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

Average Error: 15.1 → 2.2

Time: 14.0s

Precision: binary64

Cost: 2248

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) (* (* z z) (+ z 1.0)))))
   (if (<= t_0 0.0)
     (/ (* (/ y (+ z 1.0)) (/ x z)) z)
     (if (<= t_0 5e+276) t_0 (/ (/ x (* z (/ (+ z 1.0) y))) 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 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else if (t_0 <= 5e+276) {
		tmp = t_0;
	} else {
		tmp = (x / (z * ((z + 1.0) / y))) / 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 = (x * y) / ((z * z) * (z + 1.0d0))
    if (t_0 <= 0.0d0) then
        tmp = ((y / (z + 1.0d0)) * (x / z)) / z
    else if (t_0 <= 5d+276) then
        tmp = t_0
    else
        tmp = (x / (z * ((z + 1.0d0) / y))) / 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 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	} else if (t_0 <= 5e+276) {
		tmp = t_0;
	} else {
		tmp = (x / (z * ((z + 1.0) / y))) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / ((z * z) * (z + 1.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((y / (z + 1.0)) * (x / z)) / z
	elif t_0 <= 5e+276:
		tmp = t_0
	else:
		tmp = (x / (z * ((z + 1.0) / y))) / 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(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(y / Float64(z + 1.0)) * Float64(x / z)) / z);
	elseif (t_0 <= 5e+276)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(z * Float64(Float64(z + 1.0) / y))) / 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 = (x * y) / ((z * z) * (z + 1.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((y / (z + 1.0)) * (x / z)) / z;
	elseif (t_0 <= 5e+276)
		tmp = t_0;
	else
		tmp = (x / (z * ((z + 1.0) / y))) / 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[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e+276], t$95$0, N[(N[(x / N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	15.1
Target	4.1
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < -0.0

   1. Initial program 9.4

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

   2. Simplified 5.8

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

      [Start] 9.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 5.8	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.0

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

   if -0.0 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 5.00000000000000001e276

   1. Initial program 1.3

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

   if 5.00000000000000001e276 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 60.8

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

   2. Simplified 43.2

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

      [Start] 60.8	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 43.2	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.3

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	2249

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+256}\right):\\ \;\;\;\;\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.4
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-206}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.3
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	3.3
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-176}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z + z \cdot z}}}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	6.0
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 6
Error	4.1
Cost	841

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

Alternative 7
Error	3.7
Cost	841

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

Alternative 8
Error	20.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 9
Error	17.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 10
Error	17.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 11
Error	17.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	17.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 13
Error	41.8
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	42.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 15
Error	42.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 16
Error	23.5
Cost	448

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

Alternative 17
Error	45.4
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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))))