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

Percentage Accurate: 68.7% → 99.7%

Time: 17.6s

Precision: binary64

Cost: 1216

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (* (* (/ x (+ x y)) (/ 1.0 (+ x y))) (/ y (+ 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 / (x + y)) * (1.0 / (x + y))) * (y / (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 / (x + y)) * (1.0d0 / (x + y))) * (y / (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 / (x + y)) * (1.0 / (x + y))) * (y / (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 / (x + y)) * (1.0 / (x + y))) * (y / (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(x + y)) * Float64(1.0 / Float64(x + y))) * Float64(y / 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 / (x + y)) * (1.0 / (x + y))) * (y / (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[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	68.7%
Target	99.8%
Herbie	99.7%

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

Derivation

1. Initial program 70.3%

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

2. Simplified 90.8%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]
   Step-by-step derivation

   [Start] 70.3%	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   times-frac [=>] 90.8%	\[ \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]
   /-rgt-identity [<=] 90.8%	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]
   associate-/l/ [=>] 90.8%	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]
   *-lft-identity [=>] 90.8%	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 90.8%	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\frac{x}{y + x} \cdot \frac{1}{y + x}\right)} \cdot \frac{y}{x + \left(y + 1\right)} \]
   Step-by-step derivation

   [Start] 90.8%	\[ \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)} \]
   associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{x + \left(y + 1\right)} \]
   div-inv [=>] 99.8%	\[ \color{blue}{\left(\frac{x}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{y}{x + \left(y + 1\right)} \]
   +-commutative [=>] 99.8%	\[ \left(\frac{x}{\color{blue}{y + x}} \cdot \frac{1}{x + y}\right) \cdot \frac{y}{x + \left(y + 1\right)} \]
   +-commutative [=>] 99.8%	\[ \left(\frac{x}{y + x} \cdot \frac{1}{\color{blue}{y + x}}\right) \cdot \frac{y}{x + \left(y + 1\right)} \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	1216

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

Alternative 2
Accuracy	97.2%
Cost	1352

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+166}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{x + y}\right) \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 3
Accuracy	86.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(x + y\right)}}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]

Alternative 4
Accuracy	94.6%
Cost	1092

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -430000:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{x + y}\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	1088

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

Alternative 6
Accuracy	76.1%
Cost	976

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Accuracy	85.7%
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(x + y\right)}}{x}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Accuracy	85.7%
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(x + y\right)}}{x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]

Alternative 9
Accuracy	72.5%
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10
Accuracy	74.6%
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11
Accuracy	76.0%
Cost	848

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Accuracy	81.5%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Accuracy	83.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14
Accuracy	83.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(x + y\right)}}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\left(y + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15
Accuracy	81.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 16
Accuracy	64.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 2.42 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17
Accuracy	28.3%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -1750000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18
Accuracy	44.1%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 19
Accuracy	4.3%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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))))