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

Average Error: 14.3 → 2.8

Time: 10.4s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Target

Original	14.3
Target	3.8
Herbie	2.8

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

Derivation

1. Initial program 14.3

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

2. Simplified 10.6

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

   [Start] 14.3	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   times-frac [=>] 10.6	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

3. Applied egg-rr 2.7

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

4. Applied egg-rr 2.8

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

5. Final simplification 2.8

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

Alternatives

Alternative 1
Error	3.0
Cost	1736

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;t_0 \leq 10^{+43}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot \frac{z}{x}}}{z}\\ \end{array} \]

Alternative 2
Error	5.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 3
Error	4.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 7500000\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 4
Error	4.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 7500000\right):\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 5
Error	4.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 6
Error	4.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;z \leq 7500000:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z}{y}}}{z}\\ \end{array} \]

Alternative 7
Error	4.2
Cost	840

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

Alternative 8
Error	4.3
Cost	840

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

Alternative 9
Error	4.0
Cost	840

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

Alternative 10
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+123}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.46 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 11
Error	2.7
Cost	704

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

Alternative 12
Error	18.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 13
Error	18.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	18.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 15
Error	22.2
Cost	448

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

Alternative 16
Error	45.6
Cost	384

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

Reproduce

herbie shell --seed 2023016 
(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))))