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

Average Error: 14.2 → 2.9

Time: 11.8s

Precision: binary64

Cost: 1100

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{z + 1}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-128}:\\ \;\;\;\;t_0 \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (+ z 1.0))))
   (if (<= z -3.1e-128)
     (* t_0 (/ (/ x z) z))
     (if (<= z 3.4e-126)
       (/ (/ y (/ z x)) z)
       (if (<= z 1.3e+68) (/ x (/ (* z z) t_0)) (/ (* (/ x z) (/ 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 t_0 = y / (z + 1.0);
	double tmp;
	if (z <= -3.1e-128) {
		tmp = t_0 * ((x / z) / z);
	} else if (z <= 3.4e-126) {
		tmp = (y / (z / x)) / z;
	} else if (z <= 1.3e+68) {
		tmp = x / ((z * z) / t_0);
	} else {
		tmp = ((x / z) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (z + 1.0d0)
    if (z <= (-3.1d-128)) then
        tmp = t_0 * ((x / z) / z)
    else if (z <= 3.4d-126) then
        tmp = (y / (z / x)) / z
    else if (z <= 1.3d+68) then
        tmp = x / ((z * z) / t_0)
    else
        tmp = ((x / z) * (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 t_0 = y / (z + 1.0);
	double tmp;
	if (z <= -3.1e-128) {
		tmp = t_0 * ((x / z) / z);
	} else if (z <= 3.4e-126) {
		tmp = (y / (z / x)) / z;
	} else if (z <= 1.3e+68) {
		tmp = x / ((z * z) / t_0);
	} else {
		tmp = ((x / z) * (y / z)) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y / (z + 1.0)
	tmp = 0
	if z <= -3.1e-128:
		tmp = t_0 * ((x / z) / z)
	elif z <= 3.4e-126:
		tmp = (y / (z / x)) / z
	elif z <= 1.3e+68:
		tmp = x / ((z * z) / t_0)
	else:
		tmp = ((x / z) * (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)
	t_0 = Float64(y / Float64(z + 1.0))
	tmp = 0.0
	if (z <= -3.1e-128)
		tmp = Float64(t_0 * Float64(Float64(x / z) / z));
	elseif (z <= 3.4e-126)
		tmp = Float64(Float64(y / Float64(z / x)) / z);
	elseif (z <= 1.3e+68)
		tmp = Float64(x / Float64(Float64(z * z) / t_0));
	else
		tmp = Float64(Float64(Float64(x / z) * 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)
	t_0 = y / (z + 1.0);
	tmp = 0.0;
	if (z <= -3.1e-128)
		tmp = t_0 * ((x / z) / z);
	elseif (z <= 3.4e-126)
		tmp = (y / (z / x)) / z;
	elseif (z <= 1.3e+68)
		tmp = x / ((z * z) / t_0);
	else
		tmp = ((x / z) * (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_] := Block[{t$95$0 = N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.1e-128], N[(t$95$0 * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-126], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.3e+68], N[(x / N[(N[(z * z), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]

Target

Original	14.2
Target	3.7
Herbie	2.9

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

Derivation

1. Split input into 4 regimes

2. if z < -3.10000000000000003e-128

   1. Initial program 9.5

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

   2. Simplified 5.1

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

      [Start] 9.5	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 5.1	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 1.8

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

   4. Applied egg-rr 2.9

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

   if -3.10000000000000003e-128 < z < 3.4e-126

   1. Initial program 49.2

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

   2. Simplified 49.8

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

      [Start] 49.2	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 49.8	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.3

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

   4. Taylor expanded in z around 0 21.5

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

   5. Simplified 4.0

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

      [Start] 21.5	\[ \frac{\frac{y \cdot x}{z}}{z} \]
      associate-/l* [=>] 4.0	\[ \frac{\color{blue}{\frac{y}{\frac{z}{x}}}}{z} \]

   if 3.4e-126 < z < 1.2999999999999999e68

   1. Initial program 5.4

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

   2. Simplified 4.8

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

      [Start] 5.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.8	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 5.0

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

   if 1.2999999999999999e68 < z

   1. Initial program 11.4

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

   2. Simplified 4.8

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

      [Start] 11.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.8	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 0.8

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

   4. Taylor expanded in z around inf 8.5

      \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{{z}^{2}}}}{z} \]

   5. Simplified 1.9

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

      [Start] 8.5	\[ \frac{\frac{y \cdot x}{{z}^{2}}}{z} \]
      associate-/l* [=>] 5.0	\[ \frac{\color{blue}{\frac{y}{\frac{{z}^{2}}{x}}}}{z} \]
      unpow2 [=>] 5.0	\[ \frac{\frac{y}{\frac{\color{blue}{z \cdot z}}{x}}}{z} \]
      associate-*r/ [<=] 1.9	\[ \frac{\frac{y}{\color{blue}{z \cdot \frac{z}{x}}}}{z} \]

   6. Applied egg-rr 0.8

      \[\leadsto \frac{\color{blue}{\frac{y}{z} \cdot \frac{x}{z}}}{z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{\frac{y}{z + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \cdot x \leq -5 \cdot 10^{+162} \lor \neg \left(y \cdot x \leq 5 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \end{array} \]

Alternative 2
Error	3.7
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 10^{+74}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 3
Error	2.8
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 4
Error	5.8
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 5
Error	3.6
Cost	841

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

Alternative 6
Error	3.4
Cost	841

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

Alternative 7
Error	4.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 8
Error	17.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-55} \lor \neg \left(z \leq 7.6 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9
Error	17.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-55} \lor \neg \left(z \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10
Error	17.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-140} \lor \neg \left(z \leq 1.12 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 11
Error	17.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+82} \lor \neg \left(z \leq 3.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 12
Error	17.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-128} \lor \neg \left(z \leq 1.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 13
Error	2.2
Cost	704

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

Alternative 14
Error	18.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 15
Error	17.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 115000000:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 16
Error	41.9
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 17
Error	22.3
Cost	448

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

Alternative 18
Error	45.6
Cost	384

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

Alternative 19
Error	45.2
Cost	384

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

Reproduce

herbie shell --seed 2023018 
(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))))