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

Average Accuracy: 68.3% → 99.8%

Time: 19.8s

Precision: binary64

Cost: 1088

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

Math

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

↓

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

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	68.3%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 68.3%

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

2. Simplified 72.9%

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

   [Start] 68.3	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 72.9	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 72.9	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 99.8%

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

   [Start] 72.9	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)} \]
   times-frac [=>] 99.8	\[ \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{x + y}}}{x + \left(y + 1\right)} \]

4. Simplified 99.8%

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

   [Start] 99.8	\[ \frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{x + \left(y + 1\right)} \]
   *-commutative [<=] 99.8	\[ \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{x + \left(y + 1\right)} \]
   associate-*r/ [=>] 99.8	\[ \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{x + y}}}{x + \left(y + 1\right)} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	81.1%
Cost	1100

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	1088

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

Alternative 3
Accuracy	99.8%
Cost	1088

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

Alternative 4
Accuracy	70.8%
Cost	980

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	72.2%
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-187}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Accuracy	72.2%
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-90}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-189}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Accuracy	80.8%
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 8
Accuracy	81.0%
Cost	972

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	81.0%
Cost	972

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -6.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	78.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11
Accuracy	78.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Accuracy	80.9%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 13
Accuracy	80.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14
Accuracy	79.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15
Accuracy	64.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 16
Accuracy	43.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 17
Accuracy	4.3%
Cost	192

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

Alternative 18
Accuracy	25.7%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))