Average Error: 0.0 → 0.0
Time: 5.7s
Precision: binary64
Cost: 6848
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
↓
\[0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right) \]
(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))
↓
(FPCore (x y) :precision binary64 (- 0.918938533204673 (fma y (- 0.5 x) x)))
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 - fma(y, (0.5 - x), x);
}

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(0.918938533204673 - fma(y, Float64(0.5 - x), x))
end

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[(0.918938533204673 - N[(y * N[(0.5 - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673
↓
0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right)

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error27.7
Cost1248
\[\begin{array}{l} \mathbf{if}\;y \leq -0.45100847663358534:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -1.1099673577962435 \cdot 10^{-125}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.92247434841579 \cdot 10^{-197}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -4.39476807761709 \cdot 10^{-279}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1.1753641025335908 \cdot 10^{-284}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 3.9076508916098936 \cdot 10^{-210}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1.4654941647389007 \cdot 10^{-136}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 0.8347541720076823:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 2
Error28.8
Cost1116
\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+90}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1893996.518206688:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.6432814407620483 \cdot 10^{-48}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 4.723611630009177 \cdot 10^{-68}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.0076241783740685866:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+140}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 3
Error0.0
Cost704
\[0.918938533204673 + \left(x \cdot \left(y + -1\right) + y \cdot -0.5\right) \]
Alternative 4
Error1.6
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1893996.518206688:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 0.0076241783740685866:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
Alternative 5
Error10.4
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -5811.962875512381:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 0.8347541720076823:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 6
Error28.3
Cost392
\[\begin{array}{l} \mathbf{if}\;x \leq -845551160.2319802:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.0076241783740685866:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 7
Error45.5
Cost64
\[0.918938533204673 \]

Error

Reproduce

herbie shell --seed 2022296 
(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))