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

Average Error: 0.0 → 0.0

Time: 2.5s

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 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right)} \]
   Proof
   (fma.f64 y (+.f64 x -1/2) (-.f64 918938533204673/1000000000000000 x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (+.f64 x (Rewrite<= metadata-eval (neg.f64 1/2))) (-.f64 918938533204673/1000000000000000 x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 1/2) x)) (-.f64 918938533204673/1000000000000000 x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (+.f64 (neg.f64 1/2) x) (-.f64 918938533204673/1000000000000000 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (+.f64 (neg.f64 1/2) x) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 918938533204673/1000000000000000 (*.f64 (neg.f64 1) x)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (+.f64 (neg.f64 1/2) x) (+.f64 918938533204673/1000000000000000 (Rewrite<= *-commutative_binary64 (*.f64 x (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (+.f64 (neg.f64 1/2) x) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (+.f64 (neg.f64 1/2) x)) (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 y (neg.f64 1/2)) (*.f64 y x))) (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000)): 1 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y 1/2))) (*.f64 y x)) (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 y) 1/2)) (*.f64 y x)) (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (neg.f64 y) 1/2) (Rewrite<= *-commutative_binary64 (*.f64 x y))) (+.f64 (*.f64 x (neg.f64 1)) 918938533204673/1000000000000000)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 (neg.f64 y) 1/2) (*.f64 x y)) (*.f64 x (neg.f64 1))) 918938533204673/1000000000000000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (neg.f64 y) 1/2) (+.f64 (*.f64 x y) (*.f64 x (neg.f64 1))))) 918938533204673/1000000000000000): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (neg.f64 y) 1/2) (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 y (neg.f64 1))))) 918938533204673/1000000000000000): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (neg.f64 y) 1/2) (*.f64 x (Rewrite<= sub-neg_binary64 (-.f64 y 1)))) 918938533204673/1000000000000000): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x (-.f64 y 1)) (*.f64 (neg.f64 y) 1/2))) 918938533204673/1000000000000000): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x (-.f64 y 1)) (*.f64 y 1/2))) 918938533204673/1000000000000000): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	28.8
Cost	852

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+119}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1700000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 2
Error	11.1
Cost	852

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+119}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1700000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+26}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 3
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := 0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	712

\[\begin{array}{l} t_0 := 0.918938533204673 + \left(y \cdot x - x\right)\\ \mathbf{if}\;x \leq -4700000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2100000000:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.0
Cost	704

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

Alternative 6
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	28.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1700000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 760:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 8
Error	45.1
Cost	64

\[0.918938533204673 \]

Reproduce

herbie shell --seed 2022330 
(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))