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

Average Error: 14.9 → 2.5

Time: 23.4s

Precision: binary64

Cost: 1224

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z + 1}}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -2e+79)
   (/ (/ y z) (/ z (/ x (+ z 1.0))))
   (if (<= (* x y) -5e-64)
     (/ (* x y) (* (* z z) (+ z 1.0)))
     (/ (/ (/ y z) (/ z x)) (+ 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 tmp;
	if ((x * y) <= -2e+79) {
		tmp = (y / z) / (z / (x / (z + 1.0)));
	} else if ((x * y) <= -5e-64) {
		tmp = (x * y) / ((z * z) * (z + 1.0));
	} else {
		tmp = ((y / z) / (z / x)) / (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) :: tmp
    if ((x * y) <= (-2d+79)) then
        tmp = (y / z) / (z / (x / (z + 1.0d0)))
    else if ((x * y) <= (-5d-64)) then
        tmp = (x * y) / ((z * z) * (z + 1.0d0))
    else
        tmp = ((y / z) / (z / x)) / (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 tmp;
	if ((x * y) <= -2e+79) {
		tmp = (y / z) / (z / (x / (z + 1.0)));
	} else if ((x * y) <= -5e-64) {
		tmp = (x * y) / ((z * z) * (z + 1.0));
	} else {
		tmp = ((y / z) / (z / x)) / (z + 1.0);
	}
	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) <= -2e+79:
		tmp = (y / z) / (z / (x / (z + 1.0)))
	elif (x * y) <= -5e-64:
		tmp = (x * y) / ((z * z) * (z + 1.0))
	else:
		tmp = ((y / z) / (z / x)) / (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)
	tmp = 0.0
	if (Float64(x * y) <= -2e+79)
		tmp = Float64(Float64(y / z) / Float64(z / Float64(x / Float64(z + 1.0))));
	elseif (Float64(x * y) <= -5e-64)
		tmp = Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)));
	else
		tmp = Float64(Float64(Float64(y / z) / Float64(z / x)) / 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)
	tmp = 0.0;
	if ((x * y) <= -2e+79)
		tmp = (y / z) / (z / (x / (z + 1.0)));
	elseif ((x * y) <= -5e-64)
		tmp = (x * y) / ((z * z) * (z + 1.0));
	else
		tmp = ((y / z) / (z / x)) / (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_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+79], N[(N[(y / z), $MachinePrecision] / N[(z / N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-64], N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.9
Target	4.3
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999993e79

   1. Initial program 25.9

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

   2. Simplified 11.0

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

      [Start] 25.9	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      rational.json-simplify-46 [=>] 19.4	\[ \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]
      rational.json-simplify-2 [=>] 19.4	\[ \frac{\frac{\color{blue}{y \cdot x}}{z \cdot z}}{z + 1} \]
      rational.json-simplify-49 [=>] 11.0	\[ \frac{\color{blue}{x \cdot \frac{y}{z \cdot z}}}{z + 1} \]

   3. Applied egg-rr 9.6

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

   4. Simplified 3.0

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

      [Start] 9.6	\[ \frac{y}{z \cdot z} \cdot \frac{x}{z + 1} \]
      rational.json-simplify-46 [=>] 3.0	\[ \color{blue}{\frac{\frac{y}{z}}{z}} \cdot \frac{x}{z + 1} \]

   5. Applied egg-rr 3.6

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

   if -1.99999999999999993e79 < (*.f64 x y) < -5.00000000000000033e-64

   1. Initial program 2.2

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

   if -5.00000000000000033e-64 < (*.f64 x y)

   1. Initial program 14.8

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

   2. Simplified 11.8

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

      [Start] 14.8	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      rational.json-simplify-46 [=>] 13.7	\[ \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]
      rational.json-simplify-2 [=>] 13.7	\[ \frac{\frac{\color{blue}{y \cdot x}}{z \cdot z}}{z + 1} \]
      rational.json-simplify-49 [=>] 11.8	\[ \frac{\color{blue}{x \cdot \frac{y}{z \cdot z}}}{z + 1} \]

   3. Applied egg-rr 2.4

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z + 1}}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z + 1}}}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \end{array} \]

Alternative 2
Error	2.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z + 1}}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z + 1}\\ \end{array} \]

Alternative 3
Error	3.9
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	3.2
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	3.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z + 1}\\ \end{array} \]

Alternative 6
Error	3.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{z \cdot \left(\frac{z}{x} \cdot \left(z + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z + 1}\\ \end{array} \]

Alternative 7
Error	4.7
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	4.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 9
Error	2.6
Cost	968

\[\begin{array}{l} t_0 := \frac{x}{z + 1}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-113}:\\ \;\;\;\;\frac{t_0}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \end{array} \]

Alternative 10
Error	18.2
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{z \cdot z} \cdot y\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	18.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]

Alternative 12
Error	17.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]

Alternative 13
Error	17.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	43.3
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \]

Alternative 15
Error	42.9
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \]

Alternative 16
Error	42.6
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-66}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 17
Error	21.9
Cost	448

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

Alternative 18
Error	49.1
Cost	384

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

Alternative 19
Error	46.2
Cost	384

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

Reproduce

herbie shell --seed 2023064 
(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))))