Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
Cost: 576
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
↓
\[\left(0.918938533204673 + y \cdot \left(x + -0.5\right)\right) - x \]
(FPCore (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))
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;
}

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

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;
}

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

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

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

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]

\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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right)} \]
    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, 1 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)): 0 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): 4 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. 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 simplification0.0

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

Alternatives

Alternative 1
Error1.2
Cost712
\[\begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 95000000:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error1.6
Cost584
\[\begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.42:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error26.7
Cost192
\[0.918938533204673 - x \]

Error

Reproduce

herbie shell --seed 2022329 
(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))