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

Average Error: 19.8 → 0.1

Time: 9.2s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Target

Original	19.8
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 19.8

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

2. Applied egg-rr 4.0

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

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	24.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x + y \cdot 2} \cdot \frac{1}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 2
Error	24.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 243533872119.91235:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + x}}{y}\\ \end{array} \]

Alternative 3
Error	24.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.955027100959295 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 4
Error	24.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.955027100959295 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 5
Error	24.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 6
Error	33.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.062215775434316715:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 7
Error	32.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 0.062215775434316715:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	32.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 0.062215775434316715:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 9
Error	25.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 9.513062646763644 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 0.062215775434316715:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1393424343204.9263:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Error	36.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1393424343204.9263:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 12
Error	61.3
Cost	192

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

Alternative 13
Error	47.6
Cost	192

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

Alternative 14
Error	61.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(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))))