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

Percentage Accurate: 83.1% → 94.5%

Time: 14.9s

Precision: binary64

Cost: 2249

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

• 25.9% 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

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z + 1}}{z \cdot z}\\ \end{array} \]

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 1.0) (* z z)))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+283)))
     (* (/ (/ x z) z) (/ y (+ z 1.0)))
     (/ (/ (* x y) (+ z 1.0)) (* 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 t_0 = (x * y) / ((z + 1.0) * (z * z));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+283)) {
		tmp = ((x / z) / z) * (y / (z + 1.0));
	} else {
		tmp = ((x * y) / (z + 1.0)) / (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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / ((z + 1.0d0) * (z * z))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+283))) then
        tmp = ((x / z) / z) * (y / (z + 1.0d0))
    else
        tmp = ((x * y) / (z + 1.0d0)) / (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 t_0 = (x * y) / ((z + 1.0) * (z * z));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+283)) {
		tmp = ((x / z) / z) * (y / (z + 1.0));
	} else {
		tmp = ((x * y) / (z + 1.0)) / (z * 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 + 1.0) * (z * z))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+283):
		tmp = ((x / z) / z) * (y / (z + 1.0))
	else:
		tmp = ((x * y) / (z + 1.0)) / (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)
	t_0 = Float64(Float64(x * y) / Float64(Float64(z + 1.0) * Float64(z * z)))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+283))
		tmp = Float64(Float64(Float64(x / z) / z) * Float64(y / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(Float64(x * y) / Float64(z + 1.0)) / Float64(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)
	t_0 = (x * y) / ((z + 1.0) * (z * z));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+283)))
		tmp = ((x / z) / z) * (y / (z + 1.0));
	else
		tmp = ((x * y) / (z + 1.0)) / (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_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+283]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision] * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	83.1%
Target	95.9%
Herbie	94.5%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 0.0 or 1.99999999999999991e283 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 82.9%

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

   2. Simplified 91.9%

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

      [Start] 82.9%	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 91.9%	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Taylor expanded in x around 0 91.9%

      \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot \frac{y}{z + 1} \]

   4. Simplified 97.3%

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

      [Start] 91.9%	\[ \frac{x}{{z}^{2}} \cdot \frac{y}{z + 1} \]
      unpow2 [=>] 91.9%	\[ \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]
      associate-/l/ [<=] 97.3%	\[ \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

   if 0.0 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 1.99999999999999991e283

   1. Initial program 98.9%

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

   2. Simplified 88.7%

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

      [Start] 98.9%	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      associate-*l* [=>] 98.9%	\[ \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]
      times-frac [=>] 88.7%	\[ \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]
      distribute-lft-in [=>] 88.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]
      fma-def [=>] 88.7%	\[ \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]
      *-rgt-identity [=>] 88.7%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Applied egg-rr 99.0%

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

      [Start] 88.7%	\[ \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]
      frac-times [=>] 98.9%	\[ \color{blue}{\frac{x \cdot y}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
      fma-udef [=>] 98.9%	\[ \frac{x \cdot y}{z \cdot \color{blue}{\left(z \cdot z + z\right)}} \]
      distribute-lft-in [=>] 98.9%	\[ \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot z\right) + z \cdot z}} \]
      distribute-lft1-in [=>] 98.9%	\[ \frac{x \cdot y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot z\right)}} \]
      associate-/r* [=>] 99.0%	\[ \color{blue}{\frac{\frac{x \cdot y}{z + 1}}{z \cdot z}} \]
      *-commutative [=>] 99.0%	\[ \frac{\frac{\color{blue}{y \cdot x}}{z + 1}}{z \cdot z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 0 \lor \neg \left(\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z + 1}}{z \cdot z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.5%
Cost	2249

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z + 1}}{z \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	95.7%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-15}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array} \]

Alternative 3
Accuracy	94.5%
Cost	2249

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	95.2%
Cost	1232

\[\begin{array}{l} t_0 := \frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	96.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-54} \lor \neg \left(z \leq 1.42 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Accuracy	93.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -20 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 7
Accuracy	95.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -20 \lor \neg \left(z \leq 0.8\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 8
Accuracy	95.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -20 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 9
Accuracy	95.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 10
Accuracy	95.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 11
Accuracy	95.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -20:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z}}{\frac{z}{x}}\\ \end{array} \]

Alternative 12
Accuracy	78.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 13
Accuracy	79.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 14
Accuracy	79.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 15
Accuracy	40.2%
Cost	516

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16
Accuracy	40.3%
Cost	516

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 17
Accuracy	72.6%
Cost	448

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

Alternative 18
Accuracy	31.1%
Cost	320

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

Alternative 19
Accuracy	31.0%
Cost	320

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

Reproduce

herbie shell --seed 2023272 
(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))))