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

Percentage Accurate: 90.9% → 97.9%

Time: 13.0s

Precision: binary64

Cost: 13764

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{x}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-121)
   (/ (/ 1.0 (+ y (* z (* y z)))) x)
   (* (/ 1.0 (hypot 1.0 z)) (/ (/ (/ 1.0 x) y) (hypot 1.0 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 (y <= 2e-121) {
		tmp = (1.0 / (y + (z * (y * z)))) / x;
	} else {
		tmp = (1.0 / hypot(1.0, z)) * (((1.0 / x) / y) / hypot(1.0, z));
	}
	return tmp;
}

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 (y <= 2e-121) {
		tmp = (1.0 / (y + (z * (y * z)))) / x;
	} else {
		tmp = (1.0 / Math.hypot(1.0, z)) * (((1.0 / x) / y) / Math.hypot(1.0, 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 y <= 2e-121:
		tmp = (1.0 / (y + (z * (y * z)))) / x
	else:
		tmp = (1.0 / math.hypot(1.0, z)) * (((1.0 / x) / y) / math.hypot(1.0, 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 (y <= 2e-121)
		tmp = Float64(Float64(1.0 / Float64(y + Float64(z * Float64(y * z)))) / x);
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) * Float64(Float64(Float64(1.0 / x) / y) / hypot(1.0, 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 (y <= 2e-121)
		tmp = (1.0 / (y + (z * (y * z)))) / x;
	else
		tmp = (1.0 / hypot(1.0, z)) * (((1.0 / x) / y) / hypot(1.0, 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[y, 2e-121], N[(N[(1.0 / N[(y + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.9%
Target	92.9%
Herbie	97.9%

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

Derivation

1. Split input into 2 regimes

2. if y < 2e-121

   1. Initial program 90.9%

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

   2. Simplified 90.0%

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

      [Start] 90.9%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 90.0%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 90.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 90.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in x around 0 88.4%

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

   4. Simplified 95.3%

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

      [Start] 88.4%	\[ \frac{1}{y \cdot \left(\left({z}^{2} + 1\right) \cdot x\right)} \]
      associate-*r* [=>] 90.0%	\[ \frac{1}{\color{blue}{\left(y \cdot \left({z}^{2} + 1\right)\right) \cdot x}} \]
      *-commutative [<=] 90.0%	\[ \frac{1}{\color{blue}{\left(\left({z}^{2} + 1\right) \cdot y\right)} \cdot x} \]
      associate-*r* [<=] 85.6%	\[ \frac{1}{\color{blue}{\left({z}^{2} + 1\right) \cdot \left(y \cdot x\right)}} \]
      associate-/r* [=>] 86.1%	\[ \color{blue}{\frac{\frac{1}{{z}^{2} + 1}}{y \cdot x}} \]
      unpow2 [=>] 86.1%	\[ \frac{\frac{1}{\color{blue}{z \cdot z} + 1}}{y \cdot x} \]
      fma-udef [<=] 86.1%	\[ \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot x} \]
      associate-/r* [=>] 91.0%	\[ \color{blue}{\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y}}{x}} \]
      associate-/l/ [=>] 90.9%	\[ \frac{\color{blue}{\frac{1}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}}{x} \]
      fma-udef [=>] 90.9%	\[ \frac{\frac{1}{y \cdot \color{blue}{\left(z \cdot z + 1\right)}}}{x} \]
      distribute-lft-in [=>] 90.9%	\[ \frac{\frac{1}{\color{blue}{y \cdot \left(z \cdot z\right) + y \cdot 1}}}{x} \]
      *-rgt-identity [=>] 90.9%	\[ \frac{\frac{1}{y \cdot \left(z \cdot z\right) + \color{blue}{y}}}{x} \]
      fma-def [=>] 90.9%	\[ \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(y, z \cdot z, y\right)}}}{x} \]
      fma-def [<=] 90.9%	\[ \frac{\frac{1}{\color{blue}{y \cdot \left(z \cdot z\right) + y}}}{x} \]
      *-commutative [=>] 90.9%	\[ \frac{\frac{1}{\color{blue}{\left(z \cdot z\right) \cdot y} + y}}{x} \]
      associate-*r* [<=] 95.3%	\[ \frac{\frac{1}{\color{blue}{z \cdot \left(z \cdot y\right)} + y}}{x} \]
      fma-udef [<=] 95.3%	\[ \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z \cdot y, y\right)}}}{x} \]

   5. Applied egg-rr 95.3%

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

      [Start] 95.3%	\[ \frac{\frac{1}{\mathsf{fma}\left(z, z \cdot y, y\right)}}{x} \]
      fma-udef [=>] 95.3%	\[ \frac{\frac{1}{\color{blue}{z \cdot \left(z \cdot y\right) + y}}}{x} \]

   if 2e-121 < y

   1. Initial program 95.0%

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

   2. Simplified 95.2%

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

      [Start] 95.0%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 95.2%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 95.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 95.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Applied egg-rr 99.6%

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

      [Start] 95.2%	\[ \frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)} \]
      fma-udef [=>] 95.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      +-commutative [<=] 95.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)} \]
      associate-/r* [=>] 95.0%	\[ \color{blue}{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]
      associate-/r* [=>] 97.4%	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]
      add-sqr-sqrt [=>] 97.4%	\[ \frac{\frac{\frac{1}{x}}{y}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}} \]
      *-un-lft-identity [=>] 97.4%	\[ \frac{\color{blue}{1 \cdot \frac{\frac{1}{x}}{y}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}} \]
      times-frac [=>] 97.4%	\[ \color{blue}{\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}}} \]
      hypot-1-def [=>] 97.4%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}} \]
      hypot-1-def [=>] 99.6%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{x}}{y}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{x}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.9%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{x}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2
Accuracy	94.8%
Cost	1220

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

Alternative 3
Accuracy	97.2%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(y \cdot z\right) \cdot \left(z \cdot \left(-x\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	96.2%
Cost	900

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(y \cdot z\right) \cdot \left(z \cdot \left(-x\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	89.9%
Cost	836

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

Alternative 7
Accuracy	93.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot \left(y \cdot z\right)\right) \cdot x}\\ \end{array} \]

Alternative 8
Accuracy	96.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(\left(y \cdot z\right) \cdot x\right)}\\ \end{array} \]

Alternative 9
Accuracy	96.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 11
Accuracy	59.0%
Cost	320

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

Alternative 12
Accuracy	59.0%
Cost	320

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

Reproduce

herbie shell --seed 2023272 
(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)))))