?

Average Error: 30.94% → 0.17%
Time: 16.8s
Precision: binary64
Cost: 1088

?

\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
↓
\[\frac{\frac{x \cdot \frac{y}{y + x}}{y + x}}{x + \left(y + 1\right)} \]
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (/ (/ (* x (/ y (+ y x))) (+ y x)) (+ x (+ y 1.0))))
double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}
↓
double code(double x, double y) {
	return ((x * (y / (y + x))) / (y + x)) / (x + (y + 1.0));
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y / (y + x))) / (y + x)) / (x + (y + 1.0d0))
end function

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}
↓
public static double code(double x, double y) {
	return ((x * (y / (y + x))) / (y + x)) / (x + (y + 1.0));
}

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
↓
def code(x, y):
	return ((x * (y / (y + x))) / (y + x)) / (x + (y + 1.0))

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end
↓
function code(x, y)
	return Float64(Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(y + x)) / Float64(x + Float64(y + 1.0)))
end

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end
↓
function tmp = code(x, y)
	tmp = ((x * (y / (y + x))) / (y + x)) / (x + (y + 1.0));
end

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x \cdot \frac{y}{y + x}}{y + x}}{x + \left(y + 1\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.94%
Target0.22%
Herbie0.17%
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation?

  1. Initial program 30.94

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

  2. Simplified26.14

    \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}} \]
    Proof

    [Start]30.94

    \[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

    associate-/r* [=>]26.14

    \[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

    associate-+l+ [=>]26.14

    \[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

  3. Applied egg-rr0.17

    \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{x + y}}}{x + \left(y + 1\right)} \]

  4. Simplified0.17

    \[\leadsto \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{x + y}}}{x + \left(y + 1\right)} \]
    Proof

    [Start]0.17

    \[ \frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{x + \left(y + 1\right)} \]

    *-commutative [<=]0.17

    \[ \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{x + \left(y + 1\right)} \]

    associate-*r/ [=>]0.17

    \[ \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{x + y}}}{x + \left(y + 1\right)} \]

  5. Final simplification0.17

    \[\leadsto \frac{\frac{x \cdot \frac{y}{y + x}}{y + x}}{x + \left(y + 1\right)} \]

Alternatives

Alternative 1
Error37.51%
Cost1292
\[\begin{array}{l} t_0 := \frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{if}\;x \leq -0.0026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.16 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y + x}}{-1 - \left(y + x \cdot 2\right)}\\ \end{array} \]
Alternative 2
Error8.83%
Cost1220
\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)\right)}\\ \end{array} \]
Alternative 3
Error38.14%
Cost1100
\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{y + x}}{t_0}\\ \mathbf{if}\;x \leq -0.155:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]
Alternative 4
Error14.13%
Cost1092
\[\begin{array}{l} \mathbf{if}\;x \leq -6.4:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + 1}{y}\right)}\\ \end{array} \]
Alternative 5
Error38.4%
Cost1036
\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -0.0033:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -2.16 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{-y}{x \cdot \left(-1 - \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]
Alternative 6
Error38.47%
Cost972
\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + 1}\\ \mathbf{if}\;x \leq -0.43:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.16 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
Alternative 7
Error38.42%
Cost972
\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t_0}\\ \mathbf{if}\;x \leq -0.045:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]
Alternative 8
Error46.64%
Cost848
\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -54:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error46.18%
Cost848
\[\begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
Alternative 10
Error45.28%
Cost848
\[\begin{array}{l} \mathbf{if}\;x \leq -51:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
Alternative 11
Error39.39%
Cost845
\[\begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-38} \lor \neg \left(x \leq -6 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]
Alternative 12
Error39.38%
Cost845
\[\begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-38} \lor \neg \left(x \leq -2.3 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]
Alternative 13
Error38.89%
Cost844
\[\begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \leq -1.98 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
Alternative 14
Error38.7%
Cost844
\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + 1}\\ \mathbf{if}\;x \leq -0.052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
Alternative 15
Error38.68%
Cost844
\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + 1}\\ \mathbf{if}\;x \leq -0.33:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.1 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
Alternative 16
Error56.79%
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
Alternative 17
Error72.32%
Cost324
\[\begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 18
Error65.88%
Cost324
\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 19
Error95.8%
Cost192
\[\frac{0.5}{x} \]

Error

Reproduce?

herbie shell --seed 2023089 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))