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

Average Error: 19.7 → 0.1

Time: 14.4s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

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

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 / (x + y)) / (x + (y + 1.0)))) / (x + y);
}

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 / (x + y)) / (x + (y + 1.0d0)))) / (x + y)
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 / (x + y)) / (x + (y + 1.0)))) / (x + y);
}

Python

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

↓

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

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(Float64(y / Float64(x + y)) / Float64(x + Float64(y + 1.0)))) / Float64(x + y))
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 / (x + y)) / (x + (y + 1.0)))) / (x + y);
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[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

Target

Original	19.7
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 19.7

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

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] 19.7	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	18.9
Cost	1352

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{x}}{t_0}}{x + y}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{\frac{y + \left(x + 1\right)}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 2
Error	22.1
Cost	1092

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{x}}{t_0}}{x + y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 3
Error	22.2
Cost	968

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 4
Error	22.1
Cost	968

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t_0}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 5
Error	29.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Error	25.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Error	25.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	25.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Error	25.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.66 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 10
Error	24.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{y + y \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 11
Error	30.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 12
Error	29.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Error	25.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 14
Error	25.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 15
Error	25.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 16
Error	25.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 17
Error	35.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 18
Error	42.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 19
Error	61.3
Cost	192

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

Alternative 20
Error	46.7
Cost	192

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

Reproduce

herbie shell --seed 2023016 
(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))))