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

Average Error: 19.9 → 0.1

Time: 16.0s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (/ (* (/ y (+ y x)) (/ 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 ((y / (y + x)) * (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 = ((y / (y + x)) * (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 ((y / (y + x)) * (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 ((y / (y + x)) * (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(y / Float64(y + x)) * Float64(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 = ((y / (y + x)) * (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[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $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 16.8

   \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}} \]
   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* [=>] 16.8	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 16.8	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	24.3
Cost	1356

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-y}{y + x}}{-1 - \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 2
Error	19.5
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{-1}{-1 - \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{y + \left(x + 1\right)}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 3
Error	24.3
Cost	1164

\[\begin{array}{l} t_0 := \frac{\frac{-y}{y + x}}{-1 - \left(y + x\right)}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 4
Error	24.6
Cost	1108

\[\begin{array}{l} t_0 := \frac{x}{y \cdot \left(y + 1\right)}\\ t_1 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 5
Error	23.6
Cost	1104

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t_0}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{-y}{x \cdot \left(-1 - \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 6
Error	23.6
Cost	1104

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-y}{x \cdot \left(-1 - \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 7
Error	24.8
Cost	1092

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{y + x}}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 8
Error	24.6
Cost	976

\[\begin{array}{l} t_0 := \frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Error	24.5
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 10
Error	24.4
Cost	972

\[\begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t_0}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t_0}\\ \end{array} \]

Alternative 11
Error	36.7
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 12
Error	26.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Error	30.7
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 14
Error	30.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15
Error	29.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 16
Error	43.8
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-225} \lor \neg \left(y \leq 2.7 \cdot 10^{-185}\right) \land y \leq 3.4 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 17
Error	47.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00182:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18
Error	61.3
Cost	192

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

Reproduce

herbie shell --seed 2022364 
(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))))