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

Percentage Accurate: 83.2% → 97.7%

Time: 14.4s

Precision: binary64

Cost: 964

• Unsound rule application detected in e-graph. Results may not be sound.

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z + 1}{y}}}{z}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 5e-235)
   (/ (* (/ x z) (/ y z)) (+ z 1.0))
   (/ (/ (/ x (/ (+ z 1.0) y)) z) 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 tmp;
	if ((x * y) <= 5e-235) {
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	} else {
		tmp = ((x / ((z + 1.0) / y)) / z) / 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) :: tmp
    if ((x * y) <= 5d-235) then
        tmp = ((x / z) * (y / z)) / (z + 1.0d0)
    else
        tmp = ((x / ((z + 1.0d0) / y)) / z) / 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 tmp;
	if ((x * y) <= 5e-235) {
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	} else {
		tmp = ((x / ((z + 1.0) / y)) / z) / z;
	}
	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) <= 5e-235:
		tmp = ((x / z) * (y / z)) / (z + 1.0)
	else:
		tmp = ((x / ((z + 1.0) / y)) / z) / 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)
	tmp = 0.0
	if (Float64(x * y) <= 5e-235)
		tmp = Float64(Float64(Float64(x / z) * Float64(y / z)) / Float64(z + 1.0));
	else
		tmp = Float64(Float64(Float64(x / Float64(Float64(z + 1.0) / y)) / z) / 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)
	tmp = 0.0;
	if ((x * y) <= 5e-235)
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	else
		tmp = ((x / ((z + 1.0) / y)) / z) / 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_] := If[LessEqual[N[(x * y), $MachinePrecision], 5e-235], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	83.2%
Target	95.3%
Herbie	97.7%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4.9999999999999998e-235

   1. Initial program 85.0%

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

   2. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
      Step-by-step derivation

      [Start] 85.0%	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      associate-*l* [=>] 85.0%	\[ \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]
      times-frac [=>] 96.7%	\[ \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]
      distribute-lft-in [=>] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]
      fma-def [=>] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]
      *-rgt-identity [=>] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}} \]
      Step-by-step derivation

      [Start] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]
      fma-udef [=>] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z}} \]
      *-rgt-identity [<=] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{z \cdot z + \color{blue}{z \cdot 1}} \]
      distribute-lft-in [<=] 96.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot \left(z + 1\right)}} \]
      times-frac [<=] 85.0%	\[ \color{blue}{\frac{x \cdot y}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]
      associate-*l* [<=] 85.0%	\[ \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]
      associate-/r* [=>] 85.8%	\[ \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]
      times-frac [=>] 98.0%	\[ \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

   if 4.9999999999999998e-235 < (*.f64 x y)

   1. Initial program 80.6%

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

   2. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
      Step-by-step derivation

      [Start] 80.6%	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      associate-*l* [=>] 80.7%	\[ \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]
      times-frac [=>] 88.4%	\[ \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]
      distribute-lft-in [=>] 88.4%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]
      fma-def [=>] 88.4%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]
      *-rgt-identity [=>] 88.4%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]
      Step-by-step derivation

      [Start] 88.4%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]
      *-commutative [=>] 88.4%	\[ \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{x}{z}} \]
      associate-*l/ [=>] 92.0%	\[ \color{blue}{\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}} \]
      fma-udef [=>] 92.0%	\[ \frac{y \cdot \frac{x}{z}}{\color{blue}{z \cdot z + z}} \]
      distribute-lft1-in [=>] 92.0%	\[ \frac{y \cdot \frac{x}{z}}{\color{blue}{\left(z + 1\right) \cdot z}} \]
      frac-times [<=] 93.6%	\[ \color{blue}{\frac{y}{z + 1} \cdot \frac{\frac{x}{z}}{z}} \]
      associate-*r/ [=>] 93.8%	\[ \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]

   4. Applied egg-rr 95.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{\frac{z + 1}{y}}}{z}}}{z} \]
      Step-by-step derivation

      [Start] 93.8%	\[ \frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z} \]
      associate-*r/ [=>] 95.5%	\[ \frac{\color{blue}{\frac{\frac{y}{z + 1} \cdot x}{z}}}{z} \]
      clear-num [=>] 95.3%	\[ \frac{\frac{\color{blue}{\frac{1}{\frac{z + 1}{y}}} \cdot x}{z}}{z} \]
      associate-*l/ [=>] 95.3%	\[ \frac{\frac{\color{blue}{\frac{1 \cdot x}{\frac{z + 1}{y}}}}{z}}{z} \]
      *-un-lft-identity [<=] 95.3%	\[ \frac{\frac{\frac{\color{blue}{x}}{\frac{z + 1}{y}}}{z}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z + 1}{y}}}{z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z + 1}{y}}}{z}}{z}\\ \end{array} \]

Alternative 2
Accuracy	96.3%
Cost	1737

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -200000 \lor \neg \left(t_0 \leq 2 \cdot 10^{-145}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 3
Accuracy	94.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-139}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 4
Accuracy	93.1%
Cost	841

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

Alternative 5
Accuracy	95.2%
Cost	841

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

Alternative 6
Accuracy	95.0%
Cost	841

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

Alternative 7
Accuracy	95.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \end{array} \]

Alternative 8
Accuracy	97.0%
Cost	704

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

Alternative 9
Accuracy	77.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10
Accuracy	79.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Accuracy	79.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12
Accuracy	72.8%
Cost	448

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

Reproduce

herbie shell --seed 2023263 
(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))))