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

Percentage Accurate: 69.4% → 99.8%

Time: 16.2s

Precision: binary64

Cost: 1152

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ (/ (* y (/ (- x) (+ y x))) (+ y x)) (- -1.0 (+ 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 * (-x / (y + x))) / (y + x)) / (-1.0 - (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 * (-x / (y + x))) / (y + x)) / ((-1.0d0) - (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 * (-x / (y + x))) / (y + x)) / (-1.0 - (y + x));
}

Python

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

↓

def code(x, y):
	return ((y * (-x / (y + x))) / (y + x)) / (-1.0 - (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(Float64(-x) / Float64(y + x))) / Float64(y + x)) / Float64(-1.0 - 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 * (-x / (y + x))) / (y + x)) / (-1.0 - (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[((-x) / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	69.4%
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 66.1%

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

2. Simplified 81.4%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   Step-by-step derivation

   [Start] 66.1%	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-*r/ [<=] 81.4%	\[ \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]
   *-commutative [=>] 81.4%	\[ x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   distribute-rgt1-in [<=] 67.0%	\[ x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   fma-def [=>] 81.4%	\[ x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]
   cube-unmult [=>] 81.4%	\[ x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Applied egg-rr 94.9%

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

   [Start] 81.4%	\[ x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]
   associate-*r/ [=>] 66.1%	\[ \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   fma-udef [=>] 58.2%	\[ \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]
   cube-mult [=>] 58.2%	\[ \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   distribute-rgt1-in [=>] 66.1%	\[ \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   associate-+r+ [<=] 66.1%	\[ \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]
   *-commutative [<=] 66.1%	\[ \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   frac-times [<=] 88.2%	\[ \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]
   associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{x + \left(y + 1\right)} \]
   frac-2neg [=>] 99.8%	\[ \frac{\frac{x}{x + y}}{x + y} \cdot \color{blue}{\frac{-y}{-\left(x + \left(y + 1\right)\right)}} \]
   frac-times [=>] 94.9%	\[ \color{blue}{\frac{\frac{x}{x + y} \cdot \left(-y\right)}{\left(x + y\right) \cdot \left(-\left(x + \left(y + 1\right)\right)\right)}} \]
   +-commutative [=>] 94.9%	\[ \frac{\frac{x}{\color{blue}{y + x}} \cdot \left(-y\right)}{\left(x + y\right) \cdot \left(-\left(x + \left(y + 1\right)\right)\right)} \]
   +-commutative [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\color{blue}{\left(y + x\right)} \cdot \left(-\left(x + \left(y + 1\right)\right)\right)} \]
   associate-+r+ [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \left(-\color{blue}{\left(\left(x + y\right) + 1\right)}\right)} \]
   +-commutative [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \left(-\color{blue}{\left(1 + \left(x + y\right)\right)}\right)} \]
   distribute-neg-in [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\left(x + y\right)\right)\right)}} \]
   metadata-eval [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \left(\color{blue}{-1} + \left(-\left(x + y\right)\right)\right)} \]
   +-commutative [=>] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \left(-1 + \left(-\color{blue}{\left(y + x\right)}\right)\right)} \]

4. Simplified 99.9%

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

   [Start] 94.9%	\[ \frac{\frac{x}{y + x} \cdot \left(-y\right)}{\left(y + x\right) \cdot \left(-1 + \left(-\left(y + x\right)\right)\right)} \]
   associate-/r* [=>] 99.9%	\[ \color{blue}{\frac{\frac{\frac{x}{y + x} \cdot \left(-y\right)}{y + x}}{-1 + \left(-\left(y + x\right)\right)}} \]
   *-commutative [=>] 99.9%	\[ \frac{\frac{\color{blue}{\left(-y\right) \cdot \frac{x}{y + x}}}{y + x}}{-1 + \left(-\left(y + x\right)\right)} \]
   unsub-neg [=>] 99.9%	\[ \frac{\frac{\left(-y\right) \cdot \frac{x}{y + x}}{y + x}}{\color{blue}{-1 - \left(y + x\right)}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1152

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

Alternative 2
Accuracy	85.3%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 3
Accuracy	67.1%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-y}{y + x}}{-1 - \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	80.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-y}{y + x}}{-1 - \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	1088

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

Alternative 6
Accuracy	99.8%
Cost	1088

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

Alternative 7
Accuracy	62.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{-y}{y + x}}{-1 - \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	61.8%
Cost	900

\[\begin{array}{l} t_0 := -1 - \left(y + x\right)\\ \mathbf{if}\;x \leq -1.92 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{-y}{y + x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	61.5%
Cost	772

\[\begin{array}{l} \mathbf{if}\;x \leq -1.92 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{-1 - \left(y + x\right)}\\ \end{array} \]

Alternative 10
Accuracy	60.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 11
Accuracy	60.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 12
Accuracy	60.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 13
Accuracy	61.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.92 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14
Accuracy	61.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.92 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15
Accuracy	61.3%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16
Accuracy	48.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17
Accuracy	48.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 18
Accuracy	49.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.92 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 19
Accuracy	61.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 20
Accuracy	44.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 21
Accuracy	26.6%
Cost	320

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

Alternative 22
Accuracy	4.2%
Cost	192

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

Alternative 23
Accuracy	26.2%
Cost	192

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

Reproduce

herbie shell --seed 2023165 
(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))))