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

Average Error: 24% → 4.26%

Time: 11.5s

Precision: binary64

Cost: 1224

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \left(\frac{1}{y} \cdot \left(z + 1\right)\right)}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e+149)
   (/ (/ x z) (* z (* (/ 1.0 y) (+ z 1.0))))
   (if (<= (* x y) 4e+185)
     (/ (* y (/ x z)) (+ z (* z z)))
     (/ (/ x z) (* z (/ (+ z 1.0) y))))))

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 ((x * y) <= -5e+149) {
		tmp = (x / z) / (z * ((1.0 / y) * (z + 1.0)));
	} else if ((x * y) <= 4e+185) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (x / z) / (z * ((z + 1.0) / y));
	}
	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 ((x * y) <= (-5d+149)) then
        tmp = (x / z) / (z * ((1.0d0 / y) * (z + 1.0d0)))
    else if ((x * y) <= 4d+185) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (x / z) / (z * ((z + 1.0d0) / y))
    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 ((x * y) <= -5e+149) {
		tmp = (x / z) / (z * ((1.0 / y) * (z + 1.0)));
	} else if ((x * y) <= 4e+185) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (x / z) / (z * ((z + 1.0) / y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -5e+149:
		tmp = (x / z) / (z * ((1.0 / y) * (z + 1.0)))
	elif (x * y) <= 4e+185:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (x / z) / (z * ((z + 1.0) / y))
	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 (Float64(x * y) <= -5e+149)
		tmp = Float64(Float64(x / z) / Float64(z * Float64(Float64(1.0 / y) * Float64(z + 1.0))));
	elseif (Float64(x * y) <= 4e+185)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) / Float64(z * Float64(Float64(z + 1.0) / y)));
	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 ((x * y) <= -5e+149)
		tmp = (x / z) / (z * ((1.0 / y) * (z + 1.0)));
	elseif ((x * y) <= 4e+185)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (x / z) / (z * ((z + 1.0) / y));
	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[N[(x * y), $MachinePrecision], -5e+149], N[(N[(x / z), $MachinePrecision] / N[(z * N[(N[(1.0 / y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+185], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	24%
Target	6.75%
Herbie	4.26%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999999e149

   1. Initial program 50.63

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

   2. Simplified 17.74

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

      [Start] 50.63	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 17.74	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.62

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

   4. Applied egg-rr 4.68

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

   if -4.9999999999999999e149 < (*.f64 x y) < 3.9999999999999999e185

   1. Initial program 16.94

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

   2. Simplified 17.75

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

      [Start] 16.94	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 17.75	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.22

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

   if 3.9999999999999999e185 < (*.f64 x y)

   1. Initial program 61.03

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

   2. Simplified 22.91

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

      [Start] 61.03	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 22.91	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.14

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

4. Final simplification 4.26

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \left(\frac{1}{y} \cdot \left(z + 1\right)\right)}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.8%
Cost	2252

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ t_1 := \frac{\frac{x}{z}}{\frac{z}{\frac{y}{z}}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+90}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.25%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+149} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \end{array} \]

Alternative 3
Error	6.65%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{\frac{y}{z}}}\\ \end{array} \]

Alternative 4
Error	5.93%
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{\frac{z}{\frac{y}{z}}}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.11%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -33 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 6
Error	6.98%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -33 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 7
Error	6.68%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -33 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 8
Error	6.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -33 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 9
Error	28.33%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-69} \lor \neg \left(y \leq 4.6 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10
Error	28.09%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	27.56%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	27.26%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 13
Error	66.98%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	34.23%
Cost	448

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

Alternative 15
Error	72.71%
Cost	384

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

Alternative 16
Error	72.03%
Cost	384

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

Reproduce

herbie shell --seed 2023090 
(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))))