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

Average Error: 19.9 → 0.1

Time: 10.8s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (/ (+ x y) y)) (+ x y)) (+ y (+ x 1.0))))

C

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 / ((x + y) / y)) / (x + y)) / (y + (x + 1.0));
}

Fortran

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 / ((x + y) / y)) / (x + y)) / (y + (x + 1.0d0))
end function

Java

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 / ((x + y) / y)) / (x + y)) / (y + (x + 1.0));
}

Python

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

↓

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

Julia

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(Float64(x + y) / y)) / Float64(x + y)) / Float64(y + Float64(x + 1.0)))
end

MATLAB

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

↓

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

Wolfram

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[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	19.9
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 19.9

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

2. Applied egg-rr 4.1

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

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	22.7
Cost	1096

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq 2.83277814280875 \cdot 10^{-222}:\\ \;\;\;\;\frac{t_0}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.894194575092607 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 2
Error	11.3
Cost	1092

\[\begin{array}{l} t_0 := \frac{\frac{y}{x + y}}{x + y}\\ \mathbf{if}\;y \leq 3.203820540508273 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

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

Alternative 4
Error	24.6
Cost	976

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ t_1 := \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -10539221448929307000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1779216795629714 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.6
Cost	976

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -10539221448929307000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1779216795629714 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Error	25.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -9.92367623869806 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 2.1779216795629714 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Error	25.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -9.92367623869806 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 2.1779216795629714 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	25.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -9.92367623869806 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2.1779216795629714 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Error	22.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;y \leq 3.203820540508273 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 10
Error	24.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5306660110744097 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 11
Error	32.1
Cost	716

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ t_1 := \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;x \leq -2.09200666399979 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2816854703194275 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -10539221448929307000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	24.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5306660110744097 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]

Alternative 13
Error	49.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 5.185082637317824 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14
Error	61.0
Cost	320

\[\frac{1}{x + 1} \]

Alternative 15
Error	61.3
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2022300 
(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))))