Average Error: 6.0 → 6.1
Time: 1.7m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}}}
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))) / 1.0)) / ((double) (y / ((double) (((double) (((double) cbrt(1.0)) / x)) / ((double) (1.0 + ((double) (z * z))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original6.0
Target5.3
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.0

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

  3. Applied *-commutative6.0

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

  4. Applied associate-/r*5.8

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

  6. Applied *-un-lft-identity5.8

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

  7. Applied add-cube-cbrt5.8

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

  8. Applied times-frac5.8

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

  9. Applied associate-/l*5.9

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

  10. Applied associate-/l/6.2

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

  11. Simplified6.1

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

  12. Final simplification6.1

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

Reproduce

herbie shell --seed 2020114 
(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)))))