tor-parallel-relay-conn/src/test/test_prob_distr.c

/* Copyright (c) 2018-2019, The Tor Project, Inc. */
/* See LICENSE for licensing information */

/**
 * \file test_prob_distr.c
 * \brief Test probability distributions.
 * \detail
 *
 * For each probability distribution we do two kinds of tests:
 *
 * a) We do numerical deterministic testing of their cdf/icdf/sf/isf functions
 *    and the various relationships between them for each distribution. We also
 *    do deterministic tests on their sampling functions. Test vectors for
 *    these tests were computed from alternative implementations and were
 *    eyeballed to make sure they make sense
 *    (e.g. src/test/prob_distr_mpfr_ref.c computes logit(p) using GNU mpfr
 *    with 200-bit precision and is then tested in test_logit_logistic()).
 *
 * b) We do stochastic hypothesis testing (G-test) to ensure that sampling from
 *    the given distributions is distributed properly. The stochastic tests are
 *    slow and their false positive rate is not well suited for CI, so they are
 *    currently disabled-by-default and put into 'tests-slow'.
 */

#define PROB_DISTR_PRIVATE

#include "orconfig.h"

#include "test/test.h"

#include "core/or/or.h"

#include "lib/math/prob_distr.h"
#include "lib/math/fp.h"
#include "lib/crypt_ops/crypto_rand.h"
#include "test/rng_test_helpers.h"

#include <float.h>
#include <math.h>
#include <stdbool.h>
#include <stddef.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>

/**
 * Return floor(d) converted to size_t, as a workaround for complaints
 * under -Wbad-function-cast for (size_t)floor(d).
 */
static size_t
floor_to_size_t(double d)
{
  double integral_d = floor(d);
  return (size_t)integral_d;
}

/**
 * Return ceil(d) converted to size_t, as a workaround for complaints
 * under -Wbad-function-cast for (size_t)ceil(d).
 */
static size_t
ceil_to_size_t(double d)
{
  double integral_d = ceil(d);
  return (size_t)integral_d;
}

/*
 * Geometric(p) distribution, supported on {1, 2, 3, ...}.
 *
 * Compute the probability mass function Geom(n; p) of the number of
 * trials before the first success when success has probability p.
 */
static double
logpmf_geometric(unsigned n, double p)
{
  /* This is actually a check against 1, but we do >= so that the compiler
     does not raise a -Wfloat-equal */
  if (p >= 1) {
    if (n == 1)
      return 0;
    else
      return -HUGE_VAL;
  }
  return (n - 1)*log1p(-p) + log(p);
}

/**
 * Compute the logistic function, translated in output by 1/2:
 * logistichalf(x) = logistic(x) - 1/2.  Well-conditioned on the entire
 * real plane, with maximum condition number 1 at 0.
 *
 * This implementation gives relative error bounded by 5 eps.
 */
static double
logistichalf(double x)
{
  /*
   * Rewrite this with the identity
   *
   *  1/(1 + e^{-x}) - 1/2
   *  = (1 - 1/2 - e^{-x}/2)/(1 + e^{-x})
   *  = (1/2 - e^{-x}/2)/(1 + e^{-x})
   *  = (1 - e^{-x})/[2 (1 + e^{-x})]
   *  = -(e^{-x} - 1)/[2 (1 + e^{-x})],
   *
   * which we can evaluate by -expm1(-x)/[2 (1 + exp(-x))].
   *
   * Suppose exp has error d0, + has error d1, expm1 has error
   * d2, and / has error d3, so we evaluate
   *
   *  -(1 + d2) (1 + d3) (e^{-x} - 1)
   *    / [2 (1 + d1) (1 + (1 + d0) e^{-x})].
   *
   * In the denominator,
   *
   *  1 + (1 + d0) e^{-x}
   *  = 1 + e^{-x} + d0 e^{-x}
   *  = (1 + e^{-x}) (1 + d0 e^{-x}/(1 + e^{-x})),
   *
   * so the relative error of the numerator is
   *
   *  d' = d2 + d3 + d2 d3,
   * and of the denominator,
   *  d'' = d1 + d0 e^{-x}/(1 + e^{-x}) + d0 d1 e^{-x}/(1 + e^{-x})
   *      = d1 + d0 L(-x) + d0 d1 L(-x),
   *
   * where L(-x) is logistic(-x).  By Lemma 1 the relative error
   * of the quotient is bounded by
   *
   *  2|d2 + d3 + d2 d3 - d1 - d0 L(x) + d0 d1 L(x)|,
   *
   * Since 0 < L(x) < 1, this is bounded by
   *
   *  2|d2| + 2|d3| + 2|d2 d3| + 2|d1| + 2|d0| + 2|d0 d1|
   *  <= 4 eps + 2 eps^2.
   */
  if (x < log(DBL_EPSILON/8)) {
    /*
     * Avoid overflow in e^{-x}.  When x < log(eps/4), we
     * we further have x < logit(eps/4), so that
     * logistic(x) < eps/4.  Hence the relative error of
     * logistic(x) - 1/2 from -1/2 is bounded by eps/2, and
     * so the relative error of -1/2 from logistic(x) - 1/2
     * is bounded by eps.
     */
    return -0.5;
  } else {
    return -expm1(-x)/(2*(1 + exp(-x)));
  }
}

/**
 * Compute the log of the sum of the exps.  Caller should arrange the
 * array in descending order to minimize error because I don't want to
 * deal with using temporary space and the one caller in this file
 * arranges that anyway.
 *
 * Warning: This implementation does not handle infinite or NaN inputs
 * sensibly, because I don't need that here at the moment.  (NaN, or
 * -inf and +inf together, should yield NaN; +inf and finite should
 * yield +inf; otherwise all -inf should be ignored because exp(-inf) =
 * 0.)
 */
static double
logsumexp(double *A, size_t n)
{
  double maximum, sum;
  size_t i;

  if (n == 0)
    return log(0);

  maximum = A[0];
  for (i = 1; i < n; i++) {
    if (A[i] > maximum)
      maximum = A[i];
  }

  sum = 0;
  for (i = n; i --> 0;)
    sum += exp(A[i] - maximum);

  return log(sum) + maximum;
}

/**
 * Compute log(1 - e^x).  Defined only for negative x so that e^x < 1.
 * This is the complement of a probability in log space.
 */
static double
log1mexp(double x)
{

  /*
   * We want to compute log on [0, 1/2) but log1p on [1/2, +inf),
   * so partition x at -log(2) = log(1/2).
   */
  if (-log(2) < x)
    return log(-expm1(x));
  else
    return log1p(-exp(x));
}

/*
 * Tests of numerical errors in computing logit, logistic, and the
 * various cdfs, sfs, icdfs, and isfs.
 */

#define arraycount(A) (sizeof(A)/sizeof(A[0]))

/** Return relative error between <b>actual</b> and <b>expected</b>.
 *  Special cases: If <b>expected</b> is zero or infinite, return 1 if
 *  <b>actual</b> is equal to <b>expected</b> and 0 if not, since the
 *  usual notion of relative error is undefined but we only use this
 *  for testing relerr(e, a) <= bound.  If either is NaN, return NaN,
 *  which has the property that NaN <= bound is false no matter what
 *  bound is.
 *
 *  Beware: if you test !(relerr(e, a) > bound), then then the result
 *  is true when a is NaN because NaN > bound is false too.  See
 *  CHECK_RELERR for correct use to decide when to report failure.
 */
static double
relerr(double expected, double actual)
{
  /*
   * To silence -Wfloat-equal, we have to test for equality using
   * inequalities: we have (fabs(expected) <= 0) iff (expected == 0),
   * and (actual <= expected && actual >= expected) iff actual ==
   * expected whether expected is zero or infinite.
   */
  if (fabs(expected) <= 0 || tor_isinf(expected)) {
    if (actual <= expected && actual >= expected)
      return 0;
    else
      return 1;
  } else {
    return fabs((expected - actual)/expected);
  }
}

/** Check that relative error of <b>expected</b> and <b>actual</b> is within
 *  <b>relerr_bound</b>.  Caller must arrange to have i and relerr_bound in
 *  scope.  */
#define CHECK_RELERR(expected, actual) do {                                   \
  double check_expected = (expected);                                         \
  double check_actual = (actual);                                             \
  const char *str_expected = #expected;                                       \
  const char *str_actual = #actual;                                           \
  double check_relerr = relerr(expected, actual);                             \
  if (!(relerr(check_expected, check_actual) <= relerr_bound)) {              \
    log_warn(LD_GENERAL, "%s:%d: case %u: relerr(%s=%.17e, %s=%.17e)"        \
             " = %.17e > %.17e\n",                                            \
             __func__, __LINE__, (unsigned) i,                                \
             str_expected, check_expected,                                    \
             str_actual, check_actual,                                        \
             check_relerr, relerr_bound);                                     \
    ok = false;                                                               \
  }                                                                           \
} while (0)

/* Check that a <= b.
 * Caller must arrange to have i in scope.  */
#define CHECK_LE(a, b) do {                                                   \
  double check_a = (a);                                                       \
  double check_b = (b);                                                       \
  const char *str_a = #a;                                                     \
  const char *str_b = #b;                                                     \
  if (!(check_a <= check_b)) {                                                \
    log_warn(LD_GENERAL, "%s:%d: case %u: %s=%.17e > %s=%.17e\n",             \
             __func__, __LINE__, (unsigned) i,                                \
             str_a, check_a, str_b, check_b);                                 \
    ok = false;                                                               \
  }                                                                           \
} while (0)

/**
 * Test the logit and logistic functions.  Confirm that they agree with
 * the cdf, sf, icdf, and isf of the standard Logistic distribution.
 * Confirm that the sampler for the standard logistic distribution maps
 * [0, 1] into the right subinterval for the inverse transform, for
 * this implementation.
 */
static void
test_logit_logistic(void *arg)
{
  (void) arg;

  static const struct {
    double x;                   /* x = logit(p) */
    double p;                   /* p = logistic(x) */
    double phalf;               /* p - 1/2 = logistic(x) - 1/2 */
  } cases[] = {
    { -HUGE_VAL, 0, -0.5 },
    { -1000, 0, -0.5 },
    { -710, 4.47628622567513e-309, -0.5 },
    { -708, 3.307553003638408e-308, -0.5 },
    { -2, .11920292202211755, -.3807970779778824 },
    { -1.0000001, .2689414017088022, -.23105859829119776 },
    { -1, .2689414213699951, -.23105857863000487 },
    { -0.9999999, .26894144103118883, -.2310585589688111 },
    /* see src/test/prob_distr_mpfr_ref.c for computation */
    { -4.000000000537333e-5, .49999, -1.0000000000010001e-5 },
    { -4.000000000533334e-5, .49999, -.00001 },
    { -4.000000108916878e-9, .499999999, -1.0000000272292198e-9 },
    { -4e-9, .499999999, -1e-9 },
    { -4e-16, .5, -1e-16 },
    { -4e-300, .5, -1e-300 },
    { 0, .5, 0 },
    { 4e-300, .5, 1e-300 },
    { 4e-16, .5, 1e-16 },
    { 3.999999886872274e-9, .500000001, 9.999999717180685e-10 },
    { 4e-9, .500000001, 1e-9 },
    { 4.0000000005333336e-5, .50001, .00001 },
    { 8.000042667076272e-3, .502, .002 },
    { 0.9999999, .7310585589688111, .2310585589688111 },
    { 1, .7310585786300049, .23105857863000487 },
    { 1.0000001, .7310585982911977, .23105859829119774 },
    { 2, .8807970779778823, .3807970779778824 },
    { 708, 1, .5 },
    { 710, 1, .5 },
    { 1000, 1, .5 },
    { HUGE_VAL, 1, .5 },
  };
  double relerr_bound = 3e-15; /* >10eps */
  size_t i;
  bool ok = true;

  for (i = 0; i < arraycount(cases); i++) {
    double x = cases[i].x;
    double p = cases[i].p;
    double phalf = cases[i].phalf;

    /*
     * cdf is logistic, icdf is logit, and symmetry for
     * sf/isf.
     */
    CHECK_RELERR(logistic(x), cdf_logistic(x, 0, 1));
    CHECK_RELERR(logistic(-x), sf_logistic(x, 0, 1));
    CHECK_RELERR(logit(p), icdf_logistic(p, 0, 1));
    CHECK_RELERR(-logit(p), isf_logistic(p, 0, 1));

    CHECK_RELERR(cdf_logistic(x, 0, 1), cdf_logistic(x*2, 0, 2));
    CHECK_RELERR(sf_logistic(x, 0, 1), sf_logistic(x*2, 0, 2));
    CHECK_RELERR(icdf_logistic(p, 0, 1), icdf_logistic(p, 0, 2)/2);
    CHECK_RELERR(isf_logistic(p, 0, 1), isf_logistic(p, 0, 2)/2);

    CHECK_RELERR(cdf_logistic(x, 0, 1), cdf_logistic(x/2, 0, .5));
    CHECK_RELERR(sf_logistic(x, 0, 1), sf_logistic(x/2, 0, .5));
    CHECK_RELERR(icdf_logistic(p, 0, 1), icdf_logistic(p, 0,.5)*2);
    CHECK_RELERR(isf_logistic(p, 0, 1), isf_logistic(p, 0, .5)*2);

    CHECK_RELERR(cdf_logistic(x, 0, 1), cdf_logistic(x*2 + 1, 1, 2));
    CHECK_RELERR(sf_logistic(x, 0, 1), sf_logistic(x*2 + 1, 1, 2));

    /*
     * For p near 0 and p near 1/2, the arithmetic of
     * translating by 1 loses precision.
     */
    if (fabs(p) > DBL_EPSILON && fabs(p) < 0.4) {
      CHECK_RELERR(icdf_logistic(p, 0, 1),
          (icdf_logistic(p, 1, 2) - 1)/2);
      CHECK_RELERR(isf_logistic(p, 0, 1),
          (isf_logistic(p, 1, 2) - 1)/2);
    }

    CHECK_RELERR(p, logistic(x));
    CHECK_RELERR(phalf, logistichalf(x));

    /*
     * On the interior floating-point numbers, either logit or
     * logithalf had better give the correct answer.
     *
     * For probabilities near 0, we can get much finer resolution with
     * logit, and for probabilities near 1/2, we can get much finer
     * resolution with logithalf by representing them using p - 1/2.
     *
     * E.g., we can write -.00001 for phalf, and .49999 for p, but the
     * difference 1/2 - .00001 gives 1.0000000000010001e-5 in binary64
     * arithmetic.  So test logit(.49999) which should give the same
     * answer as logithalf(-1.0000000000010001e-5), namely
     * -4.000000000537333e-5, and also test logithalf(-.00001) which
     * gives -4.000000000533334e-5 instead -- but don't expect
     * logit(.49999) to give -4.000000000533334e-5 even though it looks
     * like 1/2 - .00001.
     *
     * A naive implementation of logit will just use log(p/(1 - p)) and
     * give the answer -4.000000000551673e-05 for .49999, which is
     * wrong in a lot of digits, which happens because log is
     * ill-conditioned near 1 and thus amplifies whatever relative
     * error we made in computing p/(1 - p).
     */
    if ((0 < p && p < 1) || tor_isinf(x)) {
      if (phalf >= p - 0.5 && phalf <= p - 0.5)
        CHECK_RELERR(x, logit(p));
      if (p >= 0.5 + phalf && p <= 0.5 + phalf)
        CHECK_RELERR(x, logithalf(phalf));
    }

    CHECK_RELERR(-phalf, logistichalf(-x));
    if (fabs(phalf) < 0.5 || tor_isinf(x))
      CHECK_RELERR(-x, logithalf(-phalf));
    if (p < 1 || tor_isinf(x)) {
      CHECK_RELERR(1 - p, logistic(-x));
      if (p > .75 || tor_isinf(x))
        CHECK_RELERR(-x, logit(1 - p));
    } else {
      CHECK_LE(logistic(-x), 1e-300);
    }
  }

  for (i = 0; i <= 100; i++) {
    double p0 = (double)i/100;

    CHECK_RELERR(logit(p0/(1 + M_E)), sample_logistic(0, 0, p0));
    CHECK_RELERR(-logit(p0/(1 + M_E)), sample_logistic(1, 0, p0));
    CHECK_RELERR(logithalf(p0*(0.5 - 1/(1 + M_E))),
        sample_logistic(0, 1, p0));
    CHECK_RELERR(-logithalf(p0*(0.5 - 1/(1 + M_E))),
        sample_logistic(1, 1, p0));
  }

  if (!ok)
    printf("fail logit/logistic / logistic cdf/sf\n");

  tt_assert(ok);

 done:
  ;
}

/**
 * Test the cdf, sf, icdf, and isf of the LogLogistic distribution.
 */
static void
test_log_logistic(void *arg)
{
  (void) arg;

  static const struct {
    /* x is a point in the support of the LogLogistic distribution */
    double x;
    /* 'p' is the probability that a random variable X for a given LogLogistic
     * probability ditribution will take value less-or-equal to x */
    double p;
    /* 'np' is the probability that a random variable X for a given LogLogistic
     * probability distribution will take value greater-or-equal to x. */
    double np;
  } cases[] = {
    { 0, 0, 1 },
    { 1e-300, 1e-300, 1 },
    { 1e-17, 1e-17, 1 },
    { 1e-15, 1e-15, .999999999999999 },
    { .1, .09090909090909091, .90909090909090909 },
    { .25, .2, .8 },
    { .5, .33333333333333333, .66666666666666667 },
    { .75, .42857142857142855, .5714285714285714 },
    { .9999, .49997499874993756, .5000250012500626 },
    { .99999999, .49999999749999996, .5000000025 },
    { .999999999999999, .49999999999999994, .5000000000000002 },
    { 1, .5, .5 },
  };
  double relerr_bound = 3e-15;
  size_t i;
  bool ok = true;

  for (i = 0; i < arraycount(cases); i++) {
    double x = cases[i].x;
    double p = cases[i].p;
    double np = cases[i].np;

    CHECK_RELERR(p, cdf_log_logistic(x, 1, 1));
    CHECK_RELERR(p, cdf_log_logistic(x/2, .5, 1));
    CHECK_RELERR(p, cdf_log_logistic(x*2, 2, 1));
    CHECK_RELERR(p, cdf_log_logistic(sqrt(x), 1, 2));
    CHECK_RELERR(p, cdf_log_logistic(sqrt(x)/2, .5, 2));
    CHECK_RELERR(p, cdf_log_logistic(sqrt(x)*2, 2, 2));
    if (2*sqrt(DBL_MIN) < x) {
      CHECK_RELERR(p, cdf_log_logistic(x*x, 1, .5));
      CHECK_RELERR(p, cdf_log_logistic(x*x/2, .5, .5));
      CHECK_RELERR(p, cdf_log_logistic(x*x*2, 2, .5));
    }

    CHECK_RELERR(np, sf_log_logistic(x, 1, 1));
    CHECK_RELERR(np, sf_log_logistic(x/2, .5, 1));
    CHECK_RELERR(np, sf_log_logistic(x*2, 2, 1));
    CHECK_RELERR(np, sf_log_logistic(sqrt(x), 1, 2));
    CHECK_RELERR(np, sf_log_logistic(sqrt(x)/2, .5, 2));
    CHECK_RELERR(np, sf_log_logistic(sqrt(x)*2, 2, 2));
    if (2*sqrt(DBL_MIN) < x) {
      CHECK_RELERR(np, sf_log_logistic(x*x, 1, .5));
      CHECK_RELERR(np, sf_log_logistic(x*x/2, .5, .5));
      CHECK_RELERR(np, sf_log_logistic(x*x*2, 2, .5));
    }

    CHECK_RELERR(np, cdf_log_logistic(1/x, 1, 1));
    CHECK_RELERR(np, cdf_log_logistic(1/(2*x), .5, 1));
    CHECK_RELERR(np, cdf_log_logistic(2/x, 2, 1));
    CHECK_RELERR(np, cdf_log_logistic(1/sqrt(x), 1, 2));
    CHECK_RELERR(np, cdf_log_logistic(1/(2*sqrt(x)), .5, 2));
    CHECK_RELERR(np, cdf_log_logistic(2/sqrt(x), 2, 2));
    if (2*sqrt(DBL_MIN) < x && x < 1/(2*sqrt(DBL_MIN))) {
      CHECK_RELERR(np, cdf_log_logistic(1/(x*x), 1, .5));
      CHECK_RELERR(np, cdf_log_logistic(1/(2*x*x), .5, .5));
      CHECK_RELERR(np, cdf_log_logistic(2/(x*x), 2, .5));
    }

    CHECK_RELERR(p, sf_log_logistic(1/x, 1, 1));
    CHECK_RELERR(p, sf_log_logistic(1/(2*x), .5, 1));
    CHECK_RELERR(p, sf_log_logistic(2/x, 2, 1));
    CHECK_RELERR(p, sf_log_logistic(1/sqrt(x), 1, 2));
    CHECK_RELERR(p, sf_log_logistic(1/(2*sqrt(x)), .5, 2));
    CHECK_RELERR(p, sf_log_logistic(2/sqrt(x), 2, 2));
    if (2*sqrt(DBL_MIN) < x && x < 1/(2*sqrt(DBL_MIN))) {
      CHECK_RELERR(p, sf_log_logistic(1/(x*x), 1, .5));
      CHECK_RELERR(p, sf_log_logistic(1/(2*x*x), .5, .5));
      CHECK_RELERR(p, sf_log_logistic(2/(x*x), 2, .5));
    }

    CHECK_RELERR(x, icdf_log_logistic(p, 1, 1));
    CHECK_RELERR(x/2, icdf_log_logistic(p, .5, 1));
    CHECK_RELERR(x*2, icdf_log_logistic(p, 2, 1));
    CHECK_RELERR(x, icdf_log_logistic(p, 1, 1));
    CHECK_RELERR(sqrt(x)/2, icdf_log_logistic(p, .5, 2));
    CHECK_RELERR(sqrt(x)*2, icdf_log_logistic(p, 2, 2));
    CHECK_RELERR(sqrt(x), icdf_log_logistic(p, 1, 2));
    CHECK_RELERR(x*x/2, icdf_log_logistic(p, .5, .5));
    CHECK_RELERR(x*x*2, icdf_log_logistic(p, 2, .5));

    if (np < .9) {
      CHECK_RELERR(x, isf_log_logistic(np, 1, 1));
      CHECK_RELERR(x/2, isf_log_logistic(np, .5, 1));
      CHECK_RELERR(x*2, isf_log_logistic(np, 2, 1));
      CHECK_RELERR(sqrt(x), isf_log_logistic(np, 1, 2));
      CHECK_RELERR(sqrt(x)/2, isf_log_logistic(np, .5, 2));
      CHECK_RELERR(sqrt(x)*2, isf_log_logistic(np, 2, 2));
      CHECK_RELERR(x*x, isf_log_logistic(np, 1, .5));
      CHECK_RELERR(x*x/2, isf_log_logistic(np, .5, .5));
      CHECK_RELERR(x*x*2, isf_log_logistic(np, 2, .5));

      CHECK_RELERR(1/x, icdf_log_logistic(np, 1, 1));
      CHECK_RELERR(1/(2*x), icdf_log_logistic(np, .5, 1));
      CHECK_RELERR(2/x, icdf_log_logistic(np, 2, 1));
      CHECK_RELERR(1/sqrt(x), icdf_log_logistic(np, 1, 2));
      CHECK_RELERR(1/(2*sqrt(x)),
          icdf_log_logistic(np, .5, 2));
      CHECK_RELERR(2/sqrt(x), icdf_log_logistic(np, 2, 2));
      CHECK_RELERR(1/(x*x), icdf_log_logistic(np, 1, .5));
      CHECK_RELERR(1/(2*x*x), icdf_log_logistic(np, .5, .5));
      CHECK_RELERR(2/(x*x), icdf_log_logistic(np, 2, .5));
    }

    CHECK_RELERR(1/x, isf_log_logistic(p, 1, 1));
    CHECK_RELERR(1/(2*x), isf_log_logistic(p, .5, 1));
    CHECK_RELERR(2/x, isf_log_logistic(p, 2, 1));
    CHECK_RELERR(1/sqrt(x), isf_log_logistic(p, 1, 2));
    CHECK_RELERR(1/(2*sqrt(x)), isf_log_logistic(p, .5, 2));
    CHECK_RELERR(2/sqrt(x), isf_log_logistic(p, 2, 2));
    CHECK_RELERR(1/(x*x), isf_log_logistic(p, 1, .5));
    CHECK_RELERR(1/(2*x*x), isf_log_logistic(p, .5, .5));
    CHECK_RELERR(2/(x*x), isf_log_logistic(p, 2, .5));
  }

  for (i = 0; i <= 100; i++) {
    double p0 = (double)i/100;

    CHECK_RELERR(0.5*p0/(1 - 0.5*p0), sample_log_logistic(0, p0));
    CHECK_RELERR((1 - 0.5*p0)/(0.5*p0),
        sample_log_logistic(1, p0));
  }

  if (!ok)
    printf("fail log logistic cdf/sf\n");

  tt_assert(ok);

 done:
  ;
}

/**
 * Test the cdf, sf, icdf, isf of the Weibull distribution.
 */
static void
test_weibull(void *arg)
{
  (void) arg;

  static const struct {
    /* x is a point in the support of the Weibull distribution */
    double x;
    /* 'p' is the probability that a random variable X for a given Weibull
     * probability ditribution will take value less-or-equal to x */
    double p;
    /* 'np' is the probability that a random variable X for a given Weibull
     * probability distribution will take value greater-or-equal to x. */
    double np;
  } cases[] = {
    { 0, 0, 1 },
    { 1e-300, 1e-300, 1 },
    { 1e-17, 1e-17, 1 },
    { .1, .09516258196404043, .9048374180359595 },
    { .5, .3934693402873666, .6065306597126334 },
    { .6931471805599453, .5, .5 },
    { 1, .6321205588285577, .36787944117144233 },
    { 10, .9999546000702375, 4.5399929762484854e-5 },
    { 36, .9999999999999998, 2.319522830243569e-16 },
    { 37, .9999999999999999, 8.533047625744066e-17 },
    { 38, 1, 3.1391327920480296e-17 },
    { 100, 1, 3.720075976020836e-44 },
    { 708, 1, 3.307553003638408e-308 },
    { 710, 1, 4.47628622567513e-309 },
    { 1000, 1, 0 },
    { HUGE_VAL, 1, 0 },
  };
  double relerr_bound = 3e-15;
  size_t i;
  bool ok = true;

  for (i = 0; i < arraycount(cases); i++) {
    double x = cases[i].x;
    double p = cases[i].p;
    double np = cases[i].np;

    CHECK_RELERR(p, cdf_weibull(x, 1, 1));
    CHECK_RELERR(p, cdf_weibull(x/2, .5, 1));
    CHECK_RELERR(p, cdf_weibull(x*2, 2, 1));
    /* For 0 < x < sqrt(DBL_MIN), x^2 loses lots of bits.  */
    if (x <= 0 ||
        sqrt(DBL_MIN) <= x) {
      CHECK_RELERR(p, cdf_weibull(x*x, 1, .5));
      CHECK_RELERR(p, cdf_weibull(x*x/2, .5, .5));
      CHECK_RELERR(p, cdf_weibull(x*x*2, 2, .5));
    }
    CHECK_RELERR(p, cdf_weibull(sqrt(x), 1, 2));
    CHECK_RELERR(p, cdf_weibull(sqrt(x)/2, .5, 2));
    CHECK_RELERR(p, cdf_weibull(sqrt(x)*2, 2, 2));
    CHECK_RELERR(np, sf_weibull(x, 1, 1));
    CHECK_RELERR(np, sf_weibull(x/2, .5, 1));
    CHECK_RELERR(np, sf_weibull(x*2, 2, 1));
    CHECK_RELERR(np, sf_weibull(x*x, 1, .5));
    CHECK_RELERR(np, sf_weibull(x*x/2, .5, .5));
    CHECK_RELERR(np, sf_weibull(x*x*2, 2, .5));
    if (x >= 10) {
      /*
       * exp amplifies the error of sqrt(x)^2
       * proportionally to exp(x); for large inputs
       * this is significant.
       */
      double t = -expm1(-x*(2*DBL_EPSILON + DBL_EPSILON));
      relerr_bound = t + DBL_EPSILON + t*DBL_EPSILON;
      if (relerr_bound < 3e-15)
        /*
         * The tests are written only to 16
         * decimal places anyway even if your
         * `double' is, say, i387 binary80, for
         * whatever reason.
         */
        relerr_bound = 3e-15;
      CHECK_RELERR(np, sf_weibull(sqrt(x), 1, 2));
      CHECK_RELERR(np, sf_weibull(sqrt(x)/2, .5, 2));
      CHECK_RELERR(np, sf_weibull(sqrt(x)*2, 2, 2));
    }

    if (p <= 0.75) {
      /*
       * For p near 1, not enough precision near 1 to
       * recover x.
       */
      CHECK_RELERR(x, icdf_weibull(p, 1, 1));
      CHECK_RELERR(x/2, icdf_weibull(p, .5, 1));
      CHECK_RELERR(x*2, icdf_weibull(p, 2, 1));
    }
    if (p >= 0.25 && !tor_isinf(x) && np > 0) {
      /*
       * For p near 0, not enough precision in np
       * near 1 to recover x.  For 0, isf gives inf,
       * even if p is precise enough for the icdf to
       * work.
       */
      CHECK_RELERR(x, isf_weibull(np, 1, 1));
      CHECK_RELERR(x/2, isf_weibull(np, .5, 1));
      CHECK_RELERR(x*2, isf_weibull(np, 2, 1));
    }
  }

  for (i = 0; i <= 100; i++) {
    double p0 = (double)i/100;

    CHECK_RELERR(3*sqrt(-log(p0/2)), sample_weibull(0, p0, 3, 2));
    CHECK_RELERR(3*sqrt(-log1p(-p0/2)),
        sample_weibull(1, p0, 3, 2));
  }

  if (!ok)
    printf("fail Weibull cdf/sf\n");

  tt_assert(ok);

 done:
  ;
}

/**
 * Test the cdf, sf, icdf, and isf of the generalized Pareto
 * distribution.
 */
static void
test_genpareto(void *arg)
{
  (void) arg;

  struct {
    /* xi is the 'xi' parameter of the generalized Pareto distribution, and the
     * rest are the same as in the above tests */
    double xi, x, p, np;
  } cases[] = {
    { 0, 0, 0, 1 },
    { 1e-300, .004, 3.992010656008528e-3, .9960079893439915 },
    { 1e-300, .1, .09516258196404043, .9048374180359595 },
    { 1e-300, 1, .6321205588285577, .36787944117144233 },
    { 1e-300, 10, .9999546000702375, 4.5399929762484854e-5 },
    { 1e-200, 1e-16, 9.999999999999999e-17, .9999999999999999 },
    { 1e-16, 1e-200, 9.999999999999998e-201, 1 },
    { 1e-16, 1e-16, 1e-16, 1 },
    { 1e-16, .004, 3.992010656008528e-3, .9960079893439915 },
    { 1e-16, .1, .09516258196404043, .9048374180359595 },
    { 1e-16, 1, .6321205588285577, .36787944117144233 },
    { 1e-16, 10, .9999546000702375, 4.539992976248509e-5 },
    { 1e-10, 1e-6, 9.999995000001667e-7, .9999990000005 },
    { 1e-8, 1e-8, 9.999999950000001e-9, .9999999900000001 },
    { 1, 1e-300, 1e-300, 1 },
    { 1, 1e-16, 1e-16, .9999999999999999 },
    { 1, .1, .09090909090909091, .9090909090909091 },
    { 1, 1, .5, .5 },
    { 1, 10, .9090909090909091, .0909090909090909 },
    { 1, 100, .9900990099009901, .0099009900990099 },
    { 1, 1000, .999000999000999, 9.990009990009992e-4 },
    { 10, 1e-300, 1e-300, 1 },
    { 10, 1e-16, 9.999999999999995e-17, .9999999999999999 },
    { 10, .1, .06696700846319258, .9330329915368074 },
    { 10, 1, .21320655780322778, .7867934421967723 },
    { 10, 10, .3696701667040189, .6303298332959811 },
    { 10, 100, .49886285755007337, .5011371424499267 },
    { 10, 1000, .6018968102992647, .3981031897007353 },
  };
  double xi_array[] = { -1.5, -1, -1e-30, 0, 1e-30, 1, 1.5 };
  size_t i, j;
  double relerr_bound = 3e-15;
  bool ok = true;

  for (i = 0; i < arraycount(cases); i++) {
    double xi = cases[i].xi;
    double x = cases[i].x;
    double p = cases[i].p;
    double np = cases[i].np;

    CHECK_RELERR(p, cdf_genpareto(x, 0, 1, xi));
    CHECK_RELERR(p, cdf_genpareto(x*2, 0, 2, xi));
    CHECK_RELERR(p, cdf_genpareto(x/2, 0, .5, xi));
    CHECK_RELERR(np, sf_genpareto(x, 0, 1, xi));
    CHECK_RELERR(np, sf_genpareto(x*2, 0, 2, xi));
    CHECK_RELERR(np, sf_genpareto(x/2, 0, .5, xi));

    if (p < .5) {
      CHECK_RELERR(x, icdf_genpareto(p, 0, 1, xi));
      CHECK_RELERR(x*2, icdf_genpareto(p, 0, 2, xi));
      CHECK_RELERR(x/2, icdf_genpareto(p, 0, .5, xi));
    }
    if (np < .5) {
      CHECK_RELERR(x, isf_genpareto(np, 0, 1, xi));
      CHECK_RELERR(x*2, isf_genpareto(np, 0, 2, xi));
      CHECK_RELERR(x/2, isf_genpareto(np, 0, .5, xi));
    }
  }

  for (i = 0; i < arraycount(xi_array); i++) {
    for (j = 0; j <= 100; j++) {
      double p0 = (j == 0 ? 2*DBL_MIN : (double)j/100);

      /* This is actually a check against 0, but we do <= so that the compiler
         does not raise a -Wfloat-equal */
      if (fabs(xi_array[i]) <= 0) {
        /*
         * When xi == 0, the generalized Pareto
         * distribution reduces to an
         * exponential distribution.
         */
        CHECK_RELERR(-log(p0/2),
            sample_genpareto(0, p0, 0));
        CHECK_RELERR(-log1p(-p0/2),
            sample_genpareto(1, p0, 0));
      } else {
        CHECK_RELERR(expm1(-xi_array[i]*log(p0/2))/xi_array[i],
            sample_genpareto(0, p0, xi_array[i]));
        CHECK_RELERR((j == 0 ? DBL_MIN :
                expm1(-xi_array[i]*log1p(-p0/2))/xi_array[i]),
            sample_genpareto(1, p0, xi_array[i]));
      }

      CHECK_RELERR(isf_genpareto(p0/2, 0, 1, xi_array[i]),
          sample_genpareto(0, p0, xi_array[i]));
      CHECK_RELERR(icdf_genpareto(p0/2, 0, 1, xi_array[i]),
          sample_genpareto(1, p0, xi_array[i]));
    }
  }

  tt_assert(ok);

 done:
  ;
}

/**
 * Test the deterministic sampler for uniform distribution on [a, b].
 *
 * This currently only tests whether the outcome lies within [a, b].
 */
static void
test_uniform_interval(void *arg)
{
  (void) arg;
  struct {
    /* Sample from a uniform distribution with parameters 'a' and 'b', using
     * 't' as the sampling index. */
    double t, a, b;
  } cases[] = {
    { 0, 0, 0 },
    { 0, 0, 1 },
    { 0, 1.0000000000000007, 3.999999999999995 },
    { 0, 4000, 4000 },
    { 0.42475836677491291, 4000, 4000 },
    { 0, -DBL_MAX, DBL_MAX },
    { 0.25, -DBL_MAX, DBL_MAX },
    { 0.5, -DBL_MAX, DBL_MAX },
  };
  size_t i = 0;
  bool ok = true;

  for (i = 0; i < arraycount(cases); i++) {
    double t = cases[i].t;
    double a = cases[i].a;
    double b = cases[i].b;

    CHECK_LE(a, sample_uniform_interval(t, a, b));
    CHECK_LE(sample_uniform_interval(t, a, b), b);

    CHECK_LE(a, sample_uniform_interval(1 - t, a, b));
    CHECK_LE(sample_uniform_interval(1 - t, a, b), b);

    CHECK_LE(sample_uniform_interval(t, -b, -a), -a);
    CHECK_LE(-b, sample_uniform_interval(t, -b, -a));

    CHECK_LE(sample_uniform_interval(1 - t, -b, -a), -a);
    CHECK_LE(-b, sample_uniform_interval(1 - t, -b, -a));
  }

  tt_assert(ok);

 done:
  ;
}

/********************** Stochastic tests ****************************/

/*
 * Psi test, sometimes also called G-test.  The psi test statistic,
 * suitably scaled, has chi^2 distribution, but the psi test tends to
 * have better statistical power in practice to detect deviations than
 * the chi^2 test does.  (The chi^2 test statistic is the first term of
 * the Taylor expansion of the psi test statistic.)  The psi test is
 * generic, for any CDF; particular distributions might have higher-
 * power tests to distinguish them from predictable deviations or bugs.
 *
 * We choose the psi critical value so that a single psi test has
 * probability below alpha = 1% of spuriously failing even if all the
 * code is correct.  But the false positive rate for a suite of n tests
 * is higher: 1 - Binom(0; n, alpha) = 1 - (1 - alpha)^n.  For n = 10,
 * this is about 10%, and for n = 100 it is well over 50%.
 *
 * Given that these tests will run with every CI job, we want to drive down the
 * false positive rate. We can drive it down by running each test four times,
 * and accepting it if it passes at least once; in that case, it is as if we
 * used Binom(4; 2, alpha) = alpha^4 as the false positive rate for each test,
 * and for n = 10 tests, it would be 9.99999959506e-08. If each CI build has 14
 * jobs, then the chance of a CI build failing is 1.39999903326e-06, which
 * means that a CI build will break with probability 50% after about 495106
 * builds.
 *
 * The critical value for a chi^2 distribution with 100 degrees of
 * freedom and false positive rate alpha = 1% was taken from:
 *
 *  NIST/SEMATECH e-Handbook of Statistical Methods, Section
 *  1.3.6.7.4 `Critical Values of the Chi-Square Distribution',
 *  <http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm>,
 *  retrieved 2018-10-28.
 */

static const size_t NSAMPLES = 100000;
/* Number of chances we give to the test to succeed. */
static const unsigned NTRIALS = 4;
/* Number of times we want the test to pass per NTRIALS. */
static const unsigned NPASSES_MIN = 1;

#define PSI_DF 100                          /* degrees of freedom */
static const double PSI_CRITICAL = 135.807; /* critical value, alpha = .01 */

/**
 * Perform a psi test on an array of sample counts, C, adding up to N
 * samples, and an array of log expected probabilities, logP,
 * representing the null hypothesis for the distribution of samples
 * counted.  Return false if the psi test rejects the null hypothesis,
 * true if otherwise.
 */
static bool
psi_test(const size_t C[PSI_DF], const double logP[PSI_DF], size_t N)
{
  double psi = 0;
  double c = 0;                 /* Kahan compensation */
  double t, u;
  size_t i;

  for (i = 0; i < PSI_DF; i++) {
    /*
     * c*log(c/(n*p)) = (1/n) * f*log(f/p) where f = c/n is
     * the frequency, and f*log(f/p) ---> 0 as f ---> 0, so
     * this is a reasonable choice.  Further, any mass that
     * _fails_ to turn up in this bin will inflate another
     * bin instead, so we don't really lose anything by
     * ignoring empty bins even if they have high
     * probability.
     */
    if (C[i] == 0)
      continue;
    t = C[i]*(log((double)C[i]/N) - logP[i]) - c;
    u = psi + t;
    c = (u - psi) - t;
    psi = u;
  }
  psi *= 2;

  return psi <= PSI_CRITICAL;
}

static bool
test_stochastic_geometric_impl(double p)
{
  const struct geometric geometric = {
    .base = GEOMETRIC(geometric),
    .p = p,
  };
  double logP[PSI_DF] = {0};
  unsigned ntry = NTRIALS, npass = 0;
  unsigned i;
  size_t j;

  /* Compute logP[i] = Geom(i + 1; p).  */
  for (i = 0; i < PSI_DF - 1; i++)
    logP[i] = logpmf_geometric(i + 1, p);

  /* Compute logP[n-1] = log (1 - (P[0] + P[1] + ... + P[n-2])).  */
  logP[PSI_DF - 1] = log1mexp(logsumexp(logP, PSI_DF - 1));

  while (ntry --> 0) {
    size_t C[PSI_DF] = {0};

    for (j = 0; j < NSAMPLES; j++) {
      double n_tmp = dist_sample(&geometric.base);

      /* Must be an integer.  (XXX -Wfloat-equal)  */
      tor_assert(ceil(n_tmp) <= n_tmp && ceil(n_tmp) >= n_tmp);

      /* Must be a positive integer.  */
      tor_assert(n_tmp >= 1);

      /* Probability of getting a value in the billions is negligible.  */
      tor_assert(n_tmp <= (double)UINT_MAX);

      unsigned n = (unsigned) n_tmp;

      if (n > PSI_DF)
        n = PSI_DF;
      C[n - 1]++;
    }

    if (psi_test(C, logP, NSAMPLES)) {
      if (++npass >= NPASSES_MIN)
        break;
    }
  }

  if (npass >= NPASSES_MIN) {
    /* printf("pass %s sampler\n", "geometric"); */
    return true;
  } else {
    printf("fail %s sampler\n", "geometric");
    return false;
  }
}

/**
 * Divide the support of <b>dist</b> into histogram bins in <b>logP</b>. Start
 * at the 1st percentile and ending at the 99th percentile. Pick the bin
 * boundaries using linear interpolation so that they are uniformly spaced.
 *
 * In each bin logP[i] we insert the expected log-probability that a sampled
 * value will fall into that bin. We will use this as the null hypothesis of
 * the psi test.
 *
 * Set logP[i] = log(CDF(x_i) - CDF(x_{i-1})), where x_-1 = -inf, x_n =
 * +inf, and x_i = i*(hi - lo)/(n - 2).
 */
static void
bin_cdfs(const struct dist *dist, double lo, double hi, double *logP, size_t n)
{
#define CDF(x)  dist_cdf(dist, x)
#define SF(x)   dist_sf(dist, x)
  const double w = (hi - lo)/(n - 2);
  double halfway = dist_icdf(dist, 0.5);
  double x_0, x_1;
  size_t i;
  size_t n2 = ceil_to_size_t((halfway - lo)/w);

  tor_assert(lo <= halfway);
  tor_assert(halfway <= hi);
  tor_assert(n2 <= n);

  x_1 = lo;
  logP[0] = log(CDF(x_1) - 0); /* 0 = CDF(-inf) */
  for (i = 1; i < n2; i++) {
    x_0 = x_1;
    /* do the linear interpolation */
    x_1 = (i <= n/2 ? lo + i*w : hi - (n - 2 - i)*w);
    /* set the expected log-probability */
    logP[i] = log(CDF(x_1) - CDF(x_0));
  }
  x_0 = hi;
  logP[n - 1] = log(SF(x_0) - 0); /* 0 = SF(+inf) = 1 - CDF(+inf) */

  /* In this loop we are filling out the high part of the array. We are using
   * SF because in these cases the CDF is near 1 where precision is lower. So
   * instead we are using SF near 0 where the precision is higher. We have
   * SF(t) = 1 - CDF(t).  */
  for (i = 1; i < n - n2; i++) {
    x_1 = x_0;
    /* do the linear interpolation */
    x_0 = (i <= n/2 ? hi - i*w : lo + (n - 2 - i)*w);
    /* set the expected log-probability */
    logP[n - i - 1] = log(SF(x_0) - SF(x_1));
  }
#undef SF
#undef CDF
}

/**
 * Draw NSAMPLES samples from dist, counting the number of samples x in
 * the ith bin C[i] if x_{i-1} <= x < x_i, where x_-1 = -inf, x_n =
 * +inf, and x_i = i*(hi - lo)/(n - 2).
 */
static void
bin_samples(const struct dist *dist, double lo, double hi, size_t *C, size_t n)
{
  const double w = (hi - lo)/(n - 2);
  size_t i;

  for (i = 0; i < NSAMPLES; i++) {
    double x = dist_sample(dist);
    size_t bin;

    if (x < lo)
      bin = 0;
    else if (x < hi)
      bin = 1 + floor_to_size_t((x - lo)/w);
    else
      bin = n - 1;
    tor_assert(bin < n);
    C[bin]++;
  }
}

/**
 * Carry out a Psi test on <b>dist</b>.
 *
 * Sample NSAMPLES from dist, putting them in bins from -inf to lo to
 * hi to +inf, and apply up to two psi tests.  True if at least one psi
 * test passes; false if not.  False positive rate should be bounded by
 * 0.01^2 = 0.0001.
 */
static bool
test_psi_dist_sample(const struct dist *dist)
{
  double logP[PSI_DF] = {0};
  unsigned ntry = NTRIALS, npass = 0;
  double lo = dist_icdf(dist, 1/(double)(PSI_DF + 2));
  double hi = dist_isf(dist, 1/(double)(PSI_DF + 2));

  /* Create the null hypothesis in logP */
  bin_cdfs(dist, lo, hi, logP, PSI_DF);

  /* Now run the test */
  while (ntry --> 0) {
    size_t C[PSI_DF] = {0};
    bin_samples(dist, lo, hi, C, PSI_DF);
    if (psi_test(C, logP, NSAMPLES)) {
      if (++npass >= NPASSES_MIN)
        break;
    }
  }

  /* Did we fail or succeed? */
  if (npass >= NPASSES_MIN) {
    /* printf("pass %s sampler\n", dist_name(dist));*/
    return true;
  } else {
    printf("fail %s sampler\n", dist_name(dist));
    return false;
  }
}

static void
write_stochastic_warning(void)
{
  if (tinytest_cur_test_has_failed()) {
    printf("\n"
         "NOTE: This is a stochastic test, and we expect it to fail from\n"
         "time to time, with some low probability. If you see it fail more\n"
         "than one trial in 100, though, please tell us.\n\n");
  }
}

static void
test_stochastic_uniform(void *arg)
{
  (void) arg;

  const struct uniform uniform01 = {
    .base = UNIFORM(uniform01),
    .a = 0,
    .b = 1,
  };
  const struct uniform uniform_pos = {
    .base = UNIFORM(uniform_pos),
    .a = 1.23,
    .b = 4.56,
  };
  const struct uniform uniform_neg = {
    .base = UNIFORM(uniform_neg),
    .a = -10,
    .b = -1,
  };
  const struct uniform uniform_cross = {
    .base = UNIFORM(uniform_cross),
    .a = -1.23,
    .b = 4.56,
  };
  const struct uniform uniform_subnormal = {
    .base = UNIFORM(uniform_subnormal),
    .a = 4e-324,
    .b = 4e-310,
  };
  const struct uniform uniform_subnormal_cross = {
    .base = UNIFORM(uniform_subnormal_cross),
    .a = -4e-324,
    .b = 4e-310,
  };
  bool ok = true, tests_failed = true;

  testing_enable_reproducible_rng();

  ok &= test_psi_dist_sample(&uniform01.base);
  ok &= test_psi_dist_sample(&uniform_pos.base);
  ok &= test_psi_dist_sample(&uniform_neg.base);
  ok &= test_psi_dist_sample(&uniform_cross.base);
  ok &= test_psi_dist_sample(&uniform_subnormal.base);
  ok &= test_psi_dist_sample(&uniform_subnormal_cross.base);

  tt_assert(ok);

  tests_failed = false;

 done:
  if (tests_failed) {
    write_stochastic_warning();
  }
  testing_disable_reproducible_rng();
}

static bool
test_stochastic_logistic_impl(double mu, double sigma)
{
  const struct logistic dist = {
    .base = LOGISTIC(dist),
    .mu = mu,
    .sigma = sigma,
  };

  /* XXX Consider some fancier logistic test.  */
  return test_psi_dist_sample(&dist.base);
}

static bool
test_stochastic_log_logistic_impl(double alpha, double beta)
{
  const struct log_logistic dist = {
    .base = LOG_LOGISTIC(dist),
    .alpha = alpha,
    .beta = beta,
  };

  /* XXX Consider some fancier log logistic test.  */
  return test_psi_dist_sample(&dist.base);
}

static bool
test_stochastic_weibull_impl(double lambda, double k)
{
  const struct weibull dist = {
    .base = WEIBULL(dist),
    .lambda = lambda,
    .k = k,
  };

/*
 * XXX Consider applying a Tiku-Singh test:
 *
 *    M.L. Tiku and M. Singh, `Testing the two-parameter
 *    Weibull distribution', Communications in Statistics --
 *    Theory and Methods A10(9), 1981, 907--918.
 *https://www.tandfonline.com/doi/pdf/10.1080/03610928108828082?needAccess=true
 */
  return test_psi_dist_sample(&dist.base);
}

static bool
test_stochastic_genpareto_impl(double mu, double sigma, double xi)
{
  const struct genpareto dist = {
    .base = GENPARETO(dist),
    .mu = mu,
    .sigma = sigma,
    .xi = xi,
  };

  /* XXX Consider some fancier GPD test.  */
  return test_psi_dist_sample(&dist.base);
}

static void
test_stochastic_genpareto(void *arg)
{
  bool ok = 0;
  bool tests_failed = true;
  (void) arg;

  testing_enable_reproducible_rng();

  ok = test_stochastic_genpareto_impl(0, 1, -0.25);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(0, 1, -1e-30);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(0, 1, 0);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(0, 1, 1e-30);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(0, 1, 0.25);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(-1, 1, -0.25);
  tt_assert(ok);
  ok = test_stochastic_genpareto_impl(1, 2, 0.25);
  tt_assert(ok);

  tests_failed = false;

 done:
  if (tests_failed) {
    write_stochastic_warning();
  }
  testing_disable_reproducible_rng();
}

static void
test_stochastic_geometric(void *arg)
{
  bool ok = 0;
  bool tests_failed = true;

  (void) arg;

  testing_enable_reproducible_rng();

  ok = test_stochastic_geometric_impl(0.1);
  tt_assert(ok);
  ok = test_stochastic_geometric_impl(0.5);
  tt_assert(ok);
  ok = test_stochastic_geometric_impl(0.9);
  tt_assert(ok);
  ok = test_stochastic_geometric_impl(1);
  tt_assert(ok);

  tests_failed = false;

 done:
  if (tests_failed) {
    write_stochastic_warning();
  }
  testing_disable_reproducible_rng();
}

static void
test_stochastic_logistic(void *arg)
{
  bool ok = 0;
  bool tests_failed = true;
  (void) arg;

  testing_enable_reproducible_rng();

  ok = test_stochastic_logistic_impl(0, 1);
  tt_assert(ok);
  ok = test_stochastic_logistic_impl(0, 1e-16);
  tt_assert(ok);
  ok = test_stochastic_logistic_impl(1, 10);
  tt_assert(ok);
  ok = test_stochastic_logistic_impl(-10, 100);
  tt_assert(ok);

  tests_failed = false;

 done:
  if (tests_failed) {
    write_stochastic_warning();
  }
  testing_disable_reproducible_rng();
}

static void
test_stochastic_log_logistic(void *arg)
{
  bool ok = 0;
  (void) arg;

  testing_enable_reproducible_rng();

  ok = test_stochastic_log_logistic_impl(1, 1);
  tt_assert(ok);
  ok = test_stochastic_log_logistic_impl(1, 10);
  tt_assert(ok);
  ok = test_stochastic_log_logistic_impl(M_E, 1e-1);
  tt_assert(ok);
  ok = test_stochastic_log_logistic_impl(exp(-10), 1e-2);
  tt_assert(ok);

 done:
  write_stochastic_warning();
  testing_disable_reproducible_rng();
}

static void
test_stochastic_weibull(void *arg)
{
  bool ok = 0;
  (void) arg;

  testing_enable_reproducible_rng();

  ok = test_stochastic_weibull_impl(1, 0.5);
  tt_assert(ok);
  ok = test_stochastic_weibull_impl(1, 1);
  tt_assert(ok);
  ok = test_stochastic_weibull_impl(1, 1.5);
  tt_assert(ok);
  ok = test_stochastic_weibull_impl(1, 2);
  tt_assert(ok);
  ok = test_stochastic_weibull_impl(10, 1);
  tt_assert(ok);

 done:
  write_stochastic_warning();
  testing_disable_reproducible_rng();
  UNMOCK(crypto_rand);
}

struct testcase_t prob_distr_tests[] = {
  { "logit_logistics", test_logit_logistic, TT_FORK, NULL, NULL },
  { "log_logistic", test_log_logistic, TT_FORK, NULL, NULL },
  { "weibull", test_weibull, TT_FORK, NULL, NULL },
  { "genpareto", test_genpareto, TT_FORK, NULL, NULL },
  { "uniform_interval", test_uniform_interval, TT_FORK, NULL, NULL },
  END_OF_TESTCASES
};

struct testcase_t slow_stochastic_prob_distr_tests[] = {
  { "stochastic_genpareto", test_stochastic_genpareto, TT_FORK, NULL, NULL },
  { "stochastic_geometric", test_stochastic_geometric, TT_FORK, NULL, NULL },
  { "stochastic_uniform", test_stochastic_uniform, TT_FORK, NULL, NULL },
  { "stochastic_logistic", test_stochastic_logistic, TT_FORK, NULL, NULL },
  { "stochastic_log_logistic", test_stochastic_log_logistic, TT_FORK, NULL,
    NULL },
  { "stochastic_weibull", test_stochastic_weibull, TT_FORK, NULL, NULL },
  END_OF_TESTCASES
};