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

Percentage Accurate: 68.6% → 99.3%

Time: 18.4s

Precision: binary64

Cost: 1088

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

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	68.6%
Target	99.8%
Herbie	99.3%

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

Derivation

1. Initial program 75.2%

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

2. Simplified 83.5%

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

   [Start] 75.2%	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-*r/ [<=] 83.5%	\[ \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]
   *-commutative [=>] 83.5%	\[ x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   distribute-rgt1-in [<=] 63.3%	\[ x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   fma-def [=>] 83.5%	\[ x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]
   cube-unmult [=>] 83.5%	\[ x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Applied egg-rr 92.4%

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

   [Start] 83.5%	\[ x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]
   associate-*r/ [=>] 75.2%	\[ \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   fma-udef [=>] 57.1%	\[ \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]
   cube-mult [=>] 57.1%	\[ \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   distribute-rgt1-in [=>] 75.2%	\[ \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   associate-+r+ [<=] 75.2%	\[ \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]
   *-commutative [<=] 75.2%	\[ \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   frac-times [<=] 89.5%	\[ \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]
   *-commutative [=>] 89.5%	\[ \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]
   associate-/r* [=>] 99.8%	\[ \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]
   frac-times [=>] 92.4%	\[ \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}} \]
   +-commutative [=>] 92.4%	\[ \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)} \]
   +-commutative [=>] 92.4%	\[ \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(\left(y + 1\right) + x\right)} \cdot \left(x + y\right)} \]
   associate-+l+ [=>] 92.4%	\[ \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + \left(1 + x\right)\right)} \cdot \left(x + y\right)} \]
   +-commutative [=>] 92.4%	\[ \frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

4. Simplified 99.6%

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

   [Start] 92.4%	\[ \frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)} \]
   associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{y \cdot \frac{x}{y + x}}{y + \left(1 + x\right)}}{y + x}} \]
   associate-*l/ [<=] 99.8%	\[ \frac{\color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}}{y + x} \]
   *-commutative [<=] 99.8%	\[ \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}}}{y + x} \]
   associate-/l* [=>] 99.6%	\[ \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{\frac{y}{y + \left(1 + x\right)}}}} \]
   +-commutative [=>] 99.6%	\[ \frac{\frac{x}{y + x}}{\frac{y + x}{\frac{y}{y + \color{blue}{\left(x + 1\right)}}}} \]

5. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	1088

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

Alternative 2
Accuracy	96.6%
Cost	1352

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.16 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{\frac{x + y}{\frac{y}{x + 1}}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -125:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	1228

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{t_0}{\frac{x + y}{\frac{y}{x}}}\\ \mathbf{elif}\;x \leq -4500:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	89.6%
Cost	1224

\[\begin{array}{l} t_0 := \frac{y}{x + 1}\\ \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	91.5%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{x + 1}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	84.0%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -410:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 8
Accuracy	92.3%
Cost	1092

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{\left(x + y\right) \cdot \frac{x + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	92.4%
Cost	1092

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{\frac{x + y}{\frac{y}{x + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	99.8%
Cost	1088

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

Alternative 11
Accuracy	82.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 12
Accuracy	74.7%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Accuracy	76.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-136}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14
Accuracy	80.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 15
Accuracy	81.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 16
Accuracy	82.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17
Accuracy	64.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 18
Accuracy	66.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19
Accuracy	82.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 20
Accuracy	31.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \]

Alternative 21
Accuracy	27.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 22
Accuracy	4.4%
Cost	192

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

Alternative 23
Accuracy	3.4%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023263 
(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))))