Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 90.8% → 97.7%

Time: 14.8s

Precision: binary64

Cost: 964

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+212}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+212)
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (/ (/ (/ 1.0 y) z) (* x z))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+212) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / y) / z) / (x * 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 = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
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 * z) <= 1d+212) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = ((1.0d0 / y) / z) / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+212) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / y) / z) / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

↓

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+212:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = ((1.0 / y) / z) / (x * z)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+212)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+212)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = ((1.0 / y) / z) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+212], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.8%
Target	92.6%
Herbie	97.7%

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999991e211

   1. Initial program 99.0%

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

   2. Simplified 99.0%

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

      [Start] 99.0%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 99.0%	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   if 9.9999999999999991e211 < (*.f64 z z)

   1. Initial program 75.1%

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

   2. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
      Step-by-step derivation

      [Start] 75.1%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 75.0%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 75.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 75.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in z around inf 76.9%

      \[\leadsto \frac{1}{\color{blue}{y \cdot \left({z}^{2} \cdot x\right)}} \]

   4. Simplified 85.0%

      \[\leadsto \frac{1}{\color{blue}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}} \]
      Step-by-step derivation

      [Start] 76.9%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 76.9%	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]
      *-commutative [=>] 76.9%	\[ \frac{1}{y \cdot \color{blue}{\left(x \cdot \left(z \cdot z\right)\right)}} \]
      associate-*l* [<=] 73.5%	\[ \frac{1}{\color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot z\right)}} \]
      *-commutative [=>] 73.5%	\[ \frac{1}{\color{blue}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}} \]
      associate-*l* [=>] 85.0%	\[ \frac{1}{\color{blue}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}} \]

   5. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)} \cdot 1} \]
      Step-by-step derivation

      [Start] 85.0%	\[ \frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} \]
      clear-num [=>] 85.0%	\[ \color{blue}{\frac{1}{\frac{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}{1}}} \]
      associate-/r/ [=>] 85.0%	\[ \color{blue}{\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} \cdot 1} \]
      associate-/r* [=>] 85.7%	\[ \color{blue}{\frac{\frac{1}{z}}{z \cdot \left(y \cdot x\right)}} \cdot 1 \]
      associate-*r* [=>] 94.9%	\[ \frac{\frac{1}{z}}{\color{blue}{\left(z \cdot y\right) \cdot x}} \cdot 1 \]
      *-commutative [=>] 94.9%	\[ \frac{\frac{1}{z}}{\color{blue}{x \cdot \left(z \cdot y\right)}} \cdot 1 \]

   6. Taylor expanded in z around 0 76.9%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)}} \cdot 1 \]

   7. Simplified 97.9%

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

      [Start] 76.9%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \cdot 1 \]
      associate-/r* [=>] 76.9%	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \cdot 1 \]
      unpow2 [=>] 76.9%	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \cdot 1 \]
      associate-*r* [<=] 92.8%	\[ \frac{\frac{1}{y}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \cdot 1 \]
      *-commutative [<=] 92.8%	\[ \frac{\frac{1}{y}}{z \cdot \color{blue}{\left(x \cdot z\right)}} \cdot 1 \]
      associate-/r* [=>] 97.9%	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}} \cdot 1 \]
      *-commutative [=>] 97.9%	\[ \frac{\frac{\frac{1}{y}}{z}}{\color{blue}{z \cdot x}} \cdot 1 \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+212}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+212}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	98.0%
Cost	13632

\[\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \]

Alternative 3
Accuracy	97.8%
Cost	13504

\[\frac{\frac{1}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot y} \]

Alternative 4
Accuracy	97.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

Alternative 5
Accuracy	93.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	96.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 8
Accuracy	97.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 9
Accuracy	97.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

Alternative 10
Accuracy	58.6%
Cost	320

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

Alternative 11
Accuracy	58.7%
Cost	320

\[\frac{\frac{1}{x}}{y} \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))