float normalized_diffusion_pdf(
const float r,
const float l,
const float s)
{
return normalized_diffusion_pdf(r, l / s);
}
double normalized_diffusion_pdf(
const double r,
const double l,
const double s)
{
return normalized_diffusion_pdf(r, l / s);
}
double normalized_diffusion_sample(
const double u,
const double l,
const double s,
const double eps,
const size_t max_iterations)
{
assert(u >= 0.0);
assert(u < 1.0);
const double d = l / s;
// Handle the case where u is greater than the value we consider 1 in our CDF.
if (u >= nd_cdf_rmax)
return NdCdfTableRmax * d;
// Use the CDF to find an initial interval for the root of cdf(r, 1) - u = 0.
const size_t i = sample_cdf(nd_cdf_table, nd_cdf_table + NdCdfTableSize, u);
assert(i > 0);
assert(nd_cdf_table[i - 1] <= u);
assert(nd_cdf_table[i] > u);
// Transform the cdf(r, 1) interval to cdf(r, d) using the fact that cdf(r, d) == cdf(r/d, 1).
double rmin = fit<size_t, double>(i - 1, 0, NdCdfTableSize - 1, 0.0, NdCdfTableRmax) * d;
double rmax = fit<size_t, double>(i, 0, NdCdfTableSize - 1, 0.0, NdCdfTableRmax) * d;
assert(normalized_diffusion_cdf(rmin, d) <= u);
assert(normalized_diffusion_cdf(rmax, d) > u);
double r = (rmax + rmin) * 0.5;
// Refine the root.
for (size_t i = 0; i < max_iterations; ++i)
{
// Use bisection if we go out of bounds.
if (r < rmin || r > rmax)
r = (rmax + rmin) * 0.5;
const double f = normalized_diffusion_cdf(r, d) - u;
// Convergence test.
if (abs(f) <= eps)
break;
// Update bounds.
f < 0.0 ? rmin = r : rmax = r;
// Newton step.
const double df = normalized_diffusion_pdf(r, d);
r -= f / df;
}
return r;
}
