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

Average Accuracy: 100.0% → 100.0%

Time: 3.0s

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 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   sub-neg [=>] 100.0	\[ \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y \cdot 0.5\right)\right)} + 0.918938533204673 \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]
   sub-neg [=>] 100.0	\[ \left(\left(-y \cdot 0.5\right) + x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) + 0.918938533204673 \]
   distribute-rgt-in [=>] 100.0	\[ \left(\left(-y \cdot 0.5\right) + \color{blue}{\left(y \cdot x + \left(-1\right) \cdot x\right)}\right) + 0.918938533204673 \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + \left(-1\right) \cdot x\right)} + 0.918938533204673 \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + \left(\left(-1\right) \cdot x + 0.918938533204673\right)} \]
   distribute-rgt-neg-in [=>] 100.0	\[ \left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + \left(\left(-1\right) \cdot x + 0.918938533204673\right) \]
   distribute-lft-out [=>] 100.0	\[ \color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + \left(\left(-1\right) \cdot x + 0.918938533204673\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, \left(-1\right) \cdot x + 0.918938533204673\right)} \]
   +-commutative [=>] 100.0	\[ \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, \left(-1\right) \cdot x + 0.918938533204673\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(y, x + \color{blue}{-0.5}, \left(-1\right) \cdot x + 0.918938533204673\right) \]
   +-commutative [<=] 100.0	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 + \left(-1\right) \cdot x}\right) \]
   cancel-sign-sub-inv [<=] 100.0	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 - 1 \cdot x}\right) \]
   *-lft-identity [=>] 100.0	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - \color{blue}{x}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -42 \lor \neg \left(y \leq 1.6 \cdot 10^{-13}\right):\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 - y \cdot 0.5\right) - x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	704

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

Alternative 3
Accuracy	70.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.038 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]

Alternative 4
Accuracy	69.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \end{array} \]

Alternative 5
Accuracy	42.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	71.5%
Cost	448

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

Alternative 7
Accuracy	30.0%
Cost	64

\[0.918938533204673 \]

Reproduce

herbie shell --seed 2023147 
(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))