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

Average Error: 29.87% → 0.16%

Time: 15.6s

Precision: binary64

Cost: 1088

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	29.87%
Target	0.21%
Herbie	0.16%

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

Derivation

1. Initial program 29.87

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

2. Simplified 25.51

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

   [Start] 29.87	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 25.51	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 25.51	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 0.15

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

4. Simplified 0.16

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

   [Start] 0.15	\[ \frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{x + \left(y + 1\right)} \]
   *-commutative [<=] 0.15	\[ \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{x + \left(y + 1\right)} \]
   associate-*r/ [=>] 0.16	\[ \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{x + y}}}{x + \left(y + 1\right)} \]

5. Final simplification 0.16

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

Alternatives

Alternative 1
Error	9.03%
Cost	1352

\[\begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{t_0} \cdot \frac{1}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{\frac{t_0}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 2
Error	19.05%
Cost	1100

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \end{array} \]

Alternative 3
Error	18.94%
Cost	1100

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_0}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \end{array} \]

Alternative 4
Error	21.42%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156} \lor \neg \left(x \leq -2.4 \cdot 10^{-196}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 5
Error	19.28%
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 6
Error	19.04%
Cost	972

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;x \leq -8.3 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 7
Error	20.56%
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-62} \lor \neg \left(x \leq -8.2 \cdot 10^{-156}\right) \land x \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 8
Error	20.65%
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156} \lor \neg \left(x \leq -2.4 \cdot 10^{-196}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \end{array} \]

Alternative 9
Error	16.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{y + y \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10
Error	29.31%
Cost	716

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.016:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 11
Error	27.55%
Cost	716

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.016:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Error	26.18%
Cost	716

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.016:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Error	71.97%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + 1}\\ \end{array} \]

Alternative 14
Error	38.61%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 15
Error	71.97%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 16
Error	95.91%
Cost	192

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

Reproduce

herbie shell --seed 2023088 
(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))))