from jorg =at= google =dot= com

For low or high values of P, the inverse beta function fails to converge.
Here are some inputs and results (the boost inverse-beta function is
included for context.)

a: 8000, b:2000, p: 0.995,   boost: 0.810189,  gsl:nan
a: 8000, b:2000, p: 0.005,   boost: 0.789585,  gsl:nan
a: 8000, b:2000, p: 0.99995, boost: 0.815275,  gsl:nan
a: 8000, b:2000, p: 5e-05,   boost: 0.78416,   gsl:nan
a: 630,  b:9370, p: 0.995,   boost: 0.0694215, gsl:nan
a: 630,  b:9370, p: 0.99995, boost: 0.0728637, gsl:nan
a: 5000, b:5000, p: 0.995,   boost: 0.512877,  gsl:nan
a: 5000, b:5000, p: 0.005,   boost: 0.487123,  gsl:nan
a: 5000, b:5000, p: 0.99995, boost: 0.519446,  gsl:nan
a: 5000, b:5000, p: 5e-05,   boost: 0.480554,  gsl:nan

As I looked into this bug, what appears to be going on is that the call to
bisect does an early return, thinking that the initial guess is good
enough.  But the early guess is too far away for the rest of the routine to
converge.

If I change these two lines:

  /* Do bisection to get closer */
  x = bisect (x, P, a, b, 0.01, 0.01);

To this:

  /* Do bisection to get closer */
  if (P < 0.1) {
    x = bisect (x, P, a, b, P/10, P/10);
  } else {
    x = bisect (x, P, a, b, 0.01, 0.01);
  }

Then the problem goes away.

Boilerplate follows:
Version number of GSL: 1.16, however the latest sources downloaded from git
appear to have the same problem.
Hardware and OS: HP Z620, Ubuntu Trusty
Compiler used: either gcc or clang, optimized or not.
Short program which exercises the bug:

(Code is in C++11)

#include <iostream>
#include "third_party/gsl/gsl_1_16/cdf/gsl_cdf.h"

int main(int argc, char *argv[]) {
  std::cout << "Hello World\n";
  struct Test {
    double a;
    double b;
    double p;
    double x;
  } test_cases[] = {
                 {8000, 2000, 0.995,   0.810189},
                 {8000, 2000, 0.005,   0.789585},
                 {8000, 2000, 0.99995, 0.815275},
                 {8000, 2000, 0.00005, 0.78416},
                 { 630, 9370, 0.995,   0.0694215},
                 { 630, 9370, 0.99995, 0.0728637},
                 {5000, 5000, 0.995,   0.512877},
                 {5000, 5000, 0.005,   0.487123},
                 {5000, 5000, 0.99995, 0.519446},
                 {5000, 5000, 0.00005, 0.480554},
               };
  // beta function
  for (auto t : test_cases) {
    std::cout << "a=" << t.a << " b=" << t.b << " x=" << t.x << " gsl=" <<
gsl_cdf_beta_P(t.x, t.a, t.b) << "\n";
  }
  // inverse beta function
  for (auto t : test_cases) {
    std::cout << "a=" << t.a << " b=" << t.b << " p=" << t.p << " gsl=" <<
gsl_cdf_beta_Pinv(t.p, t.a, t.b) << "\n";
  }
}