Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Average Error: 0.02% → 0.01%

Time: 4.6s

Precision: binary64

Cost: 6848

Math

\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

↓

\[\mathsf{fma}\left(y, x - 0.5, 0.918938533204673 - x\right) \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

↓

(FPCore (x y) :precision binary64 (fma y (- x 0.5) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

↓

double code(double x, double y) {
	return fma(y, (x - 0.5), (0.918938533204673 - x));
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

↓

function code(x, y)
	return fma(y, Float64(x - 0.5), Float64(0.918938533204673 - x))
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

↓

code[x_, y_] := N[(y * N[(x - 0.5), $MachinePrecision] + N[(0.918938533204673 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

2. Simplified 0.01

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right)} \]
   Proof

   [Start] 0.02	\[ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   sub-neg [=>] 0.02	\[ \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y \cdot 0.5\right)\right)} + 0.918938533204673 \]
   +-commutative [=>] 0.02	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]
   sub-neg [=>] 0.02	\[ \left(\left(-y \cdot 0.5\right) + x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) + 0.918938533204673 \]
   distribute-lft-in [=>] 0.02	\[ \left(\left(-y \cdot 0.5\right) + \color{blue}{\left(x \cdot y + x \cdot \left(-1\right)\right)}\right) + 0.918938533204673 \]
   *-commutative [<=] 0.02	\[ \left(\left(-y \cdot 0.5\right) + \left(\color{blue}{y \cdot x} + x \cdot \left(-1\right)\right)\right) + 0.918938533204673 \]
   associate-+r+ [=>] 0.02	\[ \color{blue}{\left(\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + x \cdot \left(-1\right)\right)} + 0.918938533204673 \]
   associate-+l+ [=>] 0.02	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + \left(x \cdot \left(-1\right) + 0.918938533204673\right)} \]
   distribute-rgt-neg-in [=>] 0.02	\[ \left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + \left(x \cdot \left(-1\right) + 0.918938533204673\right) \]
   distribute-lft-out [=>] 0.02	\[ \color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + \left(x \cdot \left(-1\right) + 0.918938533204673\right) \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, x \cdot \left(-1\right) + 0.918938533204673\right)} \]
   +-commutative [=>] 0.01	\[ \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, x \cdot \left(-1\right) + 0.918938533204673\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(y, x + \color{blue}{-0.5}, x \cdot \left(-1\right) + 0.918938533204673\right) \]
   +-commutative [<=] 0.01	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 + x \cdot \left(-1\right)}\right) \]
   *-commutative [=>] 0.01	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{\left(-1\right) \cdot x}\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{-1} \cdot x\right) \]
   mul-1-neg [=>] 0.01	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{\left(-x\right)}\right) \]
   unsub-neg [=>] 0.01	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 - x}\right) \]

3. Final simplification 0.01

   \[\leadsto \mathsf{fma}\left(y, x - 0.5, 0.918938533204673 - x\right) \]

Alternatives

Alternative 1
Error	43.9%
Cost	1248

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00275:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-118}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-250}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-301}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-132}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-56}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 2
Error	1.48%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -58000 \lor \neg \left(x \leq 5400000000\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + \left(y \cdot x + y \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	1.92%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-12} \lor \neg \left(x \leq 2.5 \cdot 10^{-10}\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \end{array} \]

Alternative 4
Error	0.02%
Cost	704

\[0.918938533204673 + \left(x \cdot \left(y + -1\right) + y \cdot -0.5\right) \]

Alternative 5
Error	2.72%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \lor \neg \left(y \leq 1.8\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 6
Error	2.53%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.64\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \end{array} \]

Alternative 7
Error	0.02%
Cost	576

\[\left(0.918938533204673 + y \cdot \left(x - 0.5\right)\right) - x \]

Alternative 8
Error	17.07%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -13:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 9
Error	44.35%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0037:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 5400000000:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 10
Error	69.99%
Cost	64

\[0.918938533204673 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))