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Evaluate the incomplete gamma function's series faster - #2079
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Gamma(a, z) by its series is Gamma(a) - gamma(a, z), and Gamma(a) was computed by Spouge's approximation on every evaluation: six of the nine milliseconds at a hundred digits. It is remembered now, per order and precision, since a function the integrator samples at points is asked it at one a for every z. A real a and z sum the series in real arithmetic, one product and one quotient a term where the complex arithmetic took nineteen operations, and the logarithm on the cut takes pi from the cache rather than computing it afresh. gamma(2.91, -3.3) 9.52 -> 0.92 ms, gamma(1/3, 2) 23.0 -> 2.2 ms, gamma(1/2 + i, 2 + 3i) 11.8 -> 2.7 ms; the continued fraction and the exponential integral's series are unchanged. Found by #2075's next integration rule, whose answers one declined integral sampled three thousand times. Part of #2075. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Makes the upper incomplete gamma function's series several times cheaper. Part of #2075.
Gamma(a, z)by its series isGamma(a) - gamma(a, z), andGamma(a)was computed by Spouge's approximation on every evaluation: six of the nine milliseconds at a hundred digits.Gamma(a)is remembered, per order and precision. A function the integrator samples at points is asked it at oneafor everyz.aandzsum the series in real arithmetic. That is one product and one quotient a term, where the complex arithmetic took nineteen operations.pifrom the cache rather than computing it afresh.Measured on one machine, 40 evaluations each, at the default hundred digits, against master
b21f0138:gamma(2.91, -3.3)gamma(2.91, 3.3)gamma(12/5, -30)gamma(7.71, 2.1)gamma(1/3, 2)gamma(37/10, -7/3 + 2i)gamma(1/2 + i, 2 + 3i)gamma(0, -30), the exponential integral's seriesgamma(5/2, 40), the continued fractionValues are unchanged: the 37 tests against mpmath's
gammaincat 55 digits pass, and so does the suite (16,890) and the MCP tests (329).Found by #2075's next integration rule. Its answers carry the function, and one declined integral sampled them three thousand times.
🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura