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

Average Accuracy: 68.6% → 99.8%

Time: 19.7s

Precision: binary64

Cost: 1088

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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

Target

Original	68.6%
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 68.6%

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

2. Simplified 87.4%

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

   [Start] 68.6	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   times-frac [=>] 87.4	\[ \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]
   /-rgt-identity [<=] 87.4	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]
   associate-/l/ [=>] 87.4	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]
   *-lft-identity [=>] 87.4	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 87.4	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 99.3%

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

   [Start] 87.4	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)} \]
   *-commutative [=>] 87.4	\[ \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]
   clear-num [=>] 87.3	\[ \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]
   associate-/r* [=>] 99.7	\[ \frac{1}{\frac{x + \left(y + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]
   frac-times [=>] 99.3	\[ \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]
   *-un-lft-identity [<=] 99.3	\[ \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

4. Applied egg-rr 99.8%

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

   [Start] 99.3	\[ \frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]
   div-inv [=>] 99.2	\[ \color{blue}{\frac{x}{x + y} \cdot \frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]
   *-commutative [=>] 99.2	\[ \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \cdot \frac{x}{x + y}} \]
   associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{1}{\frac{x + \left(y + 1\right)}{y}}}{x + y}} \cdot \frac{x}{x + y} \]
   clear-num [<=] 99.8	\[ \frac{\color{blue}{\frac{y}{x + \left(y + 1\right)}}}{x + y} \cdot \frac{x}{x + y} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \cdot \frac{x}{x + y} \]
   associate-+l+ [=>] 99.8	\[ \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \cdot \frac{x}{x + y} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	91.3%
Cost	1356

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq -6.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ \mathbf{elif}\;y \leq 10^{-159}:\\ \;\;\;\;y \cdot \frac{t_0}{y + x}\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y - \left(\left(-1 - x\right) - x\right)}\\ \end{array} \]

Alternative 2
Accuracy	97.1%
Cost	1352

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+160}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{y + x}{\frac{y}{y + 1}}}\\ \end{array} \]

Alternative 3
Accuracy	87.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := \frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ t_2 := \frac{t_0}{y + 1}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-260}:\\ \;\;\;\;y \cdot \frac{t_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	90.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ t_1 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-260}:\\ \;\;\;\;y \cdot \frac{t_1}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{y + 1}\\ \end{array} \]

Alternative 5
Accuracy	91.2%
Cost	1228

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := \frac{t_0}{y + x}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+46}:\\ \;\;\;\;t_1 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-259}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y - \left(\left(-1 - x\right) - x\right)}\\ \end{array} \]

Alternative 6
Accuracy	95.4%
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{x}{y + x}}{y + x}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+84}:\\ \;\;\;\;t_0 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 7
Accuracy	95.4%
Cost	1224

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{y + x}{\frac{y}{y + 1}}}\\ \end{array} \]

Alternative 8
Accuracy	91.0%
Cost	1100

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := \frac{t_0}{y + x}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+46}:\\ \;\;\;\;t_1 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-259}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \end{array} \]

Alternative 9
Accuracy	99.8%
Cost	1088

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

Alternative 10
Accuracy	79.4%
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 11
Accuracy	86.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{x}\\ \mathbf{elif}\;y \leq 0.033:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 12
Accuracy	79.2%
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+48} \lor \neg \left(x \leq -4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]

Alternative 13
Accuracy	76.8%
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;x \leq -0.38:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 14
Accuracy	81.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15
Accuracy	79.3%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16
Accuracy	79.2%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17
Accuracy	79.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.35 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18
Accuracy	71.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.74:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19
Accuracy	74.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.74:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 20
Accuracy	75.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.72:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 21
Accuracy	65.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.74:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 22
Accuracy	28.9%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -0.36:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 23
Accuracy	44.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 24
Accuracy	4.3%
Cost	192

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

Reproduce

herbie shell --seed 2023147 
(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))))