Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 14.6 → 2.6

Time: 7.7s

Precision: binary64

Cost: 704

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

Math

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

↓

\[\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z} \]

FPCore

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

↓

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

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

double code(double x, double y, double z) {
	return ((y / z) * (x / (z + 1.0))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y / z) * (x / (z + 1.0d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

public static double code(double x, double y, double z) {
	return ((y / z) * (x / (z + 1.0))) / z;
}

Python

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

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

↓

function code(x, y, z)
	return Float64(Float64(Float64(y / z) * Float64(x / Float64(z + 1.0))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

↓

function tmp = code(x, y, z)
	tmp = ((y / z) * (x / (z + 1.0))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(N[(y / z), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Target

Original	14.6
Target	3.9
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

1. Initial program 14.6

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

2. Applied egg-rr 10.9

   \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

3. Applied egg-rr 2.6

   \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}} \]

4. Final simplification 2.6

   \[\leadsto \frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z} \]

Alternatives

Alternative 1
Error	3.7
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}}\\ \mathbf{if}\;y \leq -2.6325170996799713 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.170845858513643 \cdot 10^{-255}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;y \leq 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 2
Error	4.4
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.3
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	17.2
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.5
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 7
Error	42.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.5476024964154104 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 8
Error	42.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 9
Error	22.2
Cost	448

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

Alternative 10
Error	45.6
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))