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

Average Error: 19.3 → 0.1

Time: 2.0min

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

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

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

Target

Original	19.3
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 19.3

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 16.3

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

4. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	13.7
Cost	1620

\[\begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{x}{1 + x}}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.115:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	21.5
Cost	1368

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + y}\\ t_1 := \frac{y}{\left(y - -1\right) \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 42000000000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.0
Cost	1368

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + y}\\ t_1 := \frac{\frac{x}{y - -1}}{x + y}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{\left(y - -1\right) \cdot x}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 42000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	11.9
Cost	1356

\[\begin{array}{l} t_0 := \frac{y}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \ne 0:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \end{array} \]

Alternative 5
Error	6.8
Cost	1356

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{x}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{t_0}{x + y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.125:\\ \;\;\;\;\frac{\frac{y}{x - -1} \cdot t_0}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	21.8
Cost	1244

\[\begin{array}{l} t_0 := \frac{x}{y \cdot \left(1 + x\right)}\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 116000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 7
Error	21.8
Cost	1244

\[\begin{array}{l} t_0 := \frac{x}{y \cdot \left(1 + x\right)}\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 620000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 8
Error	21.5
Cost	1240

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + y}\\ t_1 := \frac{\frac{x}{y}}{x + y}\\ t_2 := \frac{y}{\left(y - -1\right) \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 35000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	5.9
Cost	1224

\[\begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := \frac{y}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5:\\ \;\;\;\;\frac{t_0 \cdot x}{\left(y + x\right) \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \ne 0:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 10
Error	13.1
Cost	1104

\[\begin{array}{l} t_0 := \frac{y}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \ne 0:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \end{array} \]

Alternative 11
Error	15.0
Cost	1100

\[\begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	13.0
Cost	1100

\[\begin{array}{l} t_0 := \frac{y}{\left(x + y\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \end{array} \]

Alternative 13
Error	0.1
Cost	1088

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

Alternative 14
Error	0.1
Cost	1088

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

Alternative 15
Error	21.4
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 260000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	21.0
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 340000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17
Error	17.4
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	25.3
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -2000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.115:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	61.3
Cost	192

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

Alternative 20
Error	47.0
Cost	192

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

Reproduce

herbie shell --seed 2023033 
(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))))