minor gamma cleanups.

This should arguably should use `Base.@assume_effects :terminates_locally` but that only exists on 1.8+.

Author: oscardssmith

src/gamma.jl

@@ -3,44 +3,27 @@ gamma(z::Number) = _gamma(float(z))
 _gamma(x::Float32) = Float32(_gamma(Float64(x)))
 
 function _gamma(x::Float64)
+    T = Float64
     if x < 0
-        isinteger(x) && throw(DomainError(x, "NaN result for non-NaN input."))
+        (isinteger(x) || x==-Inf) && throw(DomainError(x, "NaN result for non-NaN input."))
         xp1 = abs(x) + 1.0
-        return π / sinpi(xp1) / _gammax(xp1)
-    else
-        return _gammax(x)
+        return π / (sinpi(xp1) * _gamma(xp1))
     end
-end
-# only have a Float64 implementations
-function _gammax(x)
+    isfinite(x) || return x
     if x > 11.5
-        return large_gamma(x)
-    elseif x <= 11.5
-        return small_gamma(x)
-    elseif isnan(x)
-        return x
-    end
-end
-function large_gamma(x)
-    isinf(x) && return x
-    T = Float64
-    w = inv(x)
-    s = (
-        8.333333333333331800504e-2, 3.472222222230075327854e-3, -2.681327161876304418288e-3, -2.294719747873185405699e-4,
-        7.840334842744753003862e-4, 6.989332260623193171870e-5, -5.950237554056330156018e-4, -2.363848809501759061727e-5,
-        7.147391378143610789273e-4
-    )
-    w = w * evalpoly(w, s) + one(T)
-    # lose precision on following block
-    y = exp((x))
-    # avoid overflow
-    v = x^(0.5 * x - 0.25)
-    y = v * (v / y)
+        w = inv(x)
+        s = (
+            8.333333333333331800504e-2, 3.472222222230075327854e-3, -2.681327161876304418288e-3, -2.294719747873185405699e-4,
+            7.840334842744753003862e-4, 6.989332260623193171870e-5, -5.950237554056330156018e-4, -2.363848809501759061727e-5,
+            7.147391378143610789273e-4
+        )
+        w = muladd(w, evalpoly(w, s), 1.0)
+        # avoid overflow
+        v = x ^ muladd(0.5, x, -0.25)
+        y = v * (v / exp(x))
 
-    return SQ2PI(T) * y * w
-end
-function small_gamma(x)
-    T = Float64
+        return SQ2PI(T) * y * w
+    end
     P = (
         1.000000000000000000009e0, 8.378004301573126728826e-1, 3.629515436640239168939e-1, 1.113062816019361559013e-1,
         2.385363243461108252554e-2, 4.092666828394035500949e-3, 4.542931960608009155600e-4, 4.212760487471622013093e-5
@@ -51,17 +34,17 @@ function small_gamma(x)
         -1.397148517476170440917e-5
     )
 
-    z = one(T)
+    z = 1.0
     while x >= 3.0
-        x -= one(T)
+        x -= 1.0
         z *= x
     end
     while x < 2.0
         z /= x
-        x += one(T)
+        x += 1.0
     end
 
-    x -= T(2)
+    x -= 2.0
     p = evalpoly(x, P)
     q = evalpoly(x, Q)
     return z * p / q