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Gamma.hf's derivatives keep their digits from y = 30 (#777, Gamma.hf derivatives) - #786
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…derivatives) Below y = beta x = 1000 the Gamma hazard was f / S, exp(log f - log S), whose derivatives in x and alpha are differences of terms of size 1 that cancel to a size 1 / y: alpha's was 2e-7 off mpmath's at y = 100 and 4e-5 at 1000, x's 8e-8 at 1000, and the second derivative in alpha off by more than itself. The continued fraction of the upper incomplete gamma now takes over from y = 30 (and a + 2 sqrt(a) + 1), as an autograd primitive rather than a traced loop (traced from y = 10 it made a Gamma PH fit 1.7 times slower). Its derivative in y is r (a - 1) g / y, with no cancelling terms; in alpha the fraction's own derivative, a complex step through the same Lentz loop (about the cost of the value); the second derivatives are carried along the loop (forward mode), so a Hessian is exact. Values, first and second derivatives are within 1e-14 of mpmath from y = 30 up. Six GammaPH/AFT fits take 1.01 times as long in all (1.08 at most, on data far into the tail; best of runs alternated with develop's). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01TRhKL2fJBiAAfNo9rihwts
This was referenced Oct 9, 2026
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The derivatives of log P and log Q in the shape were five-point differences of scipy's values: 1e-10 off mpmath's typically, up to 4e-9 (at a = 0.5, x = 1), wherever the Gamma's hazard is taken from them (a standardised time below 30; above it the hazard has its own fraction, #786). They are now carried along the series of P (below x = a + 1) and Legendre's continued fraction of Q (at or above it), in real forward mode, each to rounding: within 1e-14 of mpmath, median 3e-16. Where neither settles (a shape of millions with x near it), and at x of 0 or inf, they are differenced as before; the shape curvature a Hessian takes is the difference of the exact first derivative. Decided with the user: the cost is accepted. A GammaPH fit on 400 rows takes 0.32 s where it took 0.21 s (GammaAFT 0.17 to 0.24 s, a censored Gamma 0.04 to 0.05 s); the estimates are the same. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01TRhKL2fJBiAAfNo9rihwts
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Part of #777.
Below y = beta x = 1000 the Gamma hazard was f / S, exp(log f - log S),
whose derivatives in x and alpha are differences of terms of size 1
that cancel to a size 1 / y: alpha's was 2e-7 off mpmath's at y = 100
and 4e-5 at 1000, x's 8e-8 at 1000, and the second derivative in alpha
off by more than itself.
The continued fraction of the upper incomplete gamma now takes over
from y = 30 (and a + 2 sqrt(a) + 1), as an autograd primitive rather
than a traced loop (traced from y = 10 it made a Gamma PH fit 1.7 times
slower). Its derivative in y is r (a - 1) g / y, with no cancelling
terms; in alpha the fraction's own derivative, a complex step through
the same Lentz loop (about the cost of the value); the second
derivatives are carried along the loop (forward mode), so a Hessian is
exact. Values, first and second derivatives are within 1e-14 of mpmath
from y = 30 up. Six GammaPH/AFT fits take 1.01 times as long in all
(1.08 at most, on data far into the tail; best of runs alternated with
develop's).
🤖 Generated with Claude Code
https://claude.ai/code/session_01TRhKL2fJBiAAfNo9rihwts
Generated by Claude Code