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

Average Accuracy: 69.0% → 99.8%

Time: 18.4s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

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

Target

Original	69.0%
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 69.0%

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

2. Simplified 73.5%

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

   [Start] 69.0	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 73.5	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 73.5	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

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

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

Alternatives

Alternative 1
Accuracy	72.6%
Cost	1352

\[\begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-169}:\\ \;\;\;\;\frac{t_0}{1 + \left(x + y \cdot 2\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 2
Accuracy	61.6%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;\frac{1}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 3
Accuracy	62.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + \left(x + y \cdot 2\right)}\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;\frac{1}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 4
Accuracy	53.0%
Cost	980

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ t_1 := \frac{y}{x} - y\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	53.6%
Cost	980

\[\begin{array}{l} t_0 := \frac{y}{x} - y\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Accuracy	61.6%
Cost	973

\[\begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 4.3 \lor \neg \left(y \leq 1.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \end{array} \]

Alternative 7
Accuracy	61.5%
Cost	972

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;\frac{\frac{-x}{t_0}}{-y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	60.5%
Cost	844

\[\begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Accuracy	60.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10
Accuracy	61.4%
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + 1}\\ \mathbf{if}\;y \leq 1.16 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11
Accuracy	61.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Accuracy	61.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Accuracy	59.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 14
Accuracy	44.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 15
Accuracy	33.8%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 16
Accuracy	4.2%
Cost	192

\[\frac{1}{y} \]

Alternative 17
Accuracy	26.3%
Cost	192

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

Alternative 18
Accuracy	3.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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))))