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

Percentage Accurate: 90.9% → 97.3%

Time: 11.3s

Precision: binary64

Cost: 7236

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+173)
   (/ 1.0 (* x (* y (fma z z 1.0))))
   (* (/ 1.0 z) (/ (/ 1.0 y) (* 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) <= 2e+173) {
		tmp = 1.0 / (x * (y * fma(z, z, 1.0)));
	} else {
		tmp = (1.0 / z) * ((1.0 / y) / (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) <= 2e+173)
		tmp = Float64(1.0 / Float64(x * Float64(y * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y) / Float64(x * z)));
	end
	return 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], 2e+173], N[(1.0 / N[(x * N[(y * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.9%
Target	91.9%
Herbie	97.3%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e173

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 99.6%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 99.6%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 99.6%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   if 2e173 < (*.f64 z z)

   1. Initial program 70.0%

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

   2. Simplified 69.9%

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

      [Start] 70.0%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 69.9%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 69.9%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 69.9%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Applied egg-rr 85.2%

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

      [Start] 69.9%	\[ \frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)} \]
      fma-udef [=>] 69.9%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      +-commutative [<=] 69.9%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)} \]
      associate-/r* [=>] 70.0%	\[ \color{blue}{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]
      associate-/r* [=>] 68.3%	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]
      add-sqr-sqrt [=>] 68.3%	\[ \frac{\frac{\frac{1}{x}}{y}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}} \]
      *-un-lft-identity [=>] 68.3%	\[ \frac{\color{blue}{1 \cdot \frac{\frac{1}{x}}{y}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}} \]
      times-frac [=>] 68.3%	\[ \color{blue}{\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}}} \]
      hypot-1-def [=>] 68.3%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}} \]
      associate-/l/ [=>] 68.3%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{\frac{1}{y \cdot x}}}{\sqrt{1 + z \cdot z}} \]
      hypot-1-def [=>] 85.2%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \]

   4. Simplified 98.5%

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

      [Start] 85.2%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*l/ [=>] 85.3%	\[ \color{blue}{\frac{1 \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}} \]
      *-lft-identity [=>] 85.3%	\[ \frac{\color{blue}{\frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/r* [=>] 85.3%	\[ \frac{\frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/l/ [=>] 98.5%	\[ \frac{\color{blue}{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot x}}}{\mathsf{hypot}\left(1, z\right)} \]
      *-commutative [=>] 98.5%	\[ \frac{\frac{\frac{1}{y}}{\color{blue}{x \cdot \mathsf{hypot}\left(1, z\right)}}}{\mathsf{hypot}\left(1, z\right)} \]

   5. Taylor expanded in z around inf 68.8%

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

   6. Simplified 86.0%

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

      [Start] 68.8%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 68.7%	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      unpow-1 [<=] 68.7%	\[ \frac{\color{blue}{{y}^{-1}}}{{z}^{2} \cdot x} \]
      unpow2 [=>] 68.7%	\[ \frac{{y}^{-1}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-*l* [=>] 86.0%	\[ \frac{{y}^{-1}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]

   7. Applied egg-rr 98.5%

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

      [Start] 86.0%	\[ \frac{{y}^{-1}}{z \cdot \left(z \cdot x\right)} \]
      inv-pow [<=] 86.0%	\[ \frac{\color{blue}{\frac{1}{y}}}{z \cdot \left(z \cdot x\right)} \]
      *-un-lft-identity [=>] 86.0%	\[ \frac{\color{blue}{1 \cdot \frac{1}{y}}}{z \cdot \left(z \cdot x\right)} \]
      times-frac [=>] 98.5%	\[ \color{blue}{\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}} \]
      *-commutative [=>] 98.5%	\[ \frac{1}{z} \cdot \frac{\frac{1}{y}}{\color{blue}{x \cdot z}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.3%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	97.5%
Cost	13504

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

Alternative 3
Accuracy	96.6%
Cost	964

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

Alternative 4
Accuracy	97.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+20}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]

Alternative 5
Accuracy	93.8%
Cost	836

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

Alternative 6
Accuracy	96.5%
Cost	836

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

Alternative 7
Accuracy	58.1%
Cost	320

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

Reproduce

herbie shell --seed 2023271 
(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)))))