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

Average Accuracy: 75.8% → 95.7%

Time: 13.6s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -4e-160)
   (/ (* (/ y (+ z 1.0)) (/ x z)) z)
   (if (<= z 1.55e-75) (/ (/ x (/ z y)) z) (* (/ x (+ z 1.0)) (/ (/ y 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 <= -4e-160) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else if (z <= 1.55e-75) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = (x / (z + 1.0)) * ((y / 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 <= (-4d-160)) then
        tmp = ((y / (z + 1.0d0)) * (x / z)) / z
    else if (z <= 1.55d-75) then
        tmp = (x / (z / y)) / z
    else
        tmp = (x / (z + 1.0d0)) * ((y / 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 <= -4e-160) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else if (z <= 1.55e-75) {
		tmp = (x / (z / y)) / z;
	} else {
		tmp = (x / (z + 1.0)) * ((y / 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 <= -4e-160:
		tmp = ((y / (z + 1.0)) * (x / z)) / z
	elif z <= 1.55e-75:
		tmp = (x / (z / y)) / z
	else:
		tmp = (x / (z + 1.0)) * ((y / 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 <= -4e-160)
		tmp = Float64(Float64(Float64(y / Float64(z + 1.0)) * Float64(x / z)) / z);
	elseif (z <= 1.55e-75)
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	else
		tmp = Float64(Float64(x / Float64(z + 1.0)) * Float64(Float64(y / 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 <= -4e-160)
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	elseif (z <= 1.55e-75)
		tmp = (x / (z / y)) / z;
	else
		tmp = (x / (z + 1.0)) * ((y / 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, -4e-160], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.55e-75], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	75.8%
Target	93.5%
Herbie	95.7%

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

Derivation

1. Split input into 3 regimes

2. if z < -4e-160

   1. Initial program 83.9%

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

   2. Simplified 90.6%

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

      [Start] 83.9	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 90.6	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 96.5%

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

      [Start] 90.6	\[ \frac{x}{z \cdot z} \cdot \frac{y}{z + 1} \]
      *-commutative [=>] 90.6	\[ \color{blue}{\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}} \]
      associate-/r* [=>] 94.1	\[ \frac{y}{z + 1} \cdot \color{blue}{\frac{\frac{x}{z}}{z}} \]
      associate-*r/ [=>] 96.5	\[ \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]

   if -4e-160 < z < 1.55000000000000003e-75

   1. Initial program 31.8%

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

   2. Simplified 31.2%

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

      [Start] 31.8	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 31.2	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 91.6%

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

      [Start] 31.2	\[ \frac{x}{z \cdot z} \cdot \frac{y}{z + 1} \]
      *-commutative [=>] 31.2	\[ \color{blue}{\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}} \]
      associate-/r* [=>] 65.4	\[ \frac{y}{z + 1} \cdot \color{blue}{\frac{\frac{x}{z}}{z}} \]
      associate-*r/ [=>] 91.6	\[ \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]

   4. Applied egg-rr 92.4%

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

      [Start] 91.6	\[ \frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z} \]
      clear-num [=>] 91.5	\[ \frac{\color{blue}{\frac{1}{\frac{z + 1}{y}}} \cdot \frac{x}{z}}{z} \]
      frac-times [=>] 92.4	\[ \frac{\color{blue}{\frac{1 \cdot x}{\frac{z + 1}{y} \cdot z}}}{z} \]
      *-un-lft-identity [<=] 92.4	\[ \frac{\frac{\color{blue}{x}}{\frac{z + 1}{y} \cdot z}}{z} \]

   5. Taylor expanded in z around 0 92.5%

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

   if 1.55000000000000003e-75 < z

   1. Initial program 84.1%

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

   2. Taylor expanded in x around 0 84.1%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2} \cdot \left(1 + z\right)}} \]

   3. Simplified 96.1%

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

      [Start] 84.1	\[ \frac{y \cdot x}{{z}^{2} \cdot \left(1 + z\right)} \]
      unpow2 [=>] 84.1	\[ \frac{y \cdot x}{\color{blue}{\left(z \cdot z\right)} \cdot \left(1 + z\right)} \]
      times-frac [=>] 92.7	\[ \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{1 + z}} \]
      +-commutative [<=] 92.7	\[ \frac{y}{z \cdot z} \cdot \frac{x}{\color{blue}{z + 1}} \]
      *-commutative [<=] 92.7	\[ \color{blue}{\frac{x}{z + 1} \cdot \frac{y}{z \cdot z}} \]
      associate-/r* [=>] 96.1	\[ \frac{x}{z + 1} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-36} \lor \neg \left(z \leq 1.38 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 2
Accuracy	90.5%
Cost	841

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

Alternative 3
Accuracy	93.3%
Cost	841

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

Alternative 4
Accuracy	90.3%
Cost	840

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

Alternative 5
Accuracy	90.6%
Cost	840

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

Alternative 6
Accuracy	92.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \end{array} \]

Alternative 7
Accuracy	73.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-294}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 8
Accuracy	73.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 9
Accuracy	73.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-266}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Accuracy	73.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-250}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Accuracy	73.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Accuracy	73.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-249}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 13
Accuracy	95.2%
Cost	704

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

Alternative 14
Accuracy	71.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 15
Accuracy	31.8%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16
Accuracy	32.2%
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{\frac{z}{-y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 17
Accuracy	63.2%
Cost	448

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

Alternative 18
Accuracy	65.6%
Cost	448

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

Alternative 19
Accuracy	27.9%
Cost	384

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

Reproduce

herbie shell --seed 2023147 
(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))))