The commit with Bessel functions i0e and i1e placed the ifdef/endif incorrectly,
causing i0e/i1e to be undefined when EIGEN_HAS_C99_MATH=0. These functions do not
actually require C99 math, so now they are always available.
Previously, when computing the derivative, it used a relative error threshold. Now it uses an absolute error threshold. The behavior for computing the value is unchanged. This makes more sense since we do not expect the derivative to often be close to zero. This change makes the derivatives about 30% faster across the board. The error for the igamma_der_a is almost unchanged, while for gamma_sample_der_alpha it is a bit worse for float32 and unchanged for float64.
In addition to igamma(a, x), this code implements:
* igamma_der_a(a, x) = d igamma(a, x) / da -- derivative of igamma with respect to the parameter
* gamma_sample_der_alpha(alpha, sample) -- reparameterization derivative of a Gamma(alpha, 1) random variable sample with respect to the alpha parameter
The derivatives are computed by forward mode differentiation of the igamma(a, x) code. Although gamma_sample_der_alpha can be implemented via igamma_der_a, a separate function is more accurate and efficient due to analytical cancellation of some terms. All three functions are implemented by a method parameterized with "mode" that always computes the derivatives, but does not return them unless required by the mode. The compiler is expected to (and, based on benchmarks, does) skip the unnecessary computations depending on the mode.
The functions are conventionally called i0e and i1e. The exponentially scaled version is more numerically stable. The standard Bessel functions can be obtained as i0(x) = exp(|x|) i0e(x)
The code is ported from Cephes and tested against SciPy.
1. Added new packet functions using SIMD for NByOne, OneByN cases
2. Modified existing packet functions to reduce index calculations when input stride is non-SIMD
3. Added 4 test cases to cover the new packet functions
Check for nan inputs and propagate them immediately. Limit the number of internal iterations to 2000 (same number as used by scipy.special.gammainc). This prevents an infinite loop when the function is called with nan or very large arguments.
Original change by mfirgunov@google.com
If the cost is large enough then the thread count can be larger than the maximum
representable int, so just casting it to an int is undefined behavior.
Contributed by phurst@google.com.
