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

Average Error: 0.0 → 0.0

Time: 3.2s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.918938533204673d0 + (y * (x + (-0.5d0)))) - x
end function

Java

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

↓

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

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

↓

def code(x, y):
	return (0.918938533204673 + (y * (x + -0.5))) - 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 Float64(Float64(0.918938533204673 + Float64(y * Float64(x + -0.5))) - x)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

↓

function tmp = code(x, y)
	tmp = (0.918938533204673 + (y * (x + -0.5))) - 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[(N[(0.918938533204673 + N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   sub-neg [=>] 0.0	\[ \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y \cdot 0.5\right)\right)} + 0.918938533204673 \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]
   sub-neg [=>] 0.0	\[ \left(\left(-y \cdot 0.5\right) + x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) + 0.918938533204673 \]
   distribute-lft-in [=>] 0.0	\[ \left(\left(-y \cdot 0.5\right) + \color{blue}{\left(x \cdot y + x \cdot \left(-1\right)\right)}\right) + 0.918938533204673 \]
   *-commutative [<=] 0.0	\[ \left(\left(-y \cdot 0.5\right) + \left(\color{blue}{y \cdot x} + x \cdot \left(-1\right)\right)\right) + 0.918938533204673 \]
   associate-+r+ [=>] 0.0	\[ \color{blue}{\left(\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + x \cdot \left(-1\right)\right)} + 0.918938533204673 \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{\left(\left(-y \cdot 0.5\right) + y \cdot x\right) + \left(x \cdot \left(-1\right) + 0.918938533204673\right)} \]
   distribute-rgt-neg-in [=>] 0.0	\[ \left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + \left(x \cdot \left(-1\right) + 0.918938533204673\right) \]
   distribute-lft-out [=>] 0.0	\[ \color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + \left(x \cdot \left(-1\right) + 0.918938533204673\right) \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, x \cdot \left(-1\right) + 0.918938533204673\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, x \cdot \left(-1\right) + 0.918938533204673\right) \]
   metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(y, x + \color{blue}{-0.5}, x \cdot \left(-1\right) + 0.918938533204673\right) \]
   +-commutative [<=] 0.0	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 + x \cdot \left(-1\right)}\right) \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{\left(-1\right) \cdot x}\right) \]
   metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{-1} \cdot x\right) \]
   mul-1-neg [=>] 0.0	\[ \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 + \color{blue}{\left(-x\right)}\right) \]
   unsub-neg [=>] 0.0	\[ \mathsf{fma}\left(y, x + -0.5, \color{blue}{0.918938533204673 - x}\right) \]

3. Taylor expanded in y around 0 0.0

   \[\leadsto \color{blue}{\left(0.918938533204673 + y \cdot \left(x - 0.5\right)\right) - x} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-9} \lor \neg \left(y \leq 1.2 \cdot 10^{-8}\right):\\ \;\;\;\;0.918938533204673 + \left(y \cdot x + y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 2
Error	28.8
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -0.96:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 10^{+149}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3
Error	11.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -170:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+148}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 4
Error	1.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \lor \neg \left(y \leq 1.42\right):\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5
Error	1.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.36:\\ \;\;\;\;y \cdot x + y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.82:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]

Alternative 6
Error	28.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 7
Error	44.8
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023018 
(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))