value, used as placeholder */
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114
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115 /* First, the C functions that do the real work. Each of the c_*
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116 functions computes and returns the C99 Annex G recommended result
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117 and also sets errno as follows: errno = 0 if no floating-point
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118 exception is associated with the result; errno = EDOM if C99 Annex
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119 G recommends raising divide-by-zero or invalid for this result; and
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120 errno = ERANGE where the overflow floating-point signal should be
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121 raised.
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122 */
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123
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124 static Py_complex acos_special_values[7][7];
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125
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126 static Py_complex
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127 c_acos(Py_complex z)
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128 {
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129 Py_complex s1, s2, r;
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130
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131 SPECIAL_VALUE(z, acos_special_values);
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132
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133 if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
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134 /* avoid unnecessary overflow for large arguments */
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135 r.real = atan2(fabs(z.imag), z.real);
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136 /* split into cases to make sure that the branch cut has the
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137 correct continuity on systems with unsigned zeros */
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138 if (z.real < 0.) {
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139 r.imag = -copysign(log(hypot(z.real/2., z.imag/2.)) +
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140 M_LN2*2., z.imag);
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141 } else {
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142 r.imag = copysign(log(hypot(z.real/2., z.imag/2.)) +
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143 M_LN2*2., -z.imag);
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144 }
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145 } else {
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146 s1.real = 1.-z.real;
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147 s1.imag = -z.imag;
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148 s1 = c_sqrt(s1);
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149 s2.real = 1.+z.real;
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150 s2.imag = z.imag;
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151 s2 = c_sqrt(s2);
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152 r.real = 2.*atan2(s1.real, s2.real);
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153 r.imag = m_asinh(s2.real*s1.imag - s2.imag*s1.real);
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154 }
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155 errno = 0;
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156 return r;
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157 }
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158
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159 PyDoc_STRVAR(c_acos_doc,
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160 "acos(x)\n"
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161 "\n"
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162 "Return the arc cosine of x.");
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163
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164
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165 static Py_complex acosh_special_values[7][7];
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166
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167 static Py_complex
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168 c_acosh(Py_complex z)
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169 {
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170 Py_complex s1, s2, r;
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171
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172 SPECIAL_VALUE(z, acosh_special_values);
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173
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174 if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
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175 /* avoid unnecessary overflow for large arguments */
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176 r.real = log(hypot(z.real/2., z.imag/2.)) + M_LN2*2.;
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177 r.imag = atan2(z.imag, z.real);
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178 } else {
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179 s1.real = z.real - 1.;
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180 s1.imag = z.imag;
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181 s1 = c_sqrt(s1);
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182 s2.real = z.real + 1.;
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183 s2.imag = z.imag;
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184 s2 = c_sqrt(s2);
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185 r.real = m_asinh(s1.real*s2.real + s1.imag*s2.imag);
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186 r.imag = 2.*atan2(s1.imag, s2.real);
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187 }
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188 errno = 0;
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189 return r;
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190 }
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191
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192 PyDoc_STRVAR(c_acosh_doc,
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193 "acosh(x)\n"
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194 "\n"
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195 "Return the hyperbolic arccosine of x.");
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196
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197
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198 static Py_complex
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199 c_asin(Py_complex z)
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200 {
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201 /* asin(z) = -i asinh(iz) */
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202 Py_complex s, r;
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203 s.real = -z.imag;
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204 s.imag = z.real;
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205 s = c_asinh(s);
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206 r.real = s.imag;
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207 r.imag = -s.real;
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208 return r;
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209 }
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210
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211 PyDoc_STRVAR(c_asin_doc,
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212 "asin(x)\n"
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213 "\n"
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214 "Return the arc sine of x.");
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215
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216
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217 static Py_complex asinh_special_values[7][7];
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218
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219 static Py_complex
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220 c_asinh(Py_complex z)
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221 {
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222 Py_complex s1, s2, r;
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223
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224 SPECIAL_VALUE(z, asinh_special_values);
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225
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226 if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {
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227 if (z.imag >= 0.) {
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228 r.real = copysign(log(hypot(z.real/2., z.imag/2.)) +
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229 M_LN2*2., z.real);
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230 } else {
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231 r.real = -copysign(log(hypot(z.real/2., z.imag/2.)) +
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232 M_LN2*2., -z.real);
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233 }
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234 r.imag = atan2(z.imag, fabs(z.real));
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235 } else {
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236 s1.real = 1.+z.imag;
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237 s1.imag = -z.real;
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238 s1 = c_sqrt(s1);
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239 s2.real = 1.-z.imag;
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240 s2.imag = z.real;
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241 s2 = c_sqrt(s2);
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242 r.real = m_asinh(s1.real*s2.imag-s2.real*s1.imag);
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243 r.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag);
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244 }
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245 errno = 0;
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246 return r;
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247 }
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248
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249 PyDoc_STRVAR(c_asinh_doc,
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250 "asinh(x)\n"
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251 "\n"
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252 "Return the hyperbolic arc sine of x.");
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253
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254
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255 static Py_complex
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256 c_atan(Py_complex z)
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257 {
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258 /* atan(z) = -i atanh(iz) */
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259 Py_complex s, r;
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260 s.real = -z.imag;
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261 s.imag = z.real;
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262 s = c_atanh(s);
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263 r.real = s.imag;
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264 r.imag = -s.real;
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265 return r;
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266 }
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267
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268 /* Windows screws up atan2 for inf and nan, and alpha Tru64 5.1 doesn't follow
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269 C99 for atan2(0., 0.). */
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270 static double
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271 c_atan2(Py_complex z)
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272 {
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273 if (Py_IS_NAN(z.real) || Py_IS_NAN(z.imag))
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274 return Py_NAN;
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275 if (Py_IS_INFINITY(z.imag)) {
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276 if (Py_IS_INFINITY(z.real)) {
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277 if (copysign(1., z.real) == 1.)
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278 /* atan2(+-inf, +inf) == +-pi/4 */
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279 return copysign(0.25*Py_MATH_PI, z.imag);
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280 else
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281 /* atan2(+-inf, -inf) == +-pi*3/4 */
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282 return copysign(0.75*Py_MATH_PI, z.imag);
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283 }
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284 /* atan2(+-inf, x) == +-pi/2 for finite x */
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285 return copysign(0.5*Py_MATH_PI, z.imag);
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286 }
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287 if (Py_IS_INFINITY(z.real) || z.imag == 0.) {
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288 if (copysign(1., z.real) == 1.)
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289 /* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */
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290 return copysign(0., z.imag);
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291 else
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292 /* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */
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293 return copysign(Py_MATH_PI, z.imag);
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294 }
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295 return atan2(z.imag, z.real);
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296 }
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297
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298 PyDoc_STRVAR(c_atan_doc,
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299 "atan(x)\n"
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300 "\n"
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301 "Return the arc tangent of x.");
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302
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303
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304 static Py_complex atanh_special_values[7][7];
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305
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306 static Py_complex
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307 c_atanh(Py_complex z)
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308 {
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309 Py_complex r;
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310 double ay, h;
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311
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312 SPECIAL_VALUE(z, atanh_special_values);
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313
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314 /* Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z). */
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315 if (z.real < 0.) {
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316 return c_neg(c_atanh(c_neg(z)));
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317 }
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318
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319 ay = fabs(z.imag);
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320 if (z.real > CM_SQRT_LARGE_DOUBLE || ay > CM_SQRT_LARGE_DOUBLE) {
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321 /*
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322 if abs(z) is large then we use the approximation
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323 atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign
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324 of z.imag)
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325 */
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326 h = hypot(z.real/2., z.imag/2.); /* safe from overflow */
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327 r.real = z.real/4./h/h;
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328 /* the two negations in the next line cancel each other out
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329 except when working with unsigned zeros: they're there to
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330 ensure that the branch cut has the correct continuity on
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331 systems that don't support signed zeros */
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332 r.imag = -copysign(Py_MATH_PI/2., -z.imag);
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333 errno = 0;
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334 } else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) {
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335 /* C99 standard says: atanh(1+/-0.) should be inf +/- 0i */
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336 if (ay == 0.) {
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337 r.real = INF;
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338 r.imag = z.imag;
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339 errno = EDOM;
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340 } else {
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341 r.real = -log(sqrt(ay)/sqrt(hypot(ay, 2.)));
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342 r.imag = copysign(atan2(2., -ay)/2, z.imag);
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343 errno = 0;
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344 }
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345 } else {
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346 r.real = m_log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.;
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347 r.imag = -atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.;
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348 errno = 0;
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349 }
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350 return r;
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351 }
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352
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353 PyDoc_STRVAR(c_atanh_doc,
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354 "atanh(x)\n"
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355 "\n"
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356 "Return the hyperbolic arc tangent of x.");
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357
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358
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359 static Py_complex
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360 c_cos(Py_complex z)
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361 {
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362 /* cos(z) = cosh(iz) */
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363 Py_complex r;
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364 r.real = -z.imag;
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365 r.imag = z.real;
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366 r = c_cosh(r);
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367 return r;
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368 }
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369
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370 PyDoc_STRVAR(c_cos_doc,
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371 "cos(x)\n"
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372 "\n"
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373 "Return the cosine of x.");
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374
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375
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376 /* cosh(infinity + i*y) needs to be dealt with specially */
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377 static Py_complex cosh_special_values[7][7];
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378
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379 static Py_complex
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380 c_cosh(Py_complex z)
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381 {
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382 Py_complex r;
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383 double x_minus_one;
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384
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385 /* special treatment for cosh(+/-inf + iy) if y is not a NaN */
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386 if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {
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387 if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag) &&
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388 (z.imag != 0.)) {
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389 if (z.real > 0) {
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390 r.real = copysign(INF, cos(z.imag));
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391 r.imag = copysign(INF, sin(z.imag));
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392 }
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393 else {
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394 r.real = copysign(INF, cos(z.imag));
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395 r.imag = -copysign(INF, sin(z.imag));
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396 }
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397 }
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398 else {
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399 r = cosh_special_values[special_type(z.real)]
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400 [special_type(z.imag)];
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401 }
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402 /* need to set errno = EDOM if y is +/- infinity and x is not
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403 a NaN */
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404 if (Py_IS_INFINITY(z.imag) && !Py_IS_NAN(z.real))
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405 errno = EDOM;
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406 else
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407 errno = 0;
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408 return r;
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409 }
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410
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411 if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
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412 /* deal correctly with cases where cosh(z.real) overflows but
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413 cosh(z) does not. */
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414 x_minus_one = z.real - copysign(1., z.real);
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415 r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E;
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416 r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E;
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417 } else {
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418 r.real = cos(z.imag) * cosh(z.real);
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419 r.imag = sin(z.imag) * sinh(z.real);
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420 }
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421 /* detect overflow, and set errno accordingly */
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422 if (Py_IS_INFINITY(r.real) || Py_IS_INFINITY(r.imag))
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423 errno = ERANGE;
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424 else
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425 errno = 0;
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426 return r;
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427 }
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428
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429 PyDoc_STRVAR(c_cosh_doc,
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430 "cosh(x)\n"
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431 "\n"
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432 "Return the hyperbolic cosine of x.");
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433
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434
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435 /* exp(infinity + i*y) and exp(-infinity + i*y) need special treatment for
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436 finite y */
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437 static Py_complex exp_special_values[7][7];
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438
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439 static Py_complex
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440 c_exp(Py_complex z)
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441 {
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442 Py_complex r;
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443 double l;
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444
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445 if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {
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446 if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag)
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447 && (z.imag != 0.)) {
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448 if (z.real > 0) {
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449 r.real = copysign(INF, cos(z.imag));
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450 r.imag = copysign(INF, sin(z.imag));
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451 }
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452 else {
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453 r.real = copysign(0., cos(z.imag));
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454 r.imag = copysign(0., sin(z.imag));
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455 }
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456 }
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457 else {
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458 r = exp_special_values[special_type(z.real)]
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459 [special_type(z.imag)];
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460 }
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461 /* need to set errno = EDOM if y is +/- infinity and x is not
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462 a NaN and not -infinity */
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463 if (Py_IS_INFINITY(z.imag) &&
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464 (Py_IS_FINITE(z.real) ||
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465 (Py_IS_INFINITY(z.real) && z.real > 0)))
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466 errno = EDOM;
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467 else
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468 errno = 0;
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469 return r;
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470 }
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471
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472 if (z.real > CM_LOG_LARGE_DOUBLE) {
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473 l = exp(z.real-1.);
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474 r.real = l*cos(z.imag)*Py_MATH_E;
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475 r.imag = l*sin(z.imag)*Py_MATH_E;
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476 } else {
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477 l = exp(z.real);
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478 r.real = l*cos(z.imag);
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479 r.imag = l*sin(z.imag);
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480 }
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481 /* detect overflow, and set errno accordingly */
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482 if (Py_IS_INFINITY(r.real) || Py_IS_INFINITY(r.imag))
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483 errno = ERANGE;
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484 else
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485 errno = 0;
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486 return r;
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487 }
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488
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489 PyDoc_STRVAR(c_exp_doc,
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490 "exp(x)\n"
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491 "\n"
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492 "Return the exponential value e**x.");
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493
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494
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495 static Py_complex log_special_values[7][7];
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496
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497 static Py_complex
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498 c_log(Py_complex z)
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499 {
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500 /*
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501 The usual formula for the real part is log(hypot(z.real, z.imag)).
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502 There are four situations where this formula is potentially
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503 problematic:
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504
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505 (1) the absolute value of z is subnormal. Then hypot is subnormal,
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506 so has fewer than the usual number of bits of accuracy, hence may
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507 have large relative error. This then gives a large absolute error
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508 in the log. This can be solved by rescaling z by a suitable power
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509 of 2.
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510
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511 (2) the absolute value of z is greater than DBL_MAX (e.g. when both
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512 z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX)
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513 Again, rescaling solves this.
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514
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515 (3) the absolute value of z is close to 1. In this case it's
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516 difficult to achieve good accuracy, at least in part because a
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517 change of 1ulp in the real or imaginary part of z can result in a
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518 change of billions of ulps in the correctly rounded answer.
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519
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520 (4) z = 0. The simplest thing to do here is to call the
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521 floating-point log with an argument of 0, and let its behaviour
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522 (returning -infinity, signaling a floating-point exception, setting
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523 errno, or whatever) determine that of c_log. So the usual formula
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524 is fine here.
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525
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526 */
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527
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528 Py_complex r;
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529 double ax, ay, am, an, h;
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530
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531 SPECIAL_VALUE(z, log_special_values);
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532
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533 ax = fabs(z.real);
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534 ay = fabs(z.imag);
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535
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536 if (ax > CM_LARGE_DOUBLE || ay > CM_LARGE_DOUBLE) {
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537 r.real = log(hypot(ax/2., ay/2.)) + M_LN2;
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538 } else if (ax < DBL_MIN && ay < DBL_MIN) {
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539 if (ax > 0. || ay > 0.) {
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540 /* catch cases where hypot(ax, ay) is subnormal */
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541 r.real = log(hypot(ldexp(ax, DBL_MANT_DIG),
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542 ldexp(ay, DBL_MANT_DIG))) - DBL_MANT_DIG*M_LN2;
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543 }
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544 else {
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545 /* log(+/-0. +/- 0i) */
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546 r.real = -INF;
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547 r.imag = atan2(z.imag, z.real);
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548 errno = EDOM;
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549 return r;
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550 }
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551 } else {
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552 h = hypot(ax, ay);
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553 if (0.71 <= h && h <= 1.73) {
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554 am = ax > ay ? ax : ay; /* max(ax, ay) */
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555 an = ax > ay ? ay : ax; /* min(ax, ay) */
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556 r.real = m_log1p((am-1)*(am+1)+an*an)/2.;
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557 } else {
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558 r.real = log(h);
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559 }
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560 }
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561 r.imag = atan2(z.imag, z.real);
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562 errno = 0;
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563 return r;
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564 }
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565
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566
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567 static Py_complex
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568 c_log10(Py_complex z)
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569 {
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570 Py_complex r;
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571 int errno_save;
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572
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573 r = c_log(z);
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574 errno_save = errno; /* just in case the divisions affect errno */
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575 r.real = r.real / M_LN10;
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576 r.imag = r.imag / M_LN10;
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577 errno = errno_save;
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578 return r;
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579 }
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580
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581 PyDoc_STRVAR(c_log10_doc,
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582 "log10(x)\n"
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583 "\n"
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584 "Return the base-10 logarithm of x.");
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585
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586
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587 static Py_complex
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588 c_sin(Py_complex z)
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589 {
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590 /* sin(z) = -i sin(iz) */
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591 Py_complex s, r;
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592 s.real = -z.imag;
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593 s.imag = z.real;
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594 s = c_sinh(s);
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595 r.real = s.imag;
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596 r.imag = -s.real;
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597 return r;
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598 }
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599
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600 PyDoc_STRVAR(c_sin_doc,
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601 "sin(x)\n"
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602 "\n"
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603 "Return the sine of x.");
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604
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605
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606 /* sinh(infinity + i*y) needs to be dealt with specially */
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607 static Py_complex sinh_special_values[7][7];
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608
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609 static Py_complex
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610 c_sinh(Py_complex z)
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611 {
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612 Py_complex r;
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613 double x_minus_one;
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614
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615 /* special treatment for sinh(+/-inf + iy) if y is finite and
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616 nonzero */
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617 if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {
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618 if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag)
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619 && (z.imag != 0.)) {
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620 if (z.real > 0) {
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621 r.real = copysign(INF, cos(z.imag));
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622 r.imag = copysign(INF, sin(z.imag));
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623 }
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624 else {
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625 r.real = -copysign(INF, cos(z.imag));
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626 r.imag = copysign(INF, sin(z.imag));
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627 }
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628 }
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629 else {
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630 r = sinh_special_values[special_type(z.real)]
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631 [special_type(z.imag)];
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632 }
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633 /* need to set errno = EDOM if y is +/- infinity and x is not
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634 a NaN */
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635 if (Py_IS_INFINITY(z.imag) && !Py_IS_NAN(z.real))
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636 errno = EDOM;
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637 else
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638 errno = 0;
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639 return r;
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640 }
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641
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642 if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
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643 x_minus_one = z.real - copysign(1., z.real);
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644 r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E;
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645 r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E;
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646 } else {
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647 r.real = cos(z.imag) * sinh(z.real);
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648 r.imag = sin(z.imag) * cosh(z.real);
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649 }
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650 /* detect overflow, and set errno accordingly */
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651 if (Py_IS_INFINITY(r.real) || Py_IS_INFINITY(r.imag))
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652 errno = ERANGE;
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653 else
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654 errno = 0;
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655 return r;
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656 }
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657
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658 PyDoc_STRVAR(c_sinh_doc,
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659 "sinh(x)\n"
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660 "\n"
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661 "Return the hyperbolic sine of x.");
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662
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663
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664 static Py_complex sqrt_special_values[7][7];
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665
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666 static Py_complex
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667 c_sqrt(Py_complex z)
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668 {
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669 /*
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670 Method: use symmetries to reduce to the case when x = z.real and y
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671 = z.imag are nonnegative. Then the real part of the result is
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672 given by
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673
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674 s = sqrt((x + hypot(x, y))/2)
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675
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676 and the imaginary part is
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677
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678 d = (y/2)/s
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679
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680 If either x or y is very large then there's a risk of overflow in
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681 computation of the expression x + hypot(x, y). We can avoid this
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682 by rewriting the formula for s as:
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683
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684 s = 2*sqrt(x/8 + hypot(x/8, y/8))
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685
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686 This costs us two extra multiplications/divisions, but avoids the
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687 overhead of checking for x and y large.
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688
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689 If both x and y are subnormal then hypot(x, y) may also be
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690 subnormal, so will lack full precision. We solve this by rescaling
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691 x and y by a sufficiently large power of 2 to ensure that x and y
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692 are normal.
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693 */
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694
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695
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696 Py_complex r;
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697 double s,d;
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698 double ax, ay;
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699
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700 SPECIAL_VALUE(z, sqrt_special_values);
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701
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702 if (z.real == 0. && z.imag == 0.) {
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703 r.real = 0.;
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704 r.imag = z.imag;
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705 return r;
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706 }
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707
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708 ax = fabs(z.real);
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709 ay = fabs(z.imag);
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710
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711 if (ax < DBL_MIN && ay < DBL_MIN && (ax > 0. || ay > 0.)) {
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712 /* here we catch cases where hypot(ax, ay) is subnormal */
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713 ax = ldexp(ax, CM_SCALE_UP);
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714 s = ldexp(sqrt(ax + hypot(ax, ldexp(ay, CM_SCALE_UP))),
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715 CM_SCALE_DOWN);
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716 } else {
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717 ax /= 8.;
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718 s = 2.*sqrt(ax + hypot(ax, ay/8.));
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719 }
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720 d = ay/(2.*s);
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721
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722 if (z.real >= 0.) {
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723 r.real = s;
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724 r.imag = copysign(d, z.imag);
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725 } else {
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726 r.real = d;
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727 r.imag = copysign(s, z.imag);
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728 }
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729 errno = 0;
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730 return r;
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731 }
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
732
23832
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parents: 22722
diff changeset
733 PyDoc_STRVAR(c_sqrt_doc,
16707
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Tim Peters <tim.peters@gmail.com>
parents: 13817
diff changeset
734 "sqrt(x)\n"
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Tim Peters <tim.peters@gmail.com>
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735 "\n"
23832
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diff changeset
736 "Return the square root of x.");
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parents: 9588
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737
f22f769bf3d9 Added Doc strings -- by Chris Petrilli.
Guido van Rossum <guido@python.org>
parents: 9588
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738
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739 static Py_complex
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740 c_tan(Py_complex z)
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741 {
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742 /* tan(z) = -i tanh(iz) */
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743 Py_complex s, r;
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744 s.real = -z.imag;
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745 s.imag = z.real;
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746 s = c_tanh(s);
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747 r.real = s.imag;
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748 r.imag = -s.real;
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749 return r;
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750 }
969bd911e114 complex math
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751
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752 PyDoc_STRVAR(c_tan_doc,
16707
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Tim Peters <tim.peters@gmail.com>
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753 "tan(x)\n"
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Tim Peters <tim.peters@gmail.com>
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754 "\n"
23832
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diff changeset
755 "Return the tangent of x.");
9593
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Guido van Rossum <guido@python.org>
parents: 9588
diff changeset
756
f22f769bf3d9 Added Doc strings -- by Chris Petrilli.
Guido van Rossum <guido@python.org>
parents: 9588
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757
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758 /* tanh(infinity + i*y) needs to be dealt with specially */
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759 static Py_complex tanh_special_values[7][7];
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760
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761 static Py_complex
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762 c_tanh(Py_complex z)
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763 {
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764 /* Formula:
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765
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766 tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) /
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767 (1+tan(y)^2 tanh(x)^2)
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768
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769 To avoid excessive roundoff error, 1-tanh(x)^2 is better computed
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770 as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2
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771 by 4 exp(-2*x) instead, to avoid possible overflow in the
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772 computation of cosh(x).
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773
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774 */
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775
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776 Py_complex r;
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777 double tx, ty, cx, txty, denom;
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778
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diff changeset
779 /* special treatment for tanh(+/-inf + iy) if y is finite and
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diff changeset
780 nonzero */
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diff changeset
781 if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {
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diff changeset
782 if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag)
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diff changeset
783 && (z.imag != 0.)) {
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diff changeset
784 if (z.real > 0) {
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parents: 55539
diff changeset
785 r.real = 1.0;
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diff changeset
786 r.imag = copysign(0.,
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diff changeset
787 2.*sin(z.imag)*cos(z.imag));
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diff changeset
788 }
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diff changeset
789 else {
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diff changeset
790 r.real = -1.0;
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diff changeset
791 r.imag = copysign(0.,
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diff changeset
792 2.*sin(z.imag)*cos(z.imag));
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diff changeset
793 }
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794 }
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diff changeset
795 else {
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diff changeset
796 r = tanh_special_values[special_type(z.real)]
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diff changeset
797 [special_type(z.imag)];
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diff changeset
798 }
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diff changeset
799 /* need to set errno = EDOM if z.imag is +/-infinity and
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diff changeset
800 z.real is finite */
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diff changeset
801 if (Py_IS_INFINITY(z.imag) && Py_IS_FINITE(z.real))
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diff changeset
802 errno = EDOM;
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diff changeset
803 else
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diff changeset
804 errno = 0;
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diff changeset
805 return r;
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diff changeset
806 }
46477
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diff changeset
807
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diff changeset
808 /* danger of overflow in 2.*z.imag !*/
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diff changeset
809 if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
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diff changeset
810 r.real = copysign(1., z.real);
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parents: 55539
diff changeset
811 r.imag = 4.*sin(z.imag)*cos(z.imag)*exp(-2.*fabs(z.real));
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diff changeset
812 } else {
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diff changeset
813 tx = tanh(z.real);
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parents: 55539
diff changeset
814 ty = tan(z.imag);
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parents: 55539
diff changeset
815 cx = 1./cosh(z.real);
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diff changeset
816 txty = tx*ty;
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diff changeset
817 denom = 1. + txty*txty;
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diff changeset
818 r.real = tx*(1.+ty*ty)/denom;
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diff changeset
819 r.imag = ((ty/denom)*cx)*cx;
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diff changeset
820 }
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821 errno = 0;
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diff changeset
822 return r;
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Guido van Rossum <guido@python.org>
parents:
diff changeset
823 }
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
824
23832
ed1ba37a46b7 Patch #568124: Add doc string macros.
Martin v. Löwis <martin@v.loewis.de>
parents: 22722
diff changeset
825 PyDoc_STRVAR(c_tanh_doc,
16707
9696c95d09b7 Mechanical edits just so I can read it.
Tim Peters <tim.peters@gmail.com>
parents: 13817
diff changeset
826 "tanh(x)\n"
9696c95d09b7 Mechanical edits just so I can read it.
Tim Peters <tim.peters@gmail.com>
parents: 13817
diff changeset
827 "\n"
23832
ed1ba37a46b7 Patch #568124: Add doc string macros.
Martin v. Löwis <martin@v.loewis.de>
parents: 22722
diff changeset
828 "Return the hyperbolic tangent of x.");
9593
f22f769bf3d9 Added Doc strings -- by Chris Petrilli.
Guido van Rossum <guido@python.org>
parents: 9588
diff changeset
829
46477
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Christian Heimes <christian@cheimes.de>
parents: 36324
diff changeset
830
32350
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
831 static PyObject *
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
832 cmath_log(PyObject *self, PyObject *args)
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
833 {
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diff changeset
834 Py_complex x;
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diff changeset
835 Py_complex y;
32350
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
836
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diff changeset
837 if (!PyArg_ParseTuple(args, "D|D", &x, &y))
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diff changeset
838 return NULL;
32350
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
839
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840 errno = 0;
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841 PyFPE_START_PROTECT("complex function", return 0)
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842 x = c_log(x);
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843 if (PyTuple_GET_SIZE(args) == 2) {
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diff changeset
844 y = c_log(y);
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845 x = c_quot(x, y);
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846 }
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847 PyFPE_END_PROTECT(x)
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848 if (errno != 0)
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diff changeset
849 return math_error();
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diff changeset
850 return PyComplex_FromCComplex(x);
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Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
851 }
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
852
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
853 PyDoc_STRVAR(cmath_log_doc,
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
854 "log(x[, base]) -> the logarithm of x to the given base.\n\
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
855 If the base not specified, returns the natural logarithm (base e) of x.");
11c31a98dfd6 Patch #826074: cmath.log optional base argument, fixes #823209
Raymond Hettinger <python@rcn.com>
parents: 24471
diff changeset
856
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Guido van Rossum <guido@python.org>
parents:
diff changeset
857
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
858 /* And now the glue to make them available from Python: */
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
859
4639
97ec9dfbb2b6 Renamed Grandly
Roger E. Masse <rmasse@newcnri.cnri.reston.va.us>
parents: 4600
diff changeset
860 static PyObject *
13651
8a063423e0b7 Bunch of minor ANSIfications: 'void initfunc()' -> 'void initfunc(void)',
Thomas Wouters <thomas@python.org>
parents: 13493
diff changeset
861 math_error(void)
3334
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Guido van Rossum <guido@python.org>
parents:
diff changeset
862 {
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diff changeset
863 if (errno == EDOM)
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diff changeset
864 PyErr_SetString(PyExc_ValueError, "math domain error");
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diff changeset
865 else if (errno == ERANGE)
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diff changeset
866 PyErr_SetString(PyExc_OverflowError, "math range error");
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diff changeset
867 else /* Unexpected math error */
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diff changeset
868 PyErr_SetFromErrno(PyExc_ValueError);
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diff changeset
869 return NULL;
3334
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
870 }
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
871
4639
97ec9dfbb2b6 Renamed Grandly
Roger E. Masse <rmasse@newcnri.cnri.reston.va.us>
parents: 4600
diff changeset
872 static PyObject *
13493
8bacf8631a11 ANSI-fication
Peter Schneider-Kamp <nowonder@nowonder.de>
parents: 13447
diff changeset
873 math_1(PyObject *args, Py_complex (*func)(Py_complex))
3334
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parents:
diff changeset
874 {
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diff changeset
875 Py_complex x,r ;
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diff changeset
876 if (!PyArg_ParseTuple(args, "D", &x))
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diff changeset
877 return NULL;
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diff changeset
878 errno = 0;
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879 PyFPE_START_PROTECT("complex function", return 0);
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diff changeset
880 r = (*func)(x);
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diff changeset
881 PyFPE_END_PROTECT(r);
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diff changeset
882 if (errno == EDOM) {
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diff changeset
883 PyErr_SetString(PyExc_ValueError, "math domain error");
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parents: 55539
diff changeset
884 return NULL;
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diff changeset
885 }
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diff changeset
886 else if (errno == ERANGE) {
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parents: 55539
diff changeset
887 PyErr_SetString(PyExc_OverflowError, "math range error");
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diff changeset
888 return NULL;
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diff changeset
889 }
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diff changeset
890 else {
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diff changeset
891 return PyComplex_FromCComplex(r);
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diff changeset
892 }
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parents:
diff changeset
893 }
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
894
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
895 #define FUNC1(stubname, func) \
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diff changeset
896 static PyObject * stubname(PyObject *self, PyObject *args) { \
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diff changeset
897 return math_1(args, func); \
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diff changeset
898 }
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Guido van Rossum <guido@python.org>
parents:
diff changeset
899
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
900 FUNC1(cmath_acos, c_acos)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
901 FUNC1(cmath_acosh, c_acosh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
902 FUNC1(cmath_asin, c_asin)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
903 FUNC1(cmath_asinh, c_asinh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
904 FUNC1(cmath_atan, c_atan)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
905 FUNC1(cmath_atanh, c_atanh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
906 FUNC1(cmath_cos, c_cos)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
907 FUNC1(cmath_cosh, c_cosh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
908 FUNC1(cmath_exp, c_exp)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
909 FUNC1(cmath_log10, c_log10)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
910 FUNC1(cmath_sin, c_sin)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
911 FUNC1(cmath_sinh, c_sinh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
912 FUNC1(cmath_sqrt, c_sqrt)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
913 FUNC1(cmath_tan, c_tan)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
914 FUNC1(cmath_tanh, c_tanh)
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
915
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916 static PyObject *
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917 cmath_phase(PyObject *self, PyObject *args)
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918 {
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919 Py_complex z;
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920 double phi;
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921 if (!PyArg_ParseTuple(args, "D:phase", &z))
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922 return NULL;
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923 errno = 0;
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924 PyFPE_START_PROTECT("arg function", return 0)
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925 phi = c_atan2(z);
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926 PyFPE_END_PROTECT(phi)
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927 if (errno != 0)
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928 return math_error();
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929 else
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930 return PyFloat_FromDouble(phi);
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931 }
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932
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933 PyDoc_STRVAR(cmath_phase_doc,
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934 "phase(z) -> float\n\n\
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935 Return argument, also known as the phase angle, of a complex.");
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936
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937 static PyObject *
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938 cmath_polar(PyObject *self, PyObject *args)
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939 {
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940 Py_complex z;
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941 double r, phi;
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942 if (!PyArg_ParseTuple(args, "D:polar", &z))
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943 return NULL;
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944 PyFPE_START_PROTECT("polar function", return 0)
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945 phi = c_atan2(z); /* should not cause any exception */
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946 r = c_abs(z); /* sets errno to ERANGE on overflow; otherwise 0 */
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947 PyFPE_END_PROTECT(r)
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948 if (errno != 0)
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949 return math_error();
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950 else
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951 return Py_BuildValue("dd", r, phi);
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952 }
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953
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954 PyDoc_STRVAR(cmath_polar_doc,
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955 "polar(z) -> r: float, phi: float\n\n\
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956 Convert a complex from rectangular coordinates to polar coordinates. r is\n\
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957 the distance from 0 and phi the phase angle.");
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958
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959 /*
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960 rect() isn't covered by the C99 standard, but it's not too hard to
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961 figure out 'spirit of C99' rules for special value handing:
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962
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963 rect(x, t) should behave like exp(log(x) + it) for positive-signed x
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964 rect(x, t) should behave like -exp(log(-x) + it) for negative-signed x
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965 rect(nan, t) should behave like exp(nan + it), except that rect(nan, 0)
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966 gives nan +- i0 with the sign of the imaginary part unspecified.
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967
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968 */
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969
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970 static Py_complex rect_special_values[7][7];
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971
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972 static PyObject *
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973 cmath_rect(PyObject *self, PyObject *args)
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974 {
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975 Py_complex z;
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976 double r, phi;
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977 if (!PyArg_ParseTuple(args, "dd:rect", &r, &phi))
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978 return NULL;
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979 errno = 0;
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980 PyFPE_START_PROTECT("rect function", return 0)
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981
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982 /* deal with special values */
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983 if (!Py_IS_FINITE(r) || !Py_IS_FINITE(phi)) {
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984 /* if r is +/-infinity and phi is finite but nonzero then
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985 result is (+-INF +-INF i), but we need to compute cos(phi)
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986 and sin(phi) to figure out the signs. */
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987 if (Py_IS_INFINITY(r) && (Py_IS_FINITE(phi)
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988 && (phi != 0.))) {
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989 if (r > 0) {
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990 z.real = copysign(INF, cos(phi));
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991 z.imag = copysign(INF, sin(phi));
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992 }
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993 else {
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994 z.real = -copysign(INF, cos(phi));
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995 z.imag = -copysign(INF, sin(phi));
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996 }
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997 }
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998 else {
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999 z = rect_special_values[special_type(r)]
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1000 [special_type(phi)];
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1001 }
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1002 /* need to set errno = EDOM if r is a nonzero number and phi
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1003 is infinite */
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1004 if (r != 0. && !Py_IS_NAN(r) && Py_IS_INFINITY(phi))
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1005 errno = EDOM;
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1006 else
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1007 errno = 0;
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1008 }
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1009 else {
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1010 z.real = r * cos(phi);
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1011 z.imag = r * sin(phi);
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1012 errno = 0;
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1013 }
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1014
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1015 PyFPE_END_PROTECT(z)
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1016 if (errno != 0)
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1017 return math_error();
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1018 else
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1019 return PyComplex_FromCComplex(z);
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1020 }
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1021
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1022 PyDoc_STRVAR(cmath_rect_doc,
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1023 "rect(r, phi) -> z: complex\n\n\
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1024 Convert from polar coordinates to rectangular coordinates.");
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1025
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1026 static PyObject *
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1027 cmath_isnan(PyObject *self, PyObject *args)
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1028 {
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1029 Py_complex z;
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1030 if (!PyArg_ParseTuple(args, "D:isnan", &z))
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1031 return NULL;
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1032 return PyBool_FromLong(Py_IS_NAN(z.real) || Py_IS_NAN(z.imag));
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1033 }
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1034
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1035 PyDoc_STRVAR(cmath_isnan_doc,
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1036 "isnan(z) -> bool\n\
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1037 Checks if the real or imaginary part of z not a number (NaN)");
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1038
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1039 static PyObject *
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1040 cmath_isinf(PyObject *self, PyObject *args)
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1041 {
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1042 Py_complex z;
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1043 if (!PyArg_ParseTuple(args, "D:isnan", &z))
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1044 return NULL;
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1045 return PyBool_FromLong(Py_IS_INFINITY(z.real) ||
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1046 Py_IS_INFINITY(z.imag));
46477
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1047 }
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1048
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1049 PyDoc_STRVAR(cmath_isinf_doc,
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1050 "isinf(z) -> bool\n\
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1051 Checks if the real or imaginary part of z is infinite.");
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1052
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Guido van Rossum <guido@python.org>
parents:
diff changeset
1053
23832
ed1ba37a46b7 Patch #568124: Add doc string macros.
Martin v. Löwis <martin@v.loewis.de>
parents: 22722
diff changeset
1054 PyDoc_STRVAR(module_doc,
16707
9696c95d09b7 Mechanical edits just so I can read it.
Tim Peters <tim.peters@gmail.com>
parents: 13817
diff changeset
1055 "This module is always available. It provides access to mathematical\n"
23832
ed1ba37a46b7 Patch #568124: Add doc string macros.
Martin v. Löwis <martin@v.loewis.de>
parents: 22722
diff changeset
1056 "functions for complex numbers.");
9593
f22f769bf3d9 Added Doc strings -- by Chris Petrilli.
Guido van Rossum <guido@python.org>
parents: 9588
diff changeset
1057
4639
97ec9dfbb2b6 Renamed Grandly
Roger E. Masse <rmasse@newcnri.cnri.reston.va.us>
parents: 4600
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1058 static PyMethodDef cmath_methods[] = {
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1059 {"acos", cmath_acos, METH_VARARGS, c_acos_doc},
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1060 {"acosh", cmath_acosh, METH_VARARGS, c_acosh_doc},
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1061 {"asin", cmath_asin, METH_VARARGS, c_asin_doc},
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1062 {"asinh", cmath_asinh, METH_VARARGS, c_asinh_doc},
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1063 {"atan", cmath_atan, METH_VARARGS, c_atan_doc},
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1064 {"atanh", cmath_atanh, METH_VARARGS, c_atanh_doc},
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1065 {"cos", cmath_cos, METH_VARARGS, c_cos_doc},
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1066 {"cosh", cmath_cosh, METH_VARARGS, c_cosh_doc},
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1067 {"exp", cmath_exp, METH_VARARGS, c_exp_doc},
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1068 {"isinf", cmath_isinf, METH_VARARGS, cmath_isinf_doc},
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1069 {"isnan", cmath_isnan, METH_VARARGS, cmath_isnan_doc},
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1070 {"log", cmath_log, METH_VARARGS, cmath_log_doc},
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1071 {"log10", cmath_log10, METH_VARARGS, c_log10_doc},
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1072 {"phase", cmath_phase, METH_VARARGS, cmath_phase_doc},
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1073 {"polar", cmath_polar, METH_VARARGS, cmath_polar_doc},
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1074 {"rect", cmath_rect, METH_VARARGS, cmath_rect_doc},
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1075 {"sin", cmath_sin, METH_VARARGS, c_sin_doc},
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1076 {"sinh", cmath_sinh, METH_VARARGS, c_sinh_doc},
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1077 {"sqrt", cmath_sqrt, METH_VARARGS, c_sqrt_doc},
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1078 {"tan", cmath_tan, METH_VARARGS, c_tan_doc},
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1079 {"tanh", cmath_tanh, METH_VARARGS, c_tanh_doc},
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1080 {NULL, NULL} /* sentinel */
3334
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Guido van Rossum <guido@python.org>
parents:
diff changeset
1081 };
969bd911e114 complex math
Guido van Rossum <guido@python.org>
parents:
diff changeset
1082
24471
f66b3a3509c7 Excise DL_EXPORT/DL_IMPORT from Modules/*. Required adding a prototype
Mark Hammond <mhammond@skippinet.com.au>
parents: 24280
diff changeset
1083 PyMODINIT_FUNC
13651
8a063423e0b7 Bunch of minor ANSIfications: 'void initfunc()' -> 'void initfunc(void)',
Thomas Wouters <thomas@python.org>
parents: 13493
diff changeset
1084 initcmath(void)
3334
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Guido van Rossum <guido@python.org>
parents:
diff changeset
1085 {
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1086 PyObject *m;
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Tim Peters <tim.peters@gmail.com>
parents: 13817
diff changeset
1087
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1088 m = Py_InitModule3("cmath", cmath_methods, module_doc);
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1089 if (m == NULL)
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1090 return;
22722
ba68dda15df0 Use the PyModule_*() API instead of manipulating the module dictionary
Fred Drake <fdrake@acm.org>
parents: 22227
diff changeset
1091
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1092 PyModule_AddObject(m, "pi",
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1093 PyFloat_FromDouble(Py_MATH_PI));
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1094 PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
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1095
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1096 /* initialize special value tables */
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1097
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1098 #define INIT_SPECIAL_VALUES(NAME, BODY) { Py_complex* p = (Py_complex*)NAME; BODY }
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1099 #define C(REAL, IMAG) p->real = REAL; p->imag = IMAG; ++p;
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1100
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1101 INIT_SPECIAL_VALUES(acos_special_values, {
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1102 C(P34,INF) C(P,INF) C(P,INF) C(P,-INF) C(P,-INF) C(P34,-INF) C(N,INF)
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1103 C(P12,INF) C(U,U) C(U,U) C(U,U) C(U,U) C(P12,-INF) C(N,N)
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1104 C(P12,INF) C(U,U) C(P12,0.) C(P12,-0.) C(U,U) C(P12,-INF) C(P12,N)
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1105 C(P12,INF) C(U,U) C(P12,0.) C(P12,-0.) C(U,U) C(P12,-INF) C(P12,N)
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1106 C(P12,INF) C(U,U) C(U,U) C(U,U) C(U,U) C(P12,-INF) C(N,N)
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1107 C(P14,INF) C(0.,INF) C(0.,INF) C(0.,-INF) C(0.,-INF) C(P14,-INF) C(N,INF)
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1108 C(N,INF) C(N,N) C(N,N) C(N,N) C(N,N) C(N,-INF) C(N,N)
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1109 })
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1110
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1111 INIT_SPECIAL_VALUES(acosh_special_values, {
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1112 C(INF,-P34) C(INF,-P) C(INF,-P) C(INF,P) C(INF,P) C(INF,P34) C(INF,N)
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1113 C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
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1114 C(INF,-P12) C(U,U) C(0.,-P12) C(0.,P12) C(U,U) C(INF,P12) C(N,N)
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1115 C(INF,-P12) C(U,U) C(0.,-P12) C(0.,P12) C(U,U) C(INF,P12) C(N,N)
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1116 C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
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1117 C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
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1118 C(INF,N) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,N) C(N,N)
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1119 })
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1120
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1121 INIT_SPECIAL_VALUES(asinh_special_values, {
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1122 C(-INF,-P14) C(-INF,-0.) C(-INF,-0.) C(-INF,0.) C(-INF,0.) C(-INF,P14) C(-INF,N)
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1123 C(-INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(-INF,P12) C(N,N)
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1124 C(-INF,-P12) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(-INF,P12) C(N,N)
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1125 C(INF,-P12) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,P12) C(N,N)
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1126 C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
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1127 C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
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1128 C(INF,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(INF,N) C(N,N)
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1129 })
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1130
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1131 INIT_SPECIAL_VALUES(atanh_special_values, {
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1132 C(-0.,-P12) C(-0.,-P12) C(-0.,-P12) C(-0.,P12) C(-0.,P12) C(-0.,P12) C(-0.,N)
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1133 C(-0.,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(-0.,P12) C(N,N)
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1134 C(-0.,-P12) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(-0.,P12) C(-0.,N)
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1135 C(0.,-P12) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,P12) C(0.,N)
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1136 C(0.,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(0.,P12) C(N,N)
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1137 C(0.,-P12) C(0.,-P12) C(0.,-P12) C(0.,P12) C(0.,P12) C(0.,P12) C(0.,N)
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1138 C(0.,-P12) C(N,N) C(N,N) C(N,N) C(N,N) C(0.,P12) C(N,N)
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1139 })
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1140
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1141 INIT_SPECIAL_VALUES(cosh_special_values, {
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1142 C(INF,N) C(U,U) C(INF,0.) C(INF,-0.) C(U,U) C(INF,N) C(INF,N)
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1143 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1144 C(N,0.) C(U,U) C(1.,0.) C(1.,-0.) C(U,U) C(N,0.) C(N,0.)
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1145 C(N,0.) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,0.) C(N,0.)
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1146 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1147 C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
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1148 C(N,N) C(N,N) C(N,0.) C(N,0.) C(N,N) C(N,N) C(N,N)
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1149 })
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1150
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1151 INIT_SPECIAL_VALUES(exp_special_values, {
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1152 C(0.,0.) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,0.) C(0.,0.)
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1153 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1154 C(N,N) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,N) C(N,N)
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1155 C(N,N) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,N) C(N,N)
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1156 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1157 C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
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1158 C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
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1159 })
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1160
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1161 INIT_SPECIAL_VALUES(log_special_values, {
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1162 C(INF,-P34) C(INF,-P) C(INF,-P) C(INF,P) C(INF,P) C(INF,P34) C(INF,N)
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1163 C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
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1164 C(INF,-P12) C(U,U) C(-INF,-P) C(-INF,P) C(U,U) C(INF,P12) C(N,N)
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1165 C(INF,-P12) C(U,U) C(-INF,-0.) C(-INF,0.) C(U,U) C(INF,P12) C(N,N)
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1166 C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
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1167 C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
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1168 C(INF,N) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,N) C(N,N)
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1169 })
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1170
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1171 INIT_SPECIAL_VALUES(sinh_special_values, {
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1172 C(INF,N) C(U,U) C(-INF,-0.) C(-INF,0.) C(U,U) C(INF,N) C(INF,N)
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1173 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1174 C(0.,N) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(0.,N) C(0.,N)
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1175 C(0.,N) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,N) C(0.,N)
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1176 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1177 C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
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1178 C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
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1179 })
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1180
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1181 INIT_SPECIAL_VALUES(sqrt_special_values, {
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1182 C(INF,-INF) C(0.,-INF) C(0.,-INF) C(0.,INF) C(0.,INF) C(INF,INF) C(N,INF)
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1183 C(INF,-INF) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,INF) C(N,N)
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1184 C(INF,-INF) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,INF) C(N,N)
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1185 C(INF,-INF) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,INF) C(N,N)
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1186 C(INF,-INF) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,INF) C(N,N)
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1187 C(INF,-INF) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,INF) C(INF,N)
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1188 C(INF,-INF) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,INF) C(N,N)
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1189 })
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1190
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1191 INIT_SPECIAL_VALUES(tanh_special_values, {
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1192 C(-1.,0.) C(U,U) C(-1.,-0.) C(-1.,0.) C(U,U) C(-1.,0.) C(-1.,0.)
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1193 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1194 C(N,N) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(N,N) C(N,N)
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1195 C(N,N) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(N,N) C(N,N)
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1196 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1197 C(1.,0.) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(1.,0.) C(1.,0.)
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1198 C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
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1199 })
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1200
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1201 INIT_SPECIAL_VALUES(rect_special_values, {
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1202 C(INF,N) C(U,U) C(-INF,0.) C(-INF,-0.) C(U,U) C(INF,N) C(INF,N)
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1203 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1204 C(0.,0.) C(U,U) C(-0.,0.) C(-0.,-0.) C(U,U) C(0.,0.) C(0.,0.)
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1205 C(0.,0.) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,0.) C(0.,0.)
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1206 C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
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1207 C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
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1208 C(N,N) C(N,N) C(N,0.) C(N,0.) C(N,N) C(N,N) C(N,N)
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1209 })
3334
969bd911e114 complex math
Guido van Rossum <guido@python.org>
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1210 }