Average Error: 6.6 → 6.0
Time: 7.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt[3]{y}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt[3]{y}}
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) {
	return ((((cbrt((1.0 / x)) * cbrt((1.0 / x))) / sqrt((1.0 + (z * z)))) / (cbrt(y) * cbrt(y))) * ((cbrt((1.0 / x)) / sqrt((1.0 + (z * z)))) / cbrt(y)));
}

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

Original6.6
Target5.9
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied *-un-lft-identity6.6

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

  4. Applied *-un-lft-identity6.6

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

  5. Applied times-frac6.6

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

  6. Applied times-frac6.5

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

  7. Simplified6.5

    \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\]
  8. Using strategy rm

  9. Applied associate-*l/6.4

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

  10. Simplified6.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{1 + z \cdot z}}}{y}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt7.0

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

  13. Applied add-sqr-sqrt7.0

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

  14. Applied add-cube-cbrt7.2

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

  15. Applied times-frac7.2

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

  16. Applied times-frac6.0

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

  17. Final simplification6.0

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

Reproduce

herbie shell --seed 2020091 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))