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

Average Error: 23.22% → 5.3%

Time: 11.7s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x} \cdot \left(z + 1\right)}}{z}\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.35e-209)
   (/ (/ y (* (/ z x) (+ z 1.0))) z)
   (if (<= z 3.85e-94) (* x (/ (/ y z) z)) (/ (* (/ 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 (z <= 2.35e-209) {
		tmp = (y / ((z / x) * (z + 1.0))) / z;
	} else if (z <= 3.85e-94) {
		tmp = x * ((y / z) / z);
	} 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 (z <= 2.35d-209) then
        tmp = (y / ((z / x) * (z + 1.0d0))) / z
    else if (z <= 3.85d-94) then
        tmp = x * ((y / z) / z)
    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 (z <= 2.35e-209) {
		tmp = (y / ((z / x) * (z + 1.0))) / z;
	} else if (z <= 3.85e-94) {
		tmp = x * ((y / z) / z);
	} 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 z <= 2.35e-209:
		tmp = (y / ((z / x) * (z + 1.0))) / z
	elif z <= 3.85e-94:
		tmp = x * ((y / z) / z)
	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 (z <= 2.35e-209)
		tmp = Float64(Float64(y / Float64(Float64(z / x) * Float64(z + 1.0))) / z);
	elseif (z <= 3.85e-94)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	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 (z <= 2.35e-209)
		tmp = (y / ((z / x) * (z + 1.0))) / z;
	elseif (z <= 3.85e-94)
		tmp = x * ((y / z) / z);
	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[z, 2.35e-209], N[(N[(y / N[(N[(z / x), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 3.85e-94], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	23.22%
Target	6.8%
Herbie	5.3%

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

Derivation

1. Split input into 3 regimes

2. if z < 2.35e-209

   1. Initial program 27.55

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

   2. Simplified 21.35

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

      [Start] 27.55	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 21.35	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 3.75

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

   4. Applied egg-rr 4.51

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

   if 2.35e-209 < z < 3.8499999999999998e-94

   1. Initial program 44.18

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

   2. Simplified 45.4

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

      [Start] 44.18	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 45.4	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 11.29

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

   4. Taylor expanded in z around 0 18.1

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

   5. Simplified 10.49

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

      [Start] 18.1	\[ \frac{\frac{y \cdot x}{z}}{z} \]
      associate-/l* [=>] 10.49	\[ \frac{\color{blue}{\frac{y}{\frac{z}{x}}}}{z} \]

   6. Applied egg-rr 27.74

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

   if 3.8499999999999998e-94 < z

   1. Initial program 14.67

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

   2. Simplified 7.55

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

      [Start] 14.67	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 7.55	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 3.07

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

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x} \cdot \left(z + 1\right)}}{z}\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.42%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 2
Error	4.93%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-209} \lor \neg \left(z \leq 1.05 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 3
Error	7.55%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 4
Error	9.37%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 5
Error	9.18%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \lor \neg \left(z \leq 0.76\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Error	6.83%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \lor \neg \left(z \leq 0.76\right):\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 7
Error	6.59%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \lor \neg \left(z \leq 0.78\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 8
Error	6.58%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 9
Error	28.83%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-24} \lor \neg \left(z \leq 4.7 \cdot 10^{-95}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 10
Error	28.62%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-31} \lor \neg \left(z \leq 10^{-96}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 11
Error	28.39%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -50000000000000:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 12
Error	28.21%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;x \leq 1.48 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 13
Error	27.96%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	27.97%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 15
Error	68.26%
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-116}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16
Error	34.71%
Cost	448

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

Alternative 17
Error	72.47%
Cost	384

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

Reproduce

herbie shell --seed 2023121 
(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))))