Average Error: 6.0 → 4.0
Time: 4.8s
Precision: binary64
[x y]: =sort([x y])
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 1.031746313983288 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \left(\frac{\frac{1}{\sqrt[3]{x}}}{z} \cdot \left(\frac{1}{z} - \frac{1}{{z}^{3}}\right)\right)\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 1.031746313983288 \cdot 10^{+306}:\\
\;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \left(\frac{\frac{1}{\sqrt[3]{x}}}{z} \cdot \left(\frac{1}{z} - \frac{1}{{z}^{3}}\right)\right)\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 1.031746313983288e+306)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (*
    (/ (/ 1.0 (* (cbrt x) (cbrt x))) y)
    (* (/ (/ 1.0 (cbrt x)) z) (- (/ 1.0 z) (/ 1.0 (pow z 3.0)))))))
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 * (1.0 + (z * z))) <= 1.031746313983288e+306) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / (cbrt(x) * cbrt(x))) / y) * (((1.0 / cbrt(x)) / z) * ((1.0 / z) - (1.0 / pow(z, 3.0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.0
Target4.6
Herbie4.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.03174631398328803e306

    1. Initial program 2.0

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

    if 1.03174631398328803e306 < (*.f64 y (+.f64 1 (*.f64 z z)))

    1. Initial program 17.8

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary64_1000217.8

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{y \cdot \left(1 + z \cdot z\right)}\]

    4. Applied add-sqr-sqrt_binary64_998917.8

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{y \cdot \left(1 + z \cdot z\right)}\]

    5. Applied times-frac_binary64_997317.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x}}}}{y \cdot \left(1 + z \cdot z\right)}\]

    6. Applied times-frac_binary64_997312.5

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{1 + z \cdot z}}\]

    7. Simplified12.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{1 + z \cdot z}\]

    8. Simplified12.5

      \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}}\]

    9. Taylor expanded around inf 45.0

      \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \color{blue}{\left({\left(\frac{1}{x}\right)}^{0.3333333333333333} \cdot \frac{1}{{z}^{2}} - {\left(\frac{1}{x}\right)}^{0.3333333333333333} \cdot \frac{1}{{z}^{4}}\right)}\]

    10. Simplified9.8

      \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{x}}}{z} \cdot \left(\frac{1}{z} - \frac{1}{{z}^{3}}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 1.031746313983288 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \left(\frac{\frac{1}{\sqrt[3]{x}}}{z} \cdot \left(\frac{1}{z} - \frac{1}{{z}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020358 
(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)))))