Average Error: 0.0 → 0.0
Time: 5.2s
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))): 3 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))))): 2 points increase in error, 3 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
Error28.5
Cost1248
\[\begin{array}{l} \mathbf{if}\;y \leq -3.148 \cdot 10^{+55}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -0.12276131731217566:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -6.470301115903365 \cdot 10^{-72}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -1.5801748928854413 \cdot 10^{-273}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 7.892716739397749 \cdot 10^{-259}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1.0769075225572525 \cdot 10^{-153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 4.0161219423515337 \cdot 10^{-144}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 0.0032189468108935145:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 2
Error28.4
Cost1052
\[\begin{array}{l} \mathbf{if}\;x \leq -9.89106774016047 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1.0337011326188743 \cdot 10^{-61}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 2.0064217054121652 \cdot 10^{-296}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 2.753000132744273 \cdot 10^{-262}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 3.2937821325078246 \cdot 10^{-253}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 1.3686483116653674 \cdot 10^{-55}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 0.03465608800903597:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 3
Error10.2
Cost848
\[\begin{array}{l} t_0 := y \cdot x - x\\ \mathbf{if}\;y \leq -3.148 \cdot 10^{+55}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -0.12276131731217566:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.3630668289435733 \cdot 10^{-5}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4964495694362770:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 4
Error10.2
Cost848
\[\begin{array}{l} \mathbf{if}\;y \leq -3.148 \cdot 10^{+55}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -0.12276131731217566:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;y \leq 4.3630668289435733 \cdot 10^{-5}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4964495694362770:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 5
Error1.5
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0668657789567553 \cdot 10^{+21}:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 236542032794271.16:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
Alternative 6
Error0.0
Cost704
\[0.918938533204673 + \left(x \cdot \left(y + -1\right) + y \cdot -0.5\right) \]
Alternative 7
Error10.8
Cost588
\[\begin{array}{l} \mathbf{if}\;y \leq -3.148 \cdot 10^{+55}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -4956672372247334:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.0032189468108935145:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
Alternative 8
Error1.8
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -13074684.626674337:\\ \;\;\;\;y \cdot x - x\\ \mathbf{elif}\;x \leq 0.03465608800903597:\\ \;\;\;\;0.918938533204673 + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]
Alternative 9
Error0.0
Cost576
\[\left(0.918938533204673 + y \cdot \left(x + -0.5\right)\right) - x \]
Alternative 10
Error28.1
Cost392
\[\begin{array}{l} \mathbf{if}\;x \leq -13074684.626674337:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.03465608800903597:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 11
Error44.9
Cost64
\[0.918938533204673 \]

Error

Reproduce

herbie shell --seed 2022308 
(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))