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

Average Error: 15.2 → 3.0

Time: 13.7s

Precision: binary64

Cost: 1352

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

Math

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

↓

\[\begin{array}{l} t_0 := z + z \cdot z\\ \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{t_0}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{1}{z}\right) \cdot \frac{y}{z + 1}\\ \end{array} \]

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))))
   (if (<= (* x y) 5e-261)
     (* (/ y z) (/ x t_0))
     (if (<= (* x y) 2e+126)
       (/ (/ (* x y) z) t_0)
       (* (* (/ x z) (/ 1.0 z)) (/ y (+ z 1.0)))))))

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);
	double tmp;
	if ((x * y) <= 5e-261) {
		tmp = (y / z) * (x / t_0);
	} else if ((x * y) <= 2e+126) {
		tmp = ((x * y) / z) / t_0;
	} else {
		tmp = ((x / z) * (1.0 / z)) * (y / (z + 1.0));
	}
	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 = z + (z * z)
    if ((x * y) <= 5d-261) then
        tmp = (y / z) * (x / t_0)
    else if ((x * y) <= 2d+126) then
        tmp = ((x * y) / z) / t_0
    else
        tmp = ((x / z) * (1.0d0 / z)) * (y / (z + 1.0d0))
    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);
	double tmp;
	if ((x * y) <= 5e-261) {
		tmp = (y / z) * (x / t_0);
	} else if ((x * y) <= 2e+126) {
		tmp = ((x * y) / z) / t_0;
	} else {
		tmp = ((x / z) * (1.0 / z)) * (y / (z + 1.0));
	}
	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)
	tmp = 0
	if (x * y) <= 5e-261:
		tmp = (y / z) * (x / t_0)
	elif (x * y) <= 2e+126:
		tmp = ((x * y) / z) / t_0
	else:
		tmp = ((x / z) * (1.0 / z)) * (y / (z + 1.0))
	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(z + Float64(z * z))
	tmp = 0.0
	if (Float64(x * y) <= 5e-261)
		tmp = Float64(Float64(y / z) * Float64(x / t_0));
	elseif (Float64(x * y) <= 2e+126)
		tmp = Float64(Float64(Float64(x * y) / z) / t_0);
	else
		tmp = Float64(Float64(Float64(x / z) * Float64(1.0 / z)) * Float64(y / Float64(z + 1.0)));
	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);
	tmp = 0.0;
	if ((x * y) <= 5e-261)
		tmp = (y / z) * (x / t_0);
	elseif ((x * y) <= 2e+126)
		tmp = ((x * y) / z) / t_0;
	else
		tmp = ((x / z) * (1.0 / z)) * (y / (z + 1.0));
	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[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], 5e-261], N[(N[(y / z), $MachinePrecision] * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+126], N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	15.2
Target	4.0
Herbie	3.0

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 4.99999999999999981e-261

   1. Initial program 16.4

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

   2. Applied egg-rr 4.2

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

   if 4.99999999999999981e-261 < (*.f64 x y) < 1.99999999999999985e126

   1. Initial program 6.2

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

   2. Simplified 9.1

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

      [Start] 6.2	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 9.1	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 1.9

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

   4. Taylor expanded in x around 0 0.2

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

   if 1.99999999999999985e126 < (*.f64 x y)

   1. Initial program 30.5

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

   2. Simplified 11.2

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

      [Start] 30.5	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 11.2	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.7

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z + z \cdot z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{1}{z}\right) \cdot \frac{y}{z + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	2248

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

Alternative 2
Error	2.8
Cost	1736

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

Alternative 3
Error	3.2
Cost	1224

\[\begin{array}{l} t_0 := z + z \cdot z\\ \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{t_0}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 4
Error	3.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z + z \cdot z}\\ \end{array} \]

Alternative 5
Error	3.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{+243}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 6
Error	5.9
Cost	841

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

Alternative 7
Error	5.9
Cost	841

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

Alternative 8
Error	3.6
Cost	841

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

Alternative 9
Error	17.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Error	5.9
Cost	840

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

Alternative 11
Error	4.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 12
Error	17.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-124} \lor \neg \left(y \leq 7.8 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 13
Error	18.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 14
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 15
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 16
Error	4.9
Cost	704

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

Alternative 17
Error	2.5
Cost	704

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

Alternative 18
Error	17.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 19
Error	42.7
Cost	452

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

Alternative 20
Error	21.3
Cost	448

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

Alternative 21
Error	45.7
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(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))))