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

Average Error: 14.6 → 1.9

Time: 9.6s

Precision: binary64

Cost: 836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+160)
   (/ (/ y z) (* z (/ (+ z 1.0) x)))
   (/ (* (/ y (+ z 1.0)) (/ x z)) 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) {
	double tmp;
	if (y <= 2e+160) {
		tmp = (y / z) / (z * ((z + 1.0) / x));
	} else {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 2d+160) then
        tmp = (y / z) / (z * ((z + 1.0d0) / x))
    else
        tmp = ((y / (z + 1.0d0)) * (x / z)) / z
    end if
    code = tmp
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) {
	double tmp;
	if (y <= 2e+160) {
		tmp = (y / z) / (z * ((z + 1.0) / x));
	} else {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= 2e+160:
		tmp = (y / z) / (z * ((z + 1.0) / x))
	else:
		tmp = ((y / (z + 1.0)) * (x / z)) / z
	return tmp

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)
	tmp = 0.0
	if (y <= 2e+160)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(Float64(z + 1.0) / x)));
	else
		tmp = Float64(Float64(Float64(y / Float64(z + 1.0)) * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+160)
		tmp = (y / z) / (z * ((z + 1.0) / x));
	else
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[y, 2e+160], N[(N[(y / z), $MachinePrecision] / N[(z * N[(N[(z + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	14.6
Target	4.0
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000001e160

   1. Initial program 13.0

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

   2. Simplified 9.7

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
      Proof
      (*.f64 (/.f64 y (*.f64 z z)) (/.f64 x (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 (*.f64 z z) (+.f64 z 1)))): 47 points increase in error, 27 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 (*.f64 z z) (+.f64 z 1))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.7

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

   if 2.00000000000000001e160 < y

   1. Initial program 22.3

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

   2. Simplified 12.0

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      Proof
      (*.f64 (/.f64 x (*.f64 z z)) (/.f64 y (+.f64 z 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))): 48 points increase in error, 32 points decrease in error

   3. Applied egg-rr 2.8

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	1224

\[\begin{array}{l} t_0 := \frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{-295}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	4.1
Cost	972

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-152}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{x}{z} - x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.0
Cost	840

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 5
Error	4.3
Cost	840

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 7
Error	3.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{\frac{x}{z} - x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 8
Error	4.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{\frac{x}{z} - x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Error	18.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 48000:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	18.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3500:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 12
Error	42.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 13
Error	22.1
Cost	448

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

Alternative 14
Error	45.5
Cost	320

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

Reproduce

herbie shell --seed 2022329 
(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))))