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

Average Error: 14.8 → 3.3

Time: 10.4s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-14)
   (/ (/ x (* z (/ (+ z 1.0) y))) z)
   (if (<= z 6.5e-91) (/ (/ y (/ z x)) z) (/ (* y (/ (/ x z) z)) (+ 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 (z <= -1e-14) {
		tmp = (x / (z * ((z + 1.0) / y))) / z;
	} else if (z <= 6.5e-91) {
		tmp = (y / (z / x)) / z;
	} else {
		tmp = (y * ((x / z) / z)) / (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 (z <= (-1d-14)) then
        tmp = (x / (z * ((z + 1.0d0) / y))) / z
    else if (z <= 6.5d-91) then
        tmp = (y / (z / x)) / z
    else
        tmp = (y * ((x / z) / z)) / (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 (z <= -1e-14) {
		tmp = (x / (z * ((z + 1.0) / y))) / z;
	} else if (z <= 6.5e-91) {
		tmp = (y / (z / x)) / z;
	} else {
		tmp = (y * ((x / z) / z)) / (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 z <= -1e-14:
		tmp = (x / (z * ((z + 1.0) / y))) / z
	elif z <= 6.5e-91:
		tmp = (y / (z / x)) / z
	else:
		tmp = (y * ((x / z) / z)) / (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 (z <= -1e-14)
		tmp = Float64(Float64(x / Float64(z * Float64(Float64(z + 1.0) / y))) / z);
	elseif (z <= 6.5e-91)
		tmp = Float64(Float64(y / Float64(z / x)) / z);
	else
		tmp = Float64(Float64(y * Float64(Float64(x / z) / z)) / 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 (z <= -1e-14)
		tmp = (x / (z * ((z + 1.0) / y))) / z;
	elseif (z <= 6.5e-91)
		tmp = (y / (z / x)) / z;
	else
		tmp = (y * ((x / z) / z)) / (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[z, -1e-14], N[(N[(x / N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 6.5e-91], N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.8
Target	4.1
Herbie	3.3

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

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999999e-15

   1. Initial program 10.3

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

   2. Simplified 4.3

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

      [Start] 10.3	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.3	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.4

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

   if -9.99999999999999999e-15 < z < 6.5000000000000001e-91

   1. Initial program 29.4

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

   2. Simplified 30.1

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

      [Start] 29.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 30.1	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 5.7

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

   4. Taylor expanded in z around 0 15.7

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

   5. Simplified 5.4

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

      [Start] 15.7	\[ \frac{\frac{y \cdot x}{z}}{z} \]
      associate-/l* [=>] 5.4	\[ \frac{\color{blue}{\frac{y}{\frac{z}{x}}}}{z} \]

   if 6.5000000000000001e-91 < z

   1. Initial program 9.9

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

   2. Simplified 4.9

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

      [Start] 9.9	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 4.9	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.8

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	1736

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \end{array} \]

Alternative 2
Error	4.2
Cost	1736

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \end{array} \]

Alternative 3
Error	6.0
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 4
Error	6.2
Cost	841

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

Alternative 5
Error	5.2
Cost	840

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

Alternative 6
Error	4.4
Cost	840

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

Alternative 7
Error	4.5
Cost	840

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

Alternative 8
Error	17.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-63} \lor \neg \left(y \leq 1.1 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9
Error	19.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-228}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	17.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	21.8
Cost	448

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

Alternative 13
Error	45.5
Cost	384

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

Reproduce

herbie shell --seed 2023057 
(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))))