sub ltqnorm ($) {
    #
    # Lower tail quantile for standard normal distribution function.
    #
    # This function returns an approximation of the inverse cumulative
    # standard normal distribution function.  I.e., given P, it returns
    # an approximation to the X satisfying P = Pr{Z <= X} where Z is a
    # random variable from the standard normal distribution.
    #
    # The algorithm uses a minimax approximation by rational functions
    # and the result has a relative error whose absolute value is less
    # than 1.15e-9.
    #
    # Author:      Peter John Acklam
    # Time-stamp:  2000-07-19 18:26:14
    # E-mail:      pjacklam@online.no
    # WWW URL:     http://home.online.no/~pjacklam

    my $p = shift;
    die "input argument must be in (0,1)\n" unless 0 < $p && $p < 1;

    # Coefficients in rational approximations.
    my @a = (-3.969683028665376e+01,  2.209460984245205e+02,
             -2.759285104469687e+02,  1.383577518672690e+02,
             -3.066479806614716e+01,  2.506628277459239e+00);
    my @b = (-5.447609879822406e+01,  1.615858368580409e+02,
             -1.556989798598866e+02,  6.680131188771972e+01,
             -1.328068155288572e+01 );
    my @c = (-7.784894002430293e-03, -3.223964580411365e-01,
             -2.400758277161838e+00, -2.549732539343734e+00,
              4.374664141464968e+00,  2.938163982698783e+00);
    my @d = ( 7.784695709041462e-03,  3.224671290700398e-01,
              2.445134137142996e+00,  3.754408661907416e+00);

    # Define break-points.
    my $plow  = 0.02425;
    my $phigh = 1 - $plow;

    # Rational approximation for lower region:
    if ( $p < $plow ) {
       my $q  = sqrt(-2*log($p));
       return ((((($c[0]*$q+$c[1])*$q+$c[2])*$q+$c[3])*$q+$c[4])*$q+$c[5]) /
               (((($d[0]*$q+$d[1])*$q+$d[2])*$q+$d[3])*$q+1);
    }

    # Rational approximation for upper region:
    if ( $phigh < $p ) {
       my $q  = sqrt(-2*log(1-$p));
       return -((((($c[0]*$q+$c[1])*$q+$c[2])*$q+$c[3])*$q+$c[4])*$q+$c[5]) /
                (((($d[0]*$q+$d[1])*$q+$d[2])*$q+$d[3])*$q+1);
    }

    # Rational approximation for central region:
    my $q = $p - 0.5;
    my $r = $q*$q;
    return ((((($a[0]*$r+$a[1])*$r+$a[2])*$r+$a[3])*$r+$a[4])*$r+$a[5])*$q /
           ((((($b[0]*$r+$b[1])*$r+$b[2])*$r+$b[3])*$r+$b[4])*$r+1);
}