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

Average Error: 19.9 → 0.1

Time: 14.7s

Precision: binary64

Cost: 1216

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

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Target

Original	19.9
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 19.9

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

2. Simplified 16.7

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

   [Start] 19.9	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 16.7	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 16.7	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ {\left(\frac{y + x}{x}\right)}^{-1} \cdot \left(\frac{y}{x + y} \cdot \frac{1}{y + \left(x + 1\right)}\right) \]
   unpow-1 [=>] 0.2	\[ \color{blue}{\frac{1}{\frac{y + x}{x}}} \cdot \left(\frac{y}{x + y} \cdot \frac{1}{y + \left(x + 1\right)}\right) \]

6. Applied egg-rr 0.1

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

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	8.7
Cost	1360

\[\begin{array}{l} t_0 := \frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(x + y \cdot 2\right)}\\ t_1 := x + \left(1 + y\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_1}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_1}\\ \end{array} \]

Alternative 2
Error	5.4
Cost	1352

\[\begin{array}{l} t_0 := x + \left(1 + y\right)\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_0}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{\frac{y + \left(1 + x\right)}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 3
Error	3.2
Cost	1352

\[\begin{array}{l} t_0 := x + \left(1 + y\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t_0}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(\frac{y + x}{x} \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t_0}\\ \end{array} \]

Alternative 4
Error	16.0
Cost	1108

\[\begin{array}{l} \mathbf{if}\;x \leq -116:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 5
Error	0.1
Cost	1088

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

Alternative 6
Error	18.2
Cost	980

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -116:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.2
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -116:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	16.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -116:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Error	13.5
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -116:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-115} \lor \neg \left(x \leq -4.1 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 10
Error	12.1
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(1 + y\right)}\\ \end{array} \]

Alternative 11
Error	10.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{\frac{y + \left(1 + x\right)}{\frac{1}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 12
Error	10.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\frac{y + \left(1 + x\right)}{\frac{1}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \left(y + x\right)}}{\frac{y}{x}}\\ \end{array} \]

Alternative 13
Error	10.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\frac{y + \left(1 + x\right)}{\frac{1}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(1 + y\right)}\\ \end{array} \]

Alternative 14
Error	13.0
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-115} \lor \neg \left(x \leq -4.1 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 15
Error	22.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 16
Error	35.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 17
Error	61.4
Cost	192

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

Alternative 18
Error	47.3
Cost	192

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

Reproduce

herbie shell --seed 2022349 
(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))))