lib/libm/spence.c

   1 /*                                                      spence.c
   2  *
   3  *      Dilogarithm
   4  *
   5  *
   6  *
   7  * SYNOPSIS:
   8  *
   9  * double x, y, spence();
  10  *
  11  * y = spence( x );
  12  *
  13  *
  14  *
  15  * DESCRIPTION:
  16  *
  17  * Computes the integral
  18  *
  19  *                    x
  20  *                    -
  21  *                   | | log t
  22  * spence(x)  =  -   |   ----- dt
  23  *                 | |   t - 1
  24  *                  -
  25  *                  1
  26  *
  27  * for x >= 0.  A rational approximation gives the integral in
  28  * the interval (0.5, 1.5).  Transformation formulas for 1/x
  29  * and 1-x are employed outside the basic expansion range.
  30  *
  31  *
  32  *
  33  * ACCURACY:
  34  *
  35  *                      Relative error:
  36  * arithmetic   domain     # trials      peak         rms
  37  *    IEEE      0,4         30000       3.9e-15     5.4e-16
  38  *    DEC       0,4          3000       2.5e-16     4.5e-17
  39  *
  40  *
  41  */
  42 \f
  43 /*                                                      spence.c */
  44
  45
  46 /*
  47 Cephes Math Library Release 2.8:  June, 2000
  48 Copyright 1985, 1987, 1989, 2000 by Stephen L. Moshier
  49 */
  50
  51 #include "mconf.h"
  52
  53 #ifdef UNK
  54 static double A[8] = {
  55   4.65128586073990045278E-5,
  56   7.31589045238094711071E-3,
  57   1.33847639578309018650E-1,
  58   8.79691311754530315341E-1,
  59   2.71149851196553469920E0,
  60   4.25697156008121755724E0,
  61   3.29771340985225106936E0,
  62   1.00000000000000000126E0,
  63 };
  64 static double B[8] = {
  65   6.90990488912553276999E-4,
  66   2.54043763932544379113E-2,
  67   2.82974860602568089943E-1,
  68   1.41172597751831069617E0,
  69   3.63800533345137075418E0,
  70   5.03278880143316990390E0,
  71   3.54771340985225096217E0,
  72   9.99999999999999998740E-1,
  73 };
  74 #endif
  75 #ifdef DEC
  76 static unsigned short A[32] = {
  77 0034503,0013315,0034120,0157771,
  78 0036357,0135043,0016766,0150637,
  79 0037411,0007533,0005212,0161475,
  80 0040141,0031563,0023217,0120331,
  81 0040455,0104461,0007002,0155522,
  82 0040610,0034434,0065721,0120465,
  83 0040523,0006674,0105671,0054427,
  84 0040200,0000000,0000000,0000000,
  85 };
  86 static unsigned short B[32] = {
  87 0035465,0021626,0032367,0144157,
  88 0036720,0016326,0134431,0000406,
  89 0037620,0161024,0133701,0120766,
  90 0040264,0131557,0152055,0064512,
  91 0040550,0152424,0051166,0034272,
  92 0040641,0006233,0014672,0111572,
  93 0040543,0006674,0105671,0054425,
  94 0040200,0000000,0000000,0000000,
  95 };
  96 #endif
  97 #ifdef IBMPC
  98 static unsigned short A[32] = {
  99 0x1bff,0xa70a,0x62d9,0x3f08,
 100 0xda34,0x63be,0xf744,0x3f7d,
 101 0x5c68,0x6151,0x21eb,0x3fc1,
 102 0xf41b,0x64d1,0x266e,0x3fec,
 103 0x5b6a,0x21c0,0xb126,0x4005,
 104 0x3427,0x8d7a,0x0723,0x4011,
 105 0x2b23,0x9177,0x61b7,0x400a,
 106 0x0000,0x0000,0x0000,0x3ff0,
 107 };
 108 static unsigned short B[32] = {
 109 0xf90e,0xc69e,0xa472,0x3f46,
 110 0x2021,0xd723,0x039a,0x3f9a,
 111 0x343f,0x96f8,0x1c42,0x3fd2,
 112 0xad29,0xfa85,0x966d,0x3ff6,
 113 0xc717,0x8a4e,0x1aa2,0x400d,
 114 0x526f,0x6337,0x2193,0x4014,
 115 0x2b23,0x9177,0x61b7,0x400c,
 116 0x0000,0x0000,0x0000,0x3ff0,
 117 };
 118 #endif
 119 #ifdef MIEEE
 120 static unsigned short A[32] = {
 121 0x3f08,0x62d9,0xa70a,0x1bff,
 122 0x3f7d,0xf744,0x63be,0xda34,
 123 0x3fc1,0x21eb,0x6151,0x5c68,
 124 0x3fec,0x266e,0x64d1,0xf41b,
 125 0x4005,0xb126,0x21c0,0x5b6a,
 126 0x4011,0x0723,0x8d7a,0x3427,
 127 0x400a,0x61b7,0x9177,0x2b23,
 128 0x3ff0,0x0000,0x0000,0x0000,
 129 };
 130 static unsigned short B[32] = {
 131 0x3f46,0xa472,0xc69e,0xf90e,
 132 0x3f9a,0x039a,0xd723,0x2021,
 133 0x3fd2,0x1c42,0x96f8,0x343f,
 134 0x3ff6,0x966d,0xfa85,0xad29,
 135 0x400d,0x1aa2,0x8a4e,0xc717,
 136 0x4014,0x2193,0x6337,0x526f,
 137 0x400c,0x61b7,0x9177,0x2b23,
 138 0x3ff0,0x0000,0x0000,0x0000,
 139 };
 140 #endif
 141
 142 #ifdef ANSIPROT
 143 extern double fabs ( double );
 144 extern double log ( double );
 145 extern double polevl ( double, void *, int );
 146 #else
 147 double fabs(), log(), polevl();
 148 #endif
 149 extern double PI, MACHEP;
 150
 151 double spence(x)
 152 double x;
 153 {
 154 double w, y, z;
 155 int flag;
 156
 157 if( x < 0.0 )
 158         {
 159         mtherr( "spence", DOMAIN );
 160         return(0.0);
 161         }
 162
 163 if( x == 1.0 )
 164         return( 0.0 );
 165
 166 if( x == 0.0 )
 167         return( PI*PI/6.0 );
 168
 169 flag = 0;
 170
 171 if( x > 2.0 )
 172         {
 173         x = 1.0/x;
 174         flag |= 2;
 175         }
 176
 177 if( x > 1.5 )
 178         {
 179         w = (1.0/x) - 1.0;
 180         flag |= 2;
 181         }
 182
 183 else if( x < 0.5 )
 184         {
 185         w = -x;
 186         flag |= 1;
 187         }
 188
 189 else
 190         w = x - 1.0;
 191
 192
 193 y = -w * polevl( w, A, 7) / polevl( w, B, 7 );
 194
 195 if( flag & 1 )
 196         y = (PI * PI)/6.0  - log(x) * log(1.0-x) - y;
 197
 198 if( flag & 2 )
 199         {
 200         z = log(x);
 201         y = -0.5 * z * z  -  y;
 202         }
 203
 204 return( y );
 205 }