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

Average Accuracy: 70.2% → 99.8%

Time: 18.5s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ (* (/ y (+ y x)) (/ 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 ((y / (y + x)) * (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 = ((y / (y + x)) * (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 ((y / (y + x)) * (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 ((y / (y + x)) * (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(y / Float64(y + x)) * Float64(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 = ((y / (y + x)) * (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[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	70.2%
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 70.2%

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

2. Simplified 74.9%

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

   [Start] 70.2	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 74.9	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 74.9	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 99.8%

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

   [Start] 74.9	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)} \]
   times-frac [=>] 99.8	\[ \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{x + y}}}{x + \left(y + 1\right)} \]
   *-commutative [=>] 99.8	\[ \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{x + \left(y + 1\right)} \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	71.9%
Cost	1352

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{y}{x + y \cdot 2}}{t_0}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{\frac{y}{y + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + x}}{t_0}\\ \end{array} \]

Alternative 2
Accuracy	63.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \frac{\frac{1}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 3
Accuracy	63.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \frac{\frac{1}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 4
Accuracy	44.3%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-191}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.66:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 5
Accuracy	53.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 6
Accuracy	58.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Accuracy	58.9%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y}{x}}\\ \end{array} \]

Alternative 8
Accuracy	52.3%
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 9
Accuracy	52.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10
Accuracy	53.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11
Accuracy	59.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 12
Accuracy	59.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 13
Accuracy	59.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14
Accuracy	27.1%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 15
Accuracy	33.9%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 16
Accuracy	4.2%
Cost	192

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

Reproduce

herbie shell --seed 2023133 
(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))))