Incorrect `Decimal.gamma` result for non-integer values < -50

[0i00000000a7]

console.log(Decimal.gamma(-49.5))
// Expected Result: 7.322269689234166e-64
// Actual Result: 7.322269689234166e-64
console.log(Decimal.gamma(-50.5))
// Expected Result: -1.4499543939077525e-65
// Actual Result: 0
console.log(Decimal.gamma(-51.5))
// Expected Result: 2.8154454250635958e-67
// Actual Result: 0

break_eternity.js/src/index.ts

Lines 223 to 228 in f5a0513

	if (n < -50) {
	if (n === Math.trunc(n)) {
	return Number.NEGATIVE_INFINITY;
	}
	return 0;
	}

[James103]

This behavior exists since the factorial and gamma functions were first implemented in 66e215f. Back then, there was no support for numbers < 10^-308; that support was first added a month later in b44c3cf.

According to Mathematica 12.0.1 for Linux x86 (64-bit) (October 16, 2019):

In[1]:= {-#-0.5, Re[Gamma[SetPrecision[-# - 0.5, 20]]]}& /@ (10 ^ Range[10])
(* should give the following output give or take whitespace *)

Out[1]= {
	{-10.5, -2.6401218205477163162463853253112`20.30102999566398*^-7},
	{-100.5, -3.353690819807678642208099692698441`17.33267989372423*^-159},
	{-1000.5, -2.46798686733393436083164952123569`16.160317894443477*^-2569},
	{-10000.5, -1.103720391734851431352329642647032379638`15.035692994887103*^-35661},
	{-100000.5, -3.51761474591359375489308481`13.9388113665821*^-456576},
	{-1.0000005*^6, -3.801569995948928887158132159581660893058`12.85963277969308*^-5565712},
	{-1.00000005*^7, -8.2621378534271665013189016850496831`11.792686244079254*^-65657063},
	{-1.000000005*^8, -1.9426077601811394627753501`10.689929781660727*^-756570560},
	{-1.0000000005*^9, -1.00302501892504334656796`9.643529250465605*^-8565705527},
	{-1.00000000005*^10, -1.35076007339293486292249304`8.601611145922602*^-95657055191}
}

Thus, the gamma function evaluated at a large negative non-integer $-n$ outputs a value on the order of $10^{-n \log n}$.