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

Average Error: 31.09% → 0.16%

Time: 12.4s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

(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))))

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

Python

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))

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

Target

Original	31.09%
Target	0.24%
Herbie	0.16%

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

Derivation

1. Initial program 31.09

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

2. Simplified 26.65

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

   [Start] 31.09	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 26.65	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 26.65	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 0.16

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

4. Simplified 0.16

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

   [Start] 0.16	\[ \frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{x + \left(y + 1\right)} \]
   *-commutative [<=] 0.16	\[ \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{x + \left(y + 1\right)} \]
   associate-*r/ [=>] 0.16	\[ \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{x + y}}}{x + \left(y + 1\right)} \]

5. Final simplification 0.16

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

Alternatives

Alternative 1
Error	26.61%
Cost	1352

\[\begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{-214}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_1 \cdot \frac{y + x}{y}}\\ \end{array} \]

Alternative 2
Error	35.61%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;y \leq 2.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(y + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \]

Alternative 3
Error	0.16%
Cost	1088

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

Alternative 4
Error	37.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]

Alternative 5
Error	38.14%
Cost	836

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 6
Error	47.8%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 7
Error	47.12%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	38.36%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 9
Error	38.26%
Cost	708

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 10
Error	55.31%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 11
Error	39.28%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -800000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 12
Error	38.72%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -100000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 13
Error	38.47%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14
Error	73.34%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -17000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 15
Error	65.93%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 16
Error	95.75%
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))