Module special 

The gamma family’s backend for Compensated, one rung below SpecializedSpecialMath.

See the parent module for why this rung exists. In short: the gamma family is the part of thermite-special that is driven by fitted coefficients, and the width of a Compensated decides which coefficients are correct - so it needs a dispatch axis that Compensated’s single blanket backend impl does not have.

Implementing

Every method has a default, so the minimal impl is empty:

impl<V: FloatVector<Element = f64>> SpecializedCompensatedSpecialMath<Compensated<f64>>
    for Compensated<V> {}

Override a method when this width can do better than the generic series - typically by carrying a table tuned to it.

The generic algorithm

The defaults are Stirling and its derivatives, which is one expansion in one set of constants for the whole family. Shift the argument up by the recurrences until it is large ($x \gtrsim 30$ for double-double), then:

\ln\Gamma(x) \sim (x - \tfrac{1}{2})\ln x - x + \tfrac{1}{2}\ln 2\pi
    + \sum_{n \ge 1} \frac{B_{2n}}{2n(2n-1)x^{2n-1}}

\psi(x) \sim \ln x - \frac{1}{2x} - \sum_{n \ge 1} \frac{B_{2n}}{2n\,x^{2n}}
\qquad
\psi_1(x) \sim \frac{1}{x} + \frac{1}{2x^2} + \sum_{n \ge 1} \frac{B_{2n}}{x^{2n+1}}

The Bernoulli numbers are exact rationals, so unlike the minimax rationals the real f32/f64 paths use, they extend to any precision without refitting - there is no oracle to chase and no table to source. Roughly 13 terms clear $2^{-106}$ at $x > 30$, about half that for double-single, so the term count is a const off the mantissa width rather than a fixed loop.

This is why Compensated may never want Lanczos. The real paths use it because it skips the shift loop; here every operation is already an order of magnitude more expensive, so the loop costs relatively less and a 24-coefficient table at 32 digits costs a lot to source and validate.

Traits

CompensatedGammaOps

What a default body needs of Compensated<V> in order to do compensated arithmetic.

SpecializedCompensatedSpecialMath

Backend for the coefficient-bearing part of Compensated’s special math.