lib/libm/ndtri.c

   1 /*                                                      ndtri.c
   2  *
   3  *      Inverse of Normal distribution function
   4  *
   5  *
   6  *
   7  * SYNOPSIS:
   8  *
   9  * double x, y, ndtri();
  10  *
  11  * x = ndtri( y );
  12  *
  13  *
  14  *
  15  * DESCRIPTION:
  16  *
  17  * Returns the argument, x, for which the area under the
  18  * Gaussian probability density function (integrated from
  19  * minus infinity to x) is equal to y.
  20  *
  21  *
  22  * For small arguments 0 < y < exp(-2), the program computes
  23  * z = sqrt( -2.0 * log(y) );  then the approximation is
  24  * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
  25  * There are two rational functions P/Q, one for 0 < y < exp(-32)
  26  * and the other for y up to exp(-2).  For larger arguments,
  27  * w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
  28  *
  29  *
  30  * ACCURACY:
  31  *
  32  *                      Relative error:
  33  * arithmetic   domain        # trials      peak         rms
  34  *    DEC      0.125, 1         5500       9.5e-17     2.1e-17
  35  *    DEC      6e-39, 0.135     3500       5.7e-17     1.3e-17
  36  *    IEEE     0.125, 1        20000       7.2e-16     1.3e-16
  37  *    IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17
  38  *
  39  *
  40  * ERROR MESSAGES:
  41  *
  42  *   message         condition    value returned
  43  * ndtri domain       x <= 0        -MAXNUM
  44  * ndtri domain       x >= 1         MAXNUM
  45  *
  46  */
  47 \f
  48
  49 /*
  50 Cephes Math Library Release 2.8:  June, 2000
  51 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
  52 */
  53
  54 #include "mconf.h"
  55 extern double MAXNUM;
  56
  57 #ifdef UNK
  58 /* sqrt(2pi) */
  59 static double s2pi = 2.50662827463100050242E0;
  60 #endif
  61
  62 #ifdef DEC
  63 static unsigned short s2p[] = {0040440,0066230,0177661,0034055};
  64 #define s2pi *(double *)s2p
  65 #endif
  66
  67 #ifdef IBMPC
  68 static unsigned short s2p[] = {0x2706,0x1ff6,0x0d93,0x4004};
  69 #define s2pi *(double *)s2p
  70 #endif
  71
  72 #ifdef MIEEE
  73 static unsigned short s2p[] = {
  74 0x4004,0x0d93,0x1ff6,0x2706
  75 };
  76 #define s2pi *(double *)s2p
  77 #endif
  78
  79 /* approximation for 0 <= |y - 0.5| <= 3/8 */
  80 #ifdef UNK
  81 static double P0[5] = {
  82 -5.99633501014107895267E1,
  83  9.80010754185999661536E1,
  84 -5.66762857469070293439E1,
  85  1.39312609387279679503E1,
  86 -1.23916583867381258016E0,
  87 };
  88 static double Q0[8] = {
  89 /* 1.00000000000000000000E0,*/
  90  1.95448858338141759834E0,
  91  4.67627912898881538453E0,
  92  8.63602421390890590575E1,
  93 -2.25462687854119370527E2,
  94  2.00260212380060660359E2,
  95 -8.20372256168333339912E1,
  96  1.59056225126211695515E1,
  97 -1.18331621121330003142E0,
  98 };
  99 #endif
 100 #ifdef DEC
 101 static unsigned short P0[20] = {
 102 0141557,0155170,0071360,0120550,
 103 0041704,0000214,0172417,0067307,
 104 0141542,0132204,0040066,0156723,
 105 0041136,0163161,0157276,0007747,
 106 0140236,0116374,0073666,0051764,
 107 };
 108 static unsigned short Q0[32] = {
 109 /*0040200,0000000,0000000,0000000,*/
 110 0040372,0026256,0110403,0123707,
 111 0040625,0122024,0020277,0026661,
 112 0041654,0134161,0124134,0007244,
 113 0142141,0073162,0133021,0131371,
 114 0042110,0041235,0043516,0057767,
 115 0141644,0011417,0036155,0137305,
 116 0041176,0076556,0004043,0125430,
 117 0140227,0073347,0152776,0067251,
 118 };
 119 #endif
 120 #ifdef IBMPC
 121 static unsigned short P0[20] = {
 122 0x142d,0x0e5e,0xfb4f,0xc04d,
 123 0xedd9,0x9ea1,0x8011,0x4058,
 124 0xdbba,0x8806,0x5690,0xc04c,
 125 0xc1fd,0x3bd7,0xdcce,0x402b,
 126 0xca7e,0x8ef6,0xd39f,0xbff3,
 127 };
 128 static unsigned short Q0[36] = {
 129 /*0x0000,0x0000,0x0000,0x3ff0,*/
 130 0x74f9,0xd220,0x4595,0x3fff,
 131 0xe5b6,0x8417,0xb482,0x4012,
 132 0x81d4,0x350b,0x970e,0x4055,
 133 0x365f,0x56c2,0x2ece,0xc06c,
 134 0xcbff,0xa8e9,0x0853,0x4069,
 135 0xb7d9,0xe78d,0x8261,0xc054,
 136 0x7563,0xc104,0xcfad,0x402f,
 137 0xcdd5,0xfabf,0xeedc,0xbff2,
 138 };
 139 #endif
 140 #ifdef MIEEE
 141 static unsigned short P0[20] = {
 142 0xc04d,0xfb4f,0x0e5e,0x142d,
 143 0x4058,0x8011,0x9ea1,0xedd9,
 144 0xc04c,0x5690,0x8806,0xdbba,
 145 0x402b,0xdcce,0x3bd7,0xc1fd,
 146 0xbff3,0xd39f,0x8ef6,0xca7e,
 147 };
 148 static unsigned short Q0[32] = {
 149 /*0x3ff0,0x0000,0x0000,0x0000,*/
 150 0x3fff,0x4595,0xd220,0x74f9,
 151 0x4012,0xb482,0x8417,0xe5b6,
 152 0x4055,0x970e,0x350b,0x81d4,
 153 0xc06c,0x2ece,0x56c2,0x365f,
 154 0x4069,0x0853,0xa8e9,0xcbff,
 155 0xc054,0x8261,0xe78d,0xb7d9,
 156 0x402f,0xcfad,0xc104,0x7563,
 157 0xbff2,0xeedc,0xfabf,0xcdd5,
 158 };
 159 #endif
 160
 161
 162 /* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
 163  * i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
 164  */
 165 #ifdef UNK
 166 static double P1[9] = {
 167  4.05544892305962419923E0,
 168  3.15251094599893866154E1,
 169  5.71628192246421288162E1,
 170  4.40805073893200834700E1,
 171  1.46849561928858024014E1,
 172  2.18663306850790267539E0,
 173 -1.40256079171354495875E-1,
 174 -3.50424626827848203418E-2,
 175 -8.57456785154685413611E-4,
 176 };
 177 static double Q1[8] = {
 178 /*  1.00000000000000000000E0,*/
 179  1.57799883256466749731E1,
 180  4.53907635128879210584E1,
 181  4.13172038254672030440E1,
 182  1.50425385692907503408E1,
 183  2.50464946208309415979E0,
 184 -1.42182922854787788574E-1,
 185 -3.80806407691578277194E-2,
 186 -9.33259480895457427372E-4,
 187 };
 188 #endif
 189 #ifdef DEC
 190 static unsigned short P1[36] = {
 191 0040601,0143074,0150744,0073326,
 192 0041374,0031554,0113253,0146016,
 193 0041544,0123272,0012463,0176771,
 194 0041460,0051160,0103560,0156511,
 195 0041152,0172624,0117772,0030755,
 196 0040413,0170713,0151545,0176413,
 197 0137417,0117512,0022154,0131671,
 198 0137017,0104257,0071432,0007072,
 199 0135540,0143363,0063137,0036166,
 200 };
 201 static unsigned short Q1[32] = {
 202 /*0040200,0000000,0000000,0000000,*/
 203 0041174,0075325,0004736,0120326,
 204 0041465,0110044,0047561,0045567,
 205 0041445,0042321,0012142,0030340,
 206 0041160,0127074,0166076,0141051,
 207 0040440,0046055,0040745,0150400,
 208 0137421,0114146,0067330,0010621,
 209 0137033,0175162,0025555,0114351,
 210 0135564,0122773,0145750,0030357,
 211 };
 212 #endif
 213 #ifdef IBMPC
 214 static unsigned short P1[36] = {
 215 0x8edb,0x9a3c,0x38c7,0x4010,
 216 0x7982,0x92d5,0x866d,0x403f,
 217 0x7fbf,0x42a6,0x94d7,0x404c,
 218 0x1ba9,0x10ee,0x0a4e,0x4046,
 219 0x463e,0x93ff,0x5eb2,0x402d,
 220 0xbfa1,0x7a6c,0x7e39,0x4001,
 221 0x9677,0x448d,0xf3e9,0xbfc1,
 222 0x41c7,0xee63,0xf115,0xbfa1,
 223 0xe78f,0x6ccb,0x18de,0xbf4c,
 224 };
 225 static unsigned short Q1[32] = {
 226 /*0x0000,0x0000,0x0000,0x3ff0,*/
 227 0xd41b,0xa13b,0x8f5a,0x402f,
 228 0x296f,0x89ee,0xb204,0x4046,
 229 0x461c,0x228c,0xa89a,0x4044,
 230 0xd845,0x9d87,0x15c7,0x402e,
 231 0xba20,0xa83c,0x0985,0x4004,
 232 0x0232,0xcddb,0x330c,0xbfc2,
 233 0xb31d,0x456d,0x7f4e,0xbfa3,
 234 0x061e,0x797d,0x94bf,0xbf4e,
 235 };
 236 #endif
 237 #ifdef MIEEE
 238 static unsigned short P1[36] = {
 239 0x4010,0x38c7,0x9a3c,0x8edb,
 240 0x403f,0x866d,0x92d5,0x7982,
 241 0x404c,0x94d7,0x42a6,0x7fbf,
 242 0x4046,0x0a4e,0x10ee,0x1ba9,
 243 0x402d,0x5eb2,0x93ff,0x463e,
 244 0x4001,0x7e39,0x7a6c,0xbfa1,
 245 0xbfc1,0xf3e9,0x448d,0x9677,
 246 0xbfa1,0xf115,0xee63,0x41c7,
 247 0xbf4c,0x18de,0x6ccb,0xe78f,
 248 };
 249 static unsigned short Q1[32] = {
 250 /*0x3ff0,0x0000,0x0000,0x0000,*/
 251 0x402f,0x8f5a,0xa13b,0xd41b,
 252 0x4046,0xb204,0x89ee,0x296f,
 253 0x4044,0xa89a,0x228c,0x461c,
 254 0x402e,0x15c7,0x9d87,0xd845,
 255 0x4004,0x0985,0xa83c,0xba20,
 256 0xbfc2,0x330c,0xcddb,0x0232,
 257 0xbfa3,0x7f4e,0x456d,0xb31d,
 258 0xbf4e,0x94bf,0x797d,0x061e,
 259 };
 260 #endif
 261
 262 /* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
 263  * i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
 264  */
 265
 266 #ifdef UNK
 267 static double P2[9] = {
 268   3.23774891776946035970E0,
 269   6.91522889068984211695E0,
 270   3.93881025292474443415E0,
 271   1.33303460815807542389E0,
 272   2.01485389549179081538E-1,
 273   1.23716634817820021358E-2,
 274   3.01581553508235416007E-4,
 275   2.65806974686737550832E-6,
 276   6.23974539184983293730E-9,
 277 };
 278 static double Q2[8] = {
 279 /*  1.00000000000000000000E0,*/
 280   6.02427039364742014255E0,
 281   3.67983563856160859403E0,
 282   1.37702099489081330271E0,
 283   2.16236993594496635890E-1,
 284   1.34204006088543189037E-2,
 285   3.28014464682127739104E-4,
 286   2.89247864745380683936E-6,
 287   6.79019408009981274425E-9,
 288 };
 289 #endif
 290 #ifdef DEC
 291 static unsigned short P2[36] = {
 292 0040517,0033507,0036236,0125641,
 293 0040735,0044616,0014473,0140133,
 294 0040574,0012567,0114535,0102541,
 295 0040252,0120340,0143474,0150135,
 296 0037516,0051057,0115361,0031211,
 297 0036512,0131204,0101511,0125144,
 298 0035236,0016627,0043160,0140216,
 299 0033462,0060512,0060141,0010641,
 300 0031326,0062541,0101304,0077706,
 301 };
 302 static unsigned short Q2[32] = {
 303 /*0040200,0000000,0000000,0000000,*/
 304 0040700,0143322,0132137,0040501,
 305 0040553,0101155,0053221,0140257,
 306 0040260,0041071,0052573,0010004,
 307 0037535,0066472,0177261,0162330,
 308 0036533,0160475,0066666,0036132,
 309 0035253,0174533,0027771,0044027,
 310 0033502,0016147,0117666,0063671,
 311 0031351,0047455,0141663,0054751,
 312 };
 313 #endif
 314 #ifdef IBMPC
 315 static unsigned short P2[36] = {
 316 0xd574,0xe793,0xe6e8,0x4009,
 317 0x780b,0xc327,0xa931,0x401b,
 318 0xb0ac,0xf32b,0x82ae,0x400f,
 319 0x9a0c,0x18e7,0x541c,0x3ff5,
 320 0x2651,0xf35e,0xca45,0x3fc9,
 321 0x354d,0x9069,0x5650,0x3f89,
 322 0x1812,0xe8ce,0xc3b2,0x3f33,
 323 0x2234,0x4c0c,0x4c29,0x3ec6,
 324 0x8ff9,0x3058,0xccac,0x3e3a,
 325 };
 326 static unsigned short Q2[32] = {
 327 /*0x0000,0x0000,0x0000,0x3ff0,*/
 328 0xe828,0x568b,0x18da,0x4018,
 329 0x3816,0xaad2,0x704d,0x400d,
 330 0x6200,0x2aaf,0x0847,0x3ff6,
 331 0x3c9b,0x5fd6,0xada7,0x3fcb,
 332 0xc78b,0xadb6,0x7c27,0x3f8b,
 333 0x2903,0x65ff,0x7f2b,0x3f35,
 334 0xccf7,0xf3f6,0x438c,0x3ec8,
 335 0x6b3d,0xb876,0x29e5,0x3e3d,
 336 };
 337 #endif
 338 #ifdef MIEEE
 339 static unsigned short P2[36] = {
 340 0x4009,0xe6e8,0xe793,0xd574,
 341 0x401b,0xa931,0xc327,0x780b,
 342 0x400f,0x82ae,0xf32b,0xb0ac,
 343 0x3ff5,0x541c,0x18e7,0x9a0c,
 344 0x3fc9,0xca45,0xf35e,0x2651,
 345 0x3f89,0x5650,0x9069,0x354d,
 346 0x3f33,0xc3b2,0xe8ce,0x1812,
 347 0x3ec6,0x4c29,0x4c0c,0x2234,
 348 0x3e3a,0xccac,0x3058,0x8ff9,
 349 };
 350 static unsigned short Q2[32] = {
 351 /*0x3ff0,0x0000,0x0000,0x0000,*/
 352 0x4018,0x18da,0x568b,0xe828,
 353 0x400d,0x704d,0xaad2,0x3816,
 354 0x3ff6,0x0847,0x2aaf,0x6200,
 355 0x3fcb,0xada7,0x5fd6,0x3c9b,
 356 0x3f8b,0x7c27,0xadb6,0xc78b,
 357 0x3f35,0x7f2b,0x65ff,0x2903,
 358 0x3ec8,0x438c,0xf3f6,0xccf7,
 359 0x3e3d,0x29e5,0xb876,0x6b3d,
 360 };
 361 #endif
 362
 363 #ifdef ANSIPROT
 364 extern double polevl ( double, void *, int );
 365 extern double p1evl ( double, void *, int );
 366 extern double log ( double );
 367 extern double sqrt ( double );
 368 #else
 369 double polevl(), p1evl(), log(), sqrt();
 370 #endif
 371
 372 double ndtri(y0)
 373 double y0;
 374 {
 375 double x, y, z, y2, x0, x1;
 376 int code;
 377
 378 if( y0 <= 0.0 )
 379         {
 380         mtherr( "ndtri", DOMAIN );
 381         return( -MAXNUM );
 382         }
 383 if( y0 >= 1.0 )
 384         {
 385         mtherr( "ndtri", DOMAIN );
 386         return( MAXNUM );
 387         }
 388 code = 1;
 389 y = y0;
 390 if( y > (1.0 - 0.13533528323661269189) ) /* 0.135... = exp(-2) */
 391         {
 392         y = 1.0 - y;
 393         code = 0;
 394         }
 395
 396 if( y > 0.13533528323661269189 )
 397         {
 398         y = y - 0.5;
 399         y2 = y * y;
 400         x = y + y * (y2 * polevl( y2, P0, 4)/p1evl( y2, Q0, 8 ));
 401         x = x * s2pi;
 402         return(x);
 403         }
 404
 405 x = sqrt( -2.0 * log(y) );
 406 x0 = x - log(x)/x;
 407
 408 z = 1.0/x;
 409 if( x < 8.0 ) /* y > exp(-32) = 1.2664165549e-14 */
 410         x1 = z * polevl( z, P1, 8 )/p1evl( z, Q1, 8 );
 411 else
 412         x1 = z * polevl( z, P2, 8 )/p1evl( z, Q2, 8 );
 413 x = x0 - x1;
 414 if( code != 0 )
 415         x = -x;
 416 return( x );
 417 }