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

Average Error: 20.0 → 0.1

Time: 16.2s

Precision: binary64

Cost: 1088

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

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

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 / (1.0d0 + (x / y))) / (y + (x + 1.0d0))) / (x + y)
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 / (1.0 + (x / y))) / (y + (x + 1.0))) / (x + y);
}

Python

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

↓

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

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(1.0 + Float64(x / y))) / Float64(y + Float64(x + 1.0))) / Float64(x + y))
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 / (1.0 + (x / y))) / (y + (x + 1.0))) / (x + y);
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[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

Target

Original	20.0
Target	0.2
Herbie	0.1

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

Derivation

1. Initial program 20.0

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

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

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

4. Taylor expanded in x around 0 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	3.3
Cost	1352

\[\begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -2.373773625943706 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{y}{t_0}}{x + y}\\ \mathbf{elif}\;y \leq 3.43047155478845 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(1 + \frac{x}{y}\right) \cdot \left(t_0 \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{x + y}\\ \end{array} \]

Alternative 2
Error	11.4
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{if}\;x \leq -9.820027651546739 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.75206337496721 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(1 + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]

Alternative 4
Error	11.5
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{if}\;x \leq -9.820027651546739 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.75206337496721 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]

Alternative 5
Error	11.2
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{y}{x + 1}}{x}\\ \mathbf{if}\;x \leq -3.554993965577662 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.75206337496721 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]

Alternative 6
Error	15.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2955409948581024 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.988830497055325 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.001253329071841231:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Error	12.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002873400466925893:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 1.172152902457756 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	12.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002873400466925893:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 1.172152902457756 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 9
Error	12.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002873400466925893:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq 1.172152902457756 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 10
Error	20.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -51.2898923662645:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Error	12.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002873400466925893:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]

Alternative 12
Error	20.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -51.2898923662645:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.370890089858983 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 13
Error	35.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 8.988830497055325 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 14
Error	47.1
Cost	192

\[\frac{x}{y} \]

Alternative 15
Error	61.6
Cost	128

\[-y \]

Alternative 16
Error	61.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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))))