ndtri.c

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/*                                                     ndtri.c
 *
 *     Inverse of Normal distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, ndtri();
 *
 * x = ndtri( y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the argument, x, for which the area under the
 * Gaussian probability density function (integrated from
 * minus infinity to x) is equal to y.
 *
 *
 * For small arguments 0 < y < exp(-2), the program computes
 * z = sqrt( -2.0 * log(y) );  then the approximation is
 * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
 * There are two rational functions P/Q, one for 0 < y < exp(-32)
 * and the other for y up to exp(-2).  For larger arguments,
 * w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain        # trials      peak         rms
 *    IEEE     0.125, 1        20000       7.2e-16     1.3e-16
 *    IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17
 *
 *
 * ERROR MESSAGES:
 *
 *   message         condition    value returned
 * ndtri domain       x < 0        NAN
 * ndtri domain       x > 1        NAN
 *
 */

 
/*
 * Cephes Math Library Release 2.1:  January, 1989
 * Copyright 1984, 1987, 1989 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */
 
#include "mconf.h"
 
/* sqrt(2pi) */
static double s2pi = 2.50662827463100050242E0;
 
/* approximation for 0 <= |y - 0.5| <= 3/8 */
static double P0[5] = {
    -5.99633501014107895267E1,
    9.80010754185999661536E1,
    -5.66762857469070293439E1,
    1.39312609387279679503E1,
    -1.23916583867381258016E0,
};
 
static double Q0[8] = {
    /* 1.00000000000000000000E0, */
    1.95448858338141759834E0,
    4.67627912898881538453E0,
    8.63602421390890590575E1,
    -2.25462687854119370527E2,
    2.00260212380060660359E2,
    -8.20372256168333339912E1,
    1.59056225126211695515E1,
    -1.18331621121330003142E0,
};
 
/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
 * i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
 */
static double P1[9] = {
    4.05544892305962419923E0,
    3.15251094599893866154E1,
    5.71628192246421288162E1,
    4.40805073893200834700E1,
    1.46849561928858024014E1,
    2.18663306850790267539E0,
    -1.40256079171354495875E-1,
    -3.50424626827848203418E-2,
    -8.57456785154685413611E-4,
};
 
static double Q1[8] = {
    /*  1.00000000000000000000E0, */
    1.57799883256466749731E1,
    4.53907635128879210584E1,
    4.13172038254672030440E1,
    1.50425385692907503408E1,
    2.50464946208309415979E0,
    -1.42182922854787788574E-1,
    -3.80806407691578277194E-2,
    -9.33259480895457427372E-4,
};
 
/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
 * i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
 */
 
static double P2[9] = {
    3.23774891776946035970E0,
    6.91522889068984211695E0,
    3.93881025292474443415E0,
    1.33303460815807542389E0,
    2.01485389549179081538E-1,
    1.23716634817820021358E-2,
    3.01581553508235416007E-4,
    2.65806974686737550832E-6,
    6.23974539184983293730E-9,
};
 
static double Q2[8] = {
    /*  1.00000000000000000000E0, */
    6.02427039364742014255E0,
    3.67983563856160859403E0,
    1.37702099489081330271E0,
    2.16236993594496635890E-1,
    1.34204006088543189037E-2,
    3.28014464682127739104E-4,
    2.89247864745380683936E-6,
    6.79019408009981274425E-9,
};
 
double ndtri(double y0)
{
    double x, y, z, y2, x0, x1;
    int code;
 
    if (y0 == 0.0) {
        return -INFINITY;
    }
    if (y0 == 1.0) {
        return INFINITY;
    }
    if (y0 < 0.0 || y0 > 1.0) {
        sf_error("ndtri", SF_ERROR_DOMAIN, NULL);
        return NAN;
    }
    code = 1;
    y = y0;
    if (y > (1.0 - 0.13533528323661269189)) {   /* 0.135... = exp(-2) */
        y = 1.0 - y;
        code = 0;
    }
 
    if (y > 0.13533528323661269189) {
        y = y - 0.5;
        y2 = y * y;
        x = y + y * (y2 * polevl(y2, P0, 4) / p1evl(y2, Q0, 8));
        x = x * s2pi;
        return (x);
    }
 
    x = sqrt(-2.0 * log(y));
    x0 = x - log(x) / x;
 
    z = 1.0 / x;
    if (x < 8.0)                /* y > exp(-32) = 1.2664165549e-14 */
        x1 = z * polevl(z, P1, 8) / p1evl(z, Q1, 8);
    else
        x1 = z * polevl(z, P2, 8) / p1evl(z, Q2, 8);
    x = x0 - x1;
    if (code != 0)
        x = -x;
    return (x);
}