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

Average Error: 14.8 → 3.4

Time: 13.8s

Precision: binary64

Cost: 14664

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ t_1 := \frac{z + 1}{y}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{x}{z \cdot t_1}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\sqrt{z}}}{t_1 \cdot \sqrt{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))) (t_1 (/ (+ z 1.0) y)))
   (if (<= t_0 2e-258)
     (/ (/ x (* z t_1)) z)
     (if (<= t_0 2e+247)
       (/ (/ (* x y) (* z (+ z 1.0))) z)
       (/ (/ (/ x z) (sqrt z)) (* t_1 (sqrt 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 = (z * z) * (z + 1.0);
	double t_1 = (z + 1.0) / y;
	double tmp;
	if (t_0 <= 2e-258) {
		tmp = (x / (z * t_1)) / z;
	} else if (t_0 <= 2e+247) {
		tmp = ((x * y) / (z * (z + 1.0))) / z;
	} else {
		tmp = ((x / z) / sqrt(z)) / (t_1 * sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    t_1 = (z + 1.0d0) / y
    if (t_0 <= 2d-258) then
        tmp = (x / (z * t_1)) / z
    else if (t_0 <= 2d+247) then
        tmp = ((x * y) / (z * (z + 1.0d0))) / z
    else
        tmp = ((x / z) / sqrt(z)) / (t_1 * sqrt(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 = (z * z) * (z + 1.0);
	double t_1 = (z + 1.0) / y;
	double tmp;
	if (t_0 <= 2e-258) {
		tmp = (x / (z * t_1)) / z;
	} else if (t_0 <= 2e+247) {
		tmp = ((x * y) / (z * (z + 1.0))) / z;
	} else {
		tmp = ((x / z) / Math.sqrt(z)) / (t_1 * Math.sqrt(z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (z * z) * (z + 1.0)
	t_1 = (z + 1.0) / y
	tmp = 0
	if t_0 <= 2e-258:
		tmp = (x / (z * t_1)) / z
	elif t_0 <= 2e+247:
		tmp = ((x * y) / (z * (z + 1.0))) / z
	else:
		tmp = ((x / z) / math.sqrt(z)) / (t_1 * math.sqrt(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(Float64(z * z) * Float64(z + 1.0))
	t_1 = Float64(Float64(z + 1.0) / y)
	tmp = 0.0
	if (t_0 <= 2e-258)
		tmp = Float64(Float64(x / Float64(z * t_1)) / z);
	elseif (t_0 <= 2e+247)
		tmp = Float64(Float64(Float64(x * y) / Float64(z * Float64(z + 1.0))) / z);
	else
		tmp = Float64(Float64(Float64(x / z) / sqrt(z)) / Float64(t_1 * sqrt(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 = (z * z) * (z + 1.0);
	t_1 = (z + 1.0) / y;
	tmp = 0.0;
	if (t_0 <= 2e-258)
		tmp = (x / (z * t_1)) / z;
	elseif (t_0 <= 2e+247)
		tmp = ((x * y) / (z * (z + 1.0))) / z;
	else
		tmp = ((x / z) / sqrt(z)) / (t_1 * sqrt(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[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-258], N[(N[(x / N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2e+247], N[(N[(N[(x * y), $MachinePrecision] / N[(z * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	14.8
Target	4.1
Herbie	3.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.99999999999999991e-258

   1. Initial program 21.7

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

   2. Simplified 17.6

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

      [Start] 21.7	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 17.6	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 3.2

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

   if 1.99999999999999991e-258 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.9999999999999999e247

   1. Initial program 5.7

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

   2. Simplified 5.9

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

      [Start] 5.7	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 5.9	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.3

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

   4. Taylor expanded in x around 0 5.7

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

   if 1.9999999999999999e247 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 13.1

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

   2. Simplified 5.4

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

      [Start] 13.1	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 5.4	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 1.2

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\sqrt{z}}}{\frac{z + 1}{y} \cdot \sqrt{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	2248

\[\begin{array}{l} t_0 := z \cdot \frac{z + 1}{y}\\ t_1 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t_0}\\ \end{array} \]

Alternative 2
Error	4.9
Cost	2004

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ t_1 := \frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-209}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z + z \cdot z}}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{1}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.5
Cost	1736

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	6.9
Cost	1104

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -7800:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.0
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -7800:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z}{y}\right)}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 6
Error	7.0
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -7800:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 7
Error	3.5
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8
Error	18.7
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 9
Error	4.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7800 \lor \neg \left(z \leq 0.76\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 10
Error	4.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7800 \lor \neg \left(z \leq 0.76\right):\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 11
Error	6.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7800:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 12
Error	4.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7800:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 13
Error	18.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -0.52 \lor \neg \left(y \leq 2.7 \cdot 10^{+46}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 14
Error	18.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 15
Error	18.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 16
Error	17.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 17
Error	22.0
Cost	448

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

Alternative 18
Error	45.9
Cost	384

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

Reproduce

herbie shell --seed 2023059 
(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))))