Move source files and headers to separate directories.

* extent-sampling.h, gaussian-nd-random.h, nd-random.h, oracles.h,
utils.h: Move into include directory.
* extent-sampling.c, gaussian-nd-random.c, nd-random.c, oracles.c,
utils.c: Move into src directory.
* CMakeLists.txt: Set include as include directory. Look for source
files inside src directory.

5 files changed, 635 insertions, 0 deletions

diff --git a/src/extent-sampling.c b/src/extent-sampling.c
new file mode 100644
index 0000000..eff3b3f
--- /dev/null
+++ b/src/extent-sampling.c
@@ -0,0 +1,161 @@
+/*
+  This file contains all the core extent sampling routines. From time
+  to time as research progresses, I keep refactoring this code to be
+  clearer and more efficient. This mostly involves throwing out old
+  code even when such old code may be required in the future. I
+  figured that keeping old code around is not worth the trouble
+  because it impedes clarity of thought.
+*/
+
+#include <math.h>
+#include <stddef.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_integration.h>
+#include <gsl/gsl_rstat.h>
+#include <gsl/gsl_randist.h>
+#include "extent-sampling.h"
+#include "nd-random.h"
+#include "utils.h"
+
+#define VOLUME_MINIMUM_SAMPLES 100
+#define INTEGRAL_MINIMUM_SAMPLES 100
+#define CONFIDENCE_INTERVAL_FACTOR 1.96
+
+#define INTEGRATION_INTERVALS 1000
+
+#define SOLID_ANGLE_START 0.0
+#define SOLID_ANGLE_LAST 0.5
+#define SOLID_ANGLE_STEPS 100
+
+double volume (extent_oracle_t extent_oracle, double true_volume,
+               const gsl_rng* r, unsigned int dimension, double rtol,
+               gsl_rstat_workspace* stats)
+{
+  gsl_vector* x = gsl_vector_alloc(dimension);
+  double volume, error;
+  double vn = ln_volume_of_ball(dimension);
+  do {
+    random_direction_vector(r, x);
+    double extent = extent_oracle(x);
+    double volume_from_sample = exp(vn + dimension*log(extent));
+    gsl_rstat_add(volume_from_sample, stats);
+    volume = gsl_rstat_mean(stats);
+    error = CONFIDENCE_INTERVAL_FACTOR * gsl_rstat_sd_mean(stats) / volume;
+  } while ((error > rtol) || (rerror(volume, true_volume) > rtol) || (gsl_rstat_n(stats) < VOLUME_MINIMUM_SAMPLES));
+  gsl_vector_free(x);
+  return volume;
+}
+
+double volume_window (extent_oracle_t extent_oracle, double true_volume,
+                      const gsl_rng* r, unsigned int dimension, double rtol,
+                      unsigned int* number_of_samples)
+{
+  gsl_rstat_workspace* stats = gsl_rstat_alloc();
+  gsl_vector* x = gsl_vector_alloc(dimension);
+  double volume, error;
+  double vn = ln_volume_of_ball(dimension);
+  // This is the window length used in Volume.m of Lovasz-Vempala's
+  // code.
+  int window_length = 4*dimension*dimension + 500;
+  int accurate_estimates = 0;
+  do {
+    random_direction_vector(r, x);
+    double extent = extent_oracle(x);
+    double volume_from_sample = exp(vn + dimension*log(extent));
+    gsl_rstat_add(volume_from_sample, stats);
+    volume = gsl_rstat_mean(stats);
+    error = rerror(volume, true_volume);
+    if (error < rtol) accurate_estimates++;
+    else accurate_estimates = 0;
+  } while (accurate_estimates < window_length);
+  *number_of_samples = gsl_rstat_n(stats);
+  gsl_vector_free(x);
+  gsl_rstat_free(stats);
+  return volume;
+}
+
+double integral_per_direction
+(integrand_t integrand, const gsl_vector* direction, const gsl_rng* r, unsigned int n,
+ double radius, double rtol, int* neval)
+{
+  double f (double r, void* params)
+  {
+    (*neval)++;
+    return gsl_pow_uint(r, n-1) * integrand(r, direction);
+  }
+
+  gsl_function gsl_f = {&f};
+  double result, error;
+  gsl_integration_workspace *w = gsl_integration_workspace_alloc(INTEGRATION_INTERVALS);
+  gsl_integration_qag(&gsl_f, 0, radius, 0, rtol, INTEGRATION_INTERVALS, GSL_INTEG_GAUSS15, w, &result, &error);
+  gsl_integration_workspace_free(w);
+  return surface_area_of_ball(n) * result;
+}
+
+double integral
+(integrand_t integrand, extent_oracle_t extent_oracle, double true_integral,
+ const gsl_rng* r, unsigned int dimension, double rtol,
+ gsl_rstat_workspace* stats)
+{
+  gsl_vector* x = gsl_vector_alloc(dimension);
+  double integral, error;
+  do {
+    int neval = 0;
+    random_direction_vector(r, x);
+    double extent = extent_oracle(x);
+    double integral_from_sample = integral_per_direction(integrand, x, r, dimension, extent, rtol, &neval);
+    gsl_rstat_add(integral_from_sample, stats);
+    integral = gsl_rstat_mean(stats);
+    error = CONFIDENCE_INTERVAL_FACTOR * gsl_rstat_sd_mean(stats) / integral;
+  } while ((error > rtol) || (rerror(integral, true_integral) > rtol) || (gsl_rstat_n(stats) < INTEGRAL_MINIMUM_SAMPLES));
+  if (error > rtol)
+    GSL_ERROR_VAL("integral failed to converge", GSL_ETOL, integral);
+  gsl_vector_free(x);
+  return integral;
+}
+
+double volume_cone (extent_oracle_t extent_oracle, const gsl_rng* r,
+                    const gsl_vector* mean, double omega_min, double omega_max,
+                    unsigned int number_of_samples, double* variance)
+{
+  unsigned int n = mean->size;
+  gsl_vector* x = gsl_vector_alloc(n);
+  double theta_min = solid_angle_to_planar_angle(omega_min * surface_area_of_ball(n), n);
+  double theta_max = solid_angle_to_planar_angle(omega_max * surface_area_of_ball(n), n);
+  double vn = ln_volume_of_ball(n);
+  gsl_rstat_workspace* stats = gsl_rstat_alloc();
+  for (int i=0; i<number_of_samples; i++) {
+    hollow_cone_random_vector(r, mean, theta_min, theta_max, x);
+    double extent = extent_oracle(x);
+    double volume_from_sample = exp(vn + n*log(extent));
+    gsl_rstat_add(volume_from_sample, stats);
+  }
+
+  if (variance) *variance = gsl_rstat_variance(stats);
+
+  gsl_rstat_free(stats);
+  gsl_vector_free(x);
+  return gsl_rstat_mean(stats) * (omega_max - omega_min);
+}
+
+double volume_experiment
+(extent_oracle_t extent_oracle, const gsl_rng* r,
+ const gsl_vector* mean, unsigned int samples_per_cone,
+ double solid_angle_factor, double solid_angle_threshold_exponent_factor,
+ unsigned int* number_of_samples)
+{
+  int n = mean->size;
+  double volume = 0;
+  double omega_max = 0.5;
+  int N = 0;
+  do {
+    double omega_min = omega_max / solid_angle_factor;
+    volume += volume_cone(extent_oracle, r, mean, omega_min, omega_max, samples_per_cone, NULL);
+    N += samples_per_cone;
+    omega_max = omega_min;
+  } while (omega_max > pow(2, -solid_angle_threshold_exponent_factor*n));
+  volume += volume_cone(extent_oracle, r, mean, 0, omega_max, samples_per_cone, NULL);
+  N += samples_per_cone;
+  if (number_of_samples) *number_of_samples = N;
+  return volume;
+}
diff --git a/src/gaussian-nd-random.c b/src/gaussian-nd-random.c
new file mode 100644
index 0000000..16abc94
--- /dev/null
+++ b/src/gaussian-nd-random.c
@@ -0,0 +1,51 @@
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_randist.h>
+#include <gsl/gsl_sf_gamma.h>
+#include "nd-random.h"
+#include "utils.h"
+
+#define INTEGRATION_EPSABS 1e-3
+#define INTEGRATION_EPSREL 1e-3
+
+unsigned int shifted_gaussian_random_vector
+(const gsl_rng* r, const gsl_vector* mean, double theta_max, double standard_deviation, gsl_vector* x)
+{
+  for (int i=0; i<x->size; i++)
+    gsl_vector_set(x, i, gsl_ran_gaussian(r, standard_deviation));
+  gsl_vector_add(x, mean);
+  gsl_blas_dscal(1/gsl_blas_dnrm2(x), x);
+  if (angle_between_vectors(x, mean) > theta_max)
+    return 1 + shifted_gaussian_random_vector(r, mean, theta_max, standard_deviation, x);
+  else return 1;
+}
+
+double shifted_gaussian_pdf (double theta, double mean, double theta_max,
+                             double standard_deviation, unsigned int dimension, gsl_integration_workspace* w)
+{
+  // NOTE: Due to the rejection sampling, the returned result is only
+  // proportional to the actual PDF.
+  if (theta > theta_max) return 0;
+  
+  double projection = mean*cos(theta);
+  double integrand (double r, void* params)
+  {
+    return gsl_pow_uint(r, dimension - 1)
+      * gaussian_pdf((r - projection) / standard_deviation);
+  }
+  
+  gsl_function gsl_integrand = {&integrand};
+  double result, error;
+  /* gsl_integration_qagiu(&gsl_integrand, 0, INTEGRATION_EPSABS, INTEGRATION_EPSREL, */
+  /*                       w->limit, w, &result, &error); */
+  double rmax = 0.5 * (projection + sqrt(gsl_pow_2(projection) + 4*(dimension-1)*gsl_pow_2(standard_deviation)));
+  double halfwidth = 3 * standard_deviation;
+  gsl_integration_qag(&gsl_integrand,
+                      rmax - halfwidth < 0 ? 0 : rmax - halfwidth,
+                      rmax + halfwidth,
+                      INTEGRATION_EPSABS, INTEGRATION_EPSREL,
+                      w->limit, GSL_INTEG_GAUSS41, w, &result, &error);
+  return result
+    * gaussian_pdf(sqrt(gsl_pow_2(mean) - gsl_pow_2(projection)) / standard_deviation)
+    / gsl_pow_uint(M_SQRT2*sqrt(M_PI), dimension - 2)
+    / gsl_pow_uint(standard_deviation, dimension);
+}
diff --git a/src/nd-random.c b/src/nd-random.c
new file mode 100644
index 0000000..25abc7a
--- /dev/null
+++ b/src/nd-random.c
@@ -0,0 +1,101 @@
+#include <float.h>
+#include <math.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_randist.h>
+#include <gsl/gsl_sf_gamma.h>
+#include "nd-random.h"
+#include "utils.h"
+
+static double beta_inc_unnormalized (double a, double b, double x);
+static double incomplete_wallis_integral (double theta, unsigned int m);
+
+void random_direction_vector (const gsl_rng* r, gsl_vector* x)
+{
+  gsl_ran_dir_nd(r, x->size, x->data);
+}
+
+static void rotate_from_nth_canonical (gsl_vector* x, const gsl_vector* orient)
+{
+  const size_t n = x->size;
+  double xn = gsl_vector_get(x, n - 1);
+  double mun = gsl_vector_get(orient, n - 1);
+  gsl_vector_const_view orient_sub = gsl_vector_const_subvector(orient, 0, n - 1);
+  double b = gsl_blas_dnrm2(&orient_sub.vector);
+  double a = (dot_product(orient, x) - xn*mun) / b;
+  double c = mun, s = sqrt(1 - gsl_pow_2(c));
+  gsl_blas_daxpy((xn*s + a*(c - 1))/b, orient, x);
+  gsl_vector_set(x, n - 1, gsl_vector_get(x, n - 1)
+                 + xn*(c - 1) - a*s
+                 - mun*(xn*s + a*(c - 1))/b);
+}
+
+/* TODO: There is an edge case when mean is the (n-1)th canonical
+   basis vector. Fix it.
+*/
+void hollow_cone_random_vector (const gsl_rng* r, const gsl_vector* mean, double theta_min, double theta_max, gsl_vector* x)
+{
+  unsigned int n = x->size;
+  gsl_ran_dir_nd(r, n - 1, x->data);
+
+  // Generate random vector around the nth canonical basis vector
+  double omega_min = planar_angle_to_solid_angle(theta_min, n);
+  double omega_max = planar_angle_to_solid_angle(theta_max, n);
+  gsl_vector_scale(x, cos(theta_max) * tan(solid_angle_to_planar_angle(gsl_ran_flat(r, omega_min, omega_max), n)));
+  gsl_vector_set(x, n - 1, cos(theta_max));
+  gsl_vector_scale(x, 1.0/gsl_blas_dnrm2(x));
+
+  // Rotate to arbitrary basis
+  rotate_from_nth_canonical(x, mean);
+}
+
+void subsampling_random_vector (const gsl_rng* r, const gsl_vector* mean, double theta_max, gsl_vector* x)
+{
+  hollow_cone_random_vector(r, mean, 0, theta_max, x);
+}
+
+static double beta_inc_unnormalized (double a, double b, double x)
+{
+  return gsl_sf_beta_inc(a, b, x) * gsl_sf_beta(a, b);
+}
+
+static double incomplete_wallis_integral (double theta, unsigned int m)
+{
+  /**
+     @param theta 0 < theta < pi
+     @param m
+     @return \int_0^\theta \sin^m x dx
+   **/
+  if ((theta < 0) || (theta > M_PI))
+    GSL_ERROR("Incomplete Wallis integral only allows theta in [0,pi]", GSL_EDOM);
+  if (theta < M_PI_2)
+    return 0.5 * beta_inc_unnormalized((m+1)/2.0, 0.5, gsl_pow_2(sin(theta)));
+  else
+    return 0.5 * (gsl_sf_beta((m+1)/2.0, 0.5) + beta_inc_unnormalized(0.5, (m+1)/2.0, gsl_pow_2(cos(theta))));
+}
+
+double planar_angle_to_solid_angle (double planar_angle, unsigned int dimension)
+{
+  return 2 * pow(M_PI, (dimension - 1)/2.0)
+    * incomplete_wallis_integral(planar_angle, dimension - 2)
+    / gsl_sf_gamma((dimension - 1)/2.0);
+}
+
+double solid_angle_to_planar_angle (double solid_angle, unsigned int dimension)
+{
+  double f (double planar_angle, void* params)
+  {
+    return planar_angle_to_solid_angle(planar_angle, dimension) - solid_angle;
+  }
+
+  gsl_function gsl_f = {&f};
+  /* if (fabs(GSL_FN_EVAL(&gsl_f, 0))/surface_area_of_ball(dimension) < BISECTION_FUNCTION_EPSABS) */
+  /*   return 0; */
+  /* else if (fabs(GSL_FN_EVAL(&gsl_f, M_PI))/surface_area_of_ball(dimension) < BISECTION_FUNCTION_EPSABS) */
+  /*   return M_PI; */
+  /* else return bisection(&gsl_f, 0, M_PI); */
+  if (solid_angle == 0)
+    return 0;
+  else if (solid_angle == surface_area_of_ball(dimension))
+    return M_PI;
+  else return bisection(&gsl_f, 0, M_PI);
+}
diff --git a/src/oracles.c b/src/oracles.c
new file mode 100644
index 0000000..80f401f
--- /dev/null
+++ b/src/oracles.c
@@ -0,0 +1,141 @@
+#include <math.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_randist.h>
+#include "oracles.h"
+#include "utils.h"
+
+double bernoulli_extent_generator
+(const gsl_rng* r, double p, double r0, double r1)
+{
+  return gsl_ran_bernoulli(r, p) ? r1 : r0;
+}
+
+double bernoulli_true_volume (double p, double r0, double r1, unsigned int dimension)
+{
+  return volume_of_ball(dimension)
+    * (p * gsl_pow_uint(r1, dimension)
+       + (1 - p) * gsl_pow_uint(r0, dimension));
+}
+
+double uniform_extent_generator (const gsl_rng* r, double a, double b)
+{
+  return gsl_ran_flat(r, a, b);
+}
+
+// TODO: Verify the accuracy of this function for non-trivial a, b
+double uniform_true_volume (double a, double b, unsigned int dimension)
+{
+  return exp(ln_volume_of_ball(dimension) + dimension*log(b) - log(dimension + 1))
+    - exp(ln_volume_of_ball(dimension) + dimension*log(a) - log(dimension + 1));
+}
+
+double beta_extent_generator
+(const gsl_rng* r, double alpha, double beta)
+{
+  return gsl_ran_beta(r, alpha, beta);
+}
+
+double beta_true_volume (double alpha, double beta, unsigned int dimension)
+{
+  double vol = volume_of_ball(dimension);
+  for (unsigned int r=0; r<dimension; r++)
+    vol *= (alpha + r) / (alpha + beta + r);
+  return vol;
+}
+
+double symmetric_spiral_extent_oracle (const gsl_vector* x)
+{
+  double angle = atan2(gsl_vector_get(x, 1), gsl_vector_get(x, 0));
+  return fabs(angle);
+}
+
+double right_triangle_extent_oracle (const gsl_vector* x, double base, double height)
+{
+  double max_angle = atan(height / base);
+  double angle = atan2(gsl_vector_get(x, 1), gsl_vector_get(x, 0));
+  return (angle > 0) && (angle < max_angle) ? base / cos(angle) : 0;
+}
+
+double right_triangle_true_volume (double base, double height)
+{
+  return 0.5 * base * height;
+}
+
+double sphere_extent_oracle (const gsl_vector* x, double radius)
+{
+  return radius;
+}
+
+double sphere_maximum_extent (double radius)
+{
+  return radius;
+}
+
+double plane_extent_oracle (const gsl_vector* x, double displacement)
+{
+  return displacement / fabs(gsl_vector_get(x, 0));
+}
+
+double infinity_norm (const gsl_vector* x)
+{
+  double max = fabs(gsl_vector_get(x, 0));
+  for (int i=1; i<x->size; i++)
+    max = GSL_MAX(max, fabs(gsl_vector_get(x, i)));
+  return max;
+}
+
+double cube_extent_oracle (const gsl_vector* x, double edge)
+{
+  return edge / 2 / infinity_norm(x);
+}
+
+double cube_extent_oracle_with_center
+(const gsl_vector* x, const gsl_vector* center, double edge)
+{
+  double min = (0.5*edge - GSL_SIGN(gsl_vector_get(x, 0))*gsl_vector_get(center, 0))
+    / fabs(gsl_vector_get(x, 0));
+  for (int i=1; i<center->size; i++)
+    min = GSL_MIN(min, (0.5*edge - GSL_SIGN(gsl_vector_get(x, i))*gsl_vector_get(center, i))
+                  / fabs(gsl_vector_get(x, i)));
+  return min;
+}
+
+double cube_true_volume (double edge, unsigned int dimension)
+{
+  return gsl_pow_uint(edge, dimension);
+}
+
+double cube_maximum_extent (double edge, unsigned int dimension)
+{
+  return edge * sqrt(dimension) / 2;
+}
+
+double ellipsoid_extent_oracle (const gsl_vector* x, const gsl_vector* axes)
+{
+  double k = 0;
+  for (int i=0; i<axes->size; i++)
+    k += gsl_pow_2(gsl_vector_get(x, i) / gsl_vector_get(axes, i));
+  return 1/sqrt(k);
+}
+
+double ellipsoid_true_volume (const gsl_vector* axes)
+{
+  unsigned int dimension = axes->size;
+  double vol = volume_of_ball(dimension);
+  for (int i=0; i<dimension; i++)
+    vol *= gsl_vector_get(axes, i);
+  return vol;
+}
+
+double spheroid_extent_oracle (const gsl_vector* x, double eccentricity)
+{
+  unsigned int dimension = x->size;
+  gsl_vector_const_view xsub = gsl_vector_const_subvector(x, 1, dimension - 1);
+  return 1/sqrt(gsl_pow_2(gsl_blas_dnrm2(&xsub.vector))
+                + gsl_pow_2(gsl_vector_get(x, 0) / eccentricity));
+}
+
+double spheroid_true_volume (double eccentricity, unsigned int dimension)
+{
+  return volume_of_ball(dimension) * eccentricity;
+}
diff --git a/src/utils.c b/src/utils.c
new file mode 100644
index 0000000..0e0205f
--- /dev/null
+++ b/src/utils.c
@@ -0,0 +1,181 @@
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_randist.h>
+#include <gsl/gsl_sf.h>
+#include <gsl/gsl_sf_gamma.h>
+#include "utils.h"
+
+// This function exists so that guile code can access the value of
+// M_PI as provided by <gsl_math.h>.
+double pi ()
+{
+  /**
+   @return Value of pi
+  **/
+  return M_PI;
+}
+
+double volume_of_ball (unsigned int dimension)
+{
+  /**
+     @param dimension Dimension of the ball (2 for a circle, 3 for a sphere, etc.)
+     @return Volume of the unit ball
+   **/
+  return exp(ln_volume_of_ball(dimension));
+}
+
+double ln_volume_of_ball (unsigned int dimension)
+{
+  /**
+     @param dimension Dimension of the ball (2 for a circle, 3 for a sphere, etc.)
+     @return Natural logarithm of the volume of the unit ball
+   **/
+  return (dimension/2.0)*M_LNPI - gsl_sf_lngamma(1 + dimension/2.0);
+}
+
+double surface_area_of_ball (unsigned int dimension)
+{
+  /**
+     @param dimension Dimension of the ball (2 for a circle, 3 for a sphere, etc.)
+     @return Surface area of the unit ball
+   **/
+  return dimension * volume_of_ball(dimension);
+}
+
+double ln_surface_area_of_ball (unsigned int dimension)
+{
+  /**
+     @param dimension Dimension of the ball (2 for a circle, 3 for a sphere, etc.)
+     @return Natural logarithm of the surface area of the unit ball
+   **/
+  return log(dimension) + ln_volume_of_ball(dimension);
+}
+
+double lower_incomplete_gamma (double s, double x)
+{
+  return gsl_sf_gamma(s) * gsl_sf_gamma_inc_P(s, x);
+}
+
+double angle_between_vectors (const gsl_vector* x, const gsl_vector* y)
+{
+  /**
+     @param x Vector 1
+     @param y Vector 2
+     @return Angle between the two vectors in the range [0, pi]
+  **/
+  double dot_product;
+  gsl_blas_ddot(x, y, &dot_product);
+  // TODO: Is this a valid floating point comparison?
+  if (dot_product != 0)
+    return acos(dot_product / gsl_blas_dnrm2(x) / gsl_blas_dnrm2(y));
+  else return 0;
+}
+
+double dot_product (const gsl_vector* x, const gsl_vector* y)
+{
+  /**
+     @param x Vector 1
+     @param y Vector 2
+     @return Dot product between the two vectors
+  **/
+  double result;
+  gsl_blas_ddot(x, y, &result);
+  return result;
+}
+
+double gaussian_pdf (double x)
+{
+  /**
+     @param x
+     @return exp(-x^2/2) / sqrt(2*pi)
+   **/
+  return gsl_ran_gaussian_pdf(x, 1.0);
+}
+
+double gaussian_cdf (double x)
+{
+  /**
+     @param x
+     @return int_{-inf}^x gaussian_pdf(t) dt
+   **/
+  return 0.5*(1 + gsl_sf_erf(x/M_SQRT2));
+}
+
+double rerror (double approx, double exact)
+{
+  /**
+     @param approx Approximate value
+     @param exact Exact value
+     @return Relative error
+  **/
+  return fabs(1 - approx/exact);
+}
+
+double gprate (double start, double last, unsigned int n)
+{
+  /**
+     @param start First (0th) term of the geometric progression
+     @param last Last ('N-1'th) term of the geometric progression
+     @param n Number of terms in the geometric progression
+     @return The multiplicative factor of the geometric progression
+   **/
+  return pow(last/start, 1.0/(n - 1));
+}
+
+double gpterm (double a, double r, int n)
+{
+  /**
+     @param a First (0th) term of the geometric progression
+     @param r Factor of the geometric progression
+     @return The Nth term of the geometric progression, with N starting from 0
+   **/
+  return a * gsl_pow_uint(r, n);
+}
+
+#define BISECTION_EPSABS 0
+#define BISECTION_EPSREL 1e-6
+#define BISECTION_MAX_ITERATIONS 1000
+#define BISECTION_FUNCTION_EPSABS 1e-9
+double bisection (gsl_function* f, double a, double b)
+{
+  const gsl_root_fsolver_type* T = gsl_root_fsolver_bisection;
+  gsl_root_fsolver* s = gsl_root_fsolver_alloc(T);
+  double solution;
+  int status;
+
+  void error_handler (const char* reason, const char* file, int line, int gsl_errno)
+  {
+    fprintf(stderr, "Bisection error handler invoked.\n");
+    fprintf(stderr, "f(%g) = %g\n", a, GSL_FN_EVAL(f, a));
+    fprintf(stderr, "f(%g) = %g\n", b, GSL_FN_EVAL(f, b));
+    fprintf(stderr, "gsl: %s:%d: ERROR: %s\n", file, line, reason);
+    abort();
+  }
+
+  gsl_error_handler_t* old_handler = gsl_set_error_handler(error_handler);
+  gsl_root_fsolver_set(s, f, a, b);
+  do {
+    status = gsl_root_fsolver_iterate (s);
+    solution = gsl_root_fsolver_root (s);
+    a = gsl_root_fsolver_x_lower (s);
+    b = gsl_root_fsolver_x_upper (s);
+    status = gsl_root_test_interval (a, b, BISECTION_EPSABS, BISECTION_EPSREL);
+  } while (status == GSL_CONTINUE);
+  gsl_root_fsolver_free(s);
+  gsl_set_error_handler(old_handler);
+  return solution;
+}
+
+double bisection_rlimit (gsl_function* f, double a, double b)
+{
+  int sign = SIGNUM(GSL_FN_EVAL(f, a));
+  for (int i=0; i<BISECTION_MAX_ITERATIONS; i++)
+    if (sign*GSL_FN_EVAL(f, b) > 0) b *= 2;
+    else return b;
+  fprintf(stderr, "Bisection bracketing failed\n");
+  exit(1);
+}
+#undef BISECTION_EPSABS
+#undef BISECTION_EPSREL
+#undef BISECTION_MAX_ITERATIONS
+#undef BISECTION_FUNCTION_EPSABS