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

Average Accuracy: 69.7% → 99.8%

Time: 14.6s

Precision: binary64

Cost: 1088.00

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

Math

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

↓

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

FPCore

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

↓

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

Target

Original	69.7%
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 69.7%

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

2. Simplified 74.5%

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

   [Start] 69.7	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 74.5	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 74.5	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	95.3%
Cost	1352.00

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{y}{x + y \cdot 2}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{\frac{y}{x + y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 2
Accuracy	83.3%
Cost	1229.00

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{-81} \lor \neg \left(y \leq 1.35 \cdot 10^{-40}\right) \land y \leq 1.16 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{x + y \cdot 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	1088.00

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

Alternative 4
Accuracy	99.8%
Cost	1088.00

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

Alternative 5
Accuracy	81.4%
Cost	976.00

\[\begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ t_1 := \frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{if}\;y \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Accuracy	83.0%
Cost	973.00

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-81} \lor \neg \left(y \leq 1.22 \cdot 10^{-40}\right) \land y \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 7
Accuracy	78.6%
Cost	844.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Accuracy	77.3%
Cost	844.00

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 10^{+18}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	844.00

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{\frac{x}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10
Accuracy	81.4%
Cost	844.00

\[\begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 7.4 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 11
Accuracy	82.7%
Cost	844.00

\[\begin{array}{l} t_0 := \frac{\frac{y}{x + 1}}{x}\\ \mathbf{if}\;y \leq 8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 12
Accuracy	68.9%
Cost	716.00

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-206}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 13
Accuracy	70.9%
Cost	716.00

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14
Accuracy	60.0%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 15
Accuracy	27.8%
Cost	324.00

\[\begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 16
Accuracy	4.3%
Cost	192.00

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

Reproduce

herbie shell --seed 2023096 
(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))))