[Rafael-SOWNet]

Evaluates gamma(a, z) in the left half-plane by a series whose terms do not cancel there, and makes each part of the value on the cut accurate to its own size. Part of #2075.

Speed. gamma(a, z) was Gamma(a) less the lower function, summed everywhere as z^a e^(-z) sum_k z^k/(a)_(k + 1) (DLMF 8.7.1). On the negative axis its terms alternate, are e^|z| in size and sum to e^(-|z|). So twice |z|/ln 10 guard digits were carried, and Gamma(a) was computed afresh at that precision. In the left half-plane the lower function is now z^a sum_k (-z)^k/(k! (a + k)) (DLMF 8.7.3), as E_1 already was. Its terms share a sign on that axis, and it loses (|z| + Re z)/ln 10 digits.

at 100 digits	master	this
gamma(1/3, -143)	9,560 ms	6 ms
gamma(1/2, -100 + 5i)	5,568 ms	9 ms
gamma(-5/2, -60)	2,354 ms	6 ms
gamma(1/3, -30)	1,671 ms	3 ms

The first is the evaluation that made #2087 change a test's sign, e^(c (d + f x)^3) to e^(-c (d + f x)^3), to keep it under a second.

Two defects on the cut, which the guard digits had hidden. Each part of the value should be accurate to its own size, and on the negative axis one part is often exactly Gamma(a) or a multiple of pi.

• z^a was e^(a ln|z| + i a pi) with a pi rounded, so cos(pi/2) came out as 10^(-W) where it is 0. Times the lower function's e^143, that was the real part of gamma(1/2, -143), which is sqrt(pi). The angle is now reduced exactly before pi multiplies it, and the cosine is exactly 0 at a half-whole a.
• Past |z| = (P + 5) ln 10 the continued fraction is used, and on the cut its coefficients are real. It converges to e^(i pi a) times a real function, the mean of the two sides, and falls short of the value from above by -i e^(i pi a) sin(pi a) Gamma(a). That is below the precision of the modulus, but it is the whole of the real part at a half-whole a, and of the imaginary part at a = -n, where it is -i (-1)^n pi/n!. On master the real part of gamma(1/2, -500) was 2.9e100, where it is sqrt(pi).

Checked part by part against mpmath at 300 to 500 digits. The grid is 16 orders (half-whole, whole, complex and odd-denominator ones among them) at 20 points of the left half-plane, on both sides of the threshold:

	master	this
240 points short of the threshold: worst part	9e-98 of its size	9e-98 of its size
80 points past it: parts wrong	18, by up to 1e113 times their size	0
the grid's time	48.4 s	3.4 s

The worst part on either side is gamma(7/3, -5 - 2i)'s, and comes from 7/3 itself being rounded to 100 digits.

Measured on Rubi's suite, with the harness that reads GAMMA, against this branch's base 3dbae246, the whole corpus each side:

	3dbae246	this
solved	48,875 of 49,052	48,880
wrong	0	0
timeouts	7	2
rows lost		0

All five gains were timeouts of the harness's own check. After the library answers, it evaluates the answer again at 300 and 2000 digits, and for 6.1.1:205-207 and 6.1.3:98/100 that put gamma(a, z) in the left half-plane. They now finish in 0.08 to 1.9 s. The time per stream is 1.006, 1.005 and 0.922 of the base's. Two rows' allocation grew in the run, 6.1.7:196 and 8.1:302, and alone they are the same on both builds: 1.079 GB each, and 1.406 against 1.408 GB.

Eight tests added: three far into the left half-plane, four past the threshold on the cut, and gamma(1/3, -143) with the real root. The suite passes (16,947), and so do the MCP tests (329).

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura