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

Percentage Accurate: 90.5% → 97.6%

Time: 10.1s

Precision: binary64

Cost: 7236

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

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+145)
   (/ (/ (/ 1.0 (fma z z 1.0)) x) y)
   (* (/ (/ 1.0 y) z) (/ (/ 1.0 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+145) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x) / y;
	} else {
		tmp = ((1.0 / y) / z) * ((1.0 / 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+145)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) * Float64(Float64(1.0 / 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+145], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.5%
Target	91.5%
Herbie	97.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e145

   1. Initial program 98.5%

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

   2. Simplified 98.0%

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

      [Start] 98.5%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 98.0%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 98.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 98.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Applied egg-rr 98.9%

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

      [Start] 98.0%	\[ \frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)} \]
      fma-udef [=>] 98.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      +-commutative [<=] 98.0%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)} \]
      associate-/r* [=>] 98.5%	\[ \color{blue}{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]
      associate-/r* [=>] 97.3%	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]
      div-inv [=>] 97.2%	\[ \frac{\color{blue}{\frac{1}{x} \cdot \frac{1}{y}}}{1 + z \cdot z} \]
      add-sqr-sqrt [=>] 97.1%	\[ \frac{\frac{1}{x} \cdot \frac{1}{y}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}} \]
      times-frac [=>] 98.9%	\[ \color{blue}{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}} \]
      hypot-1-def [=>] 98.9%	\[ \frac{\frac{1}{x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}} \]
      hypot-1-def [=>] 98.9%	\[ \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \]

   4. Taylor expanded in x around 0 99.2%

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

   5. Simplified 99.7%

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

      [Start] 99.2%	\[ \frac{1}{y \cdot \left(\left({z}^{2} + 1\right) \cdot x\right)} \]
      associate-*r* [=>] 98.0%	\[ \frac{1}{\color{blue}{\left(y \cdot \left({z}^{2} + 1\right)\right) \cdot x}} \]
      *-commutative [<=] 98.0%	\[ \frac{1}{\color{blue}{\left(\left({z}^{2} + 1\right) \cdot y\right)} \cdot x} \]
      associate-*r* [<=] 97.4%	\[ \frac{1}{\color{blue}{\left({z}^{2} + 1\right) \cdot \left(y \cdot x\right)}} \]
      associate-/r* [=>] 97.6%	\[ \color{blue}{\frac{\frac{1}{{z}^{2} + 1}}{y \cdot x}} \]
      unpow2 [=>] 97.6%	\[ \frac{\frac{1}{\color{blue}{z \cdot z} + 1}}{y \cdot x} \]
      fma-udef [<=] 97.6%	\[ \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot x} \]
      *-commutative [<=] 97.6%	\[ \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{x \cdot y}} \]
      associate-/r* [=>] 99.7%	\[ \color{blue}{\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}} \]

   if 2e145 < (*.f64 z z)

   1. Initial program 76.8%

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

   2. Simplified 76.7%

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

      [Start] 76.8%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 76.7%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 76.7%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 76.7%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in z around inf 76.7%

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

   4. Simplified 87.4%

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

      [Start] 76.7%	\[ \frac{1}{x \cdot \left(y \cdot {z}^{2}\right)} \]
      unpow2 [=>] 76.7%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)} \]
      associate-*r* [=>] 87.4%	\[ \frac{1}{x \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}} \]
      *-commutative [=>] 87.4%	\[ \frac{1}{x \cdot \left(\color{blue}{\left(z \cdot y\right)} \cdot z\right)} \]

   5. Applied egg-rr 95.7%

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

      [Start] 87.4%	\[ \frac{1}{x \cdot \left(\left(z \cdot y\right) \cdot z\right)} \]
      associate-/r* [=>] 87.5%	\[ \color{blue}{\frac{\frac{1}{x}}{\left(z \cdot y\right) \cdot z}} \]
      *-un-lft-identity [=>] 87.5%	\[ \frac{\color{blue}{1 \cdot \frac{1}{x}}}{\left(z \cdot y\right) \cdot z} \]
      *-commutative [=>] 87.5%	\[ \frac{1 \cdot \frac{1}{x}}{\color{blue}{\left(y \cdot z\right)} \cdot z} \]
      times-frac [=>] 95.4%	\[ \color{blue}{\frac{1}{y \cdot z} \cdot \frac{\frac{1}{x}}{z}} \]
      associate-/r* [=>] 95.7%	\[ \color{blue}{\frac{\frac{1}{y}}{z}} \cdot \frac{\frac{1}{x}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 2
Accuracy	97.7%
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	97.5%
Cost	13504

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

Alternative 4
Accuracy	96.9%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 5
Accuracy	97.5%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+57}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 6
Accuracy	97.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+57}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 7
Accuracy	97.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+57}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 8
Accuracy	89.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	93.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	93.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 14
Accuracy	96.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{x \cdot z}\\ \end{array} \]

Alternative 15
Accuracy	69.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 2.9 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{-1}{y \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \]

Alternative 16
Accuracy	70.3%
Cost	712

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{-1}{t_0}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 17
Accuracy	70.1%
Cost	712

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

Alternative 18
Accuracy	59.0%
Cost	320

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

Alternative 19
Accuracy	59.0%
Cost	320

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

Reproduce

herbie shell --seed 2023229 
(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)))))