Average Error: 6.3 → 5.9
Time: 4.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{y}}{\left(\sqrt[3]{1 + z \cdot z} \cdot \sqrt[3]{1 + z \cdot z}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{y}}{\left(\sqrt[3]{1 + z \cdot z} \cdot \sqrt[3]{1 + z \cdot z}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{1 + z \cdot z}}
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 (((sqrt(1.0) / y) / ((cbrt((1.0 + (z * z))) * cbrt((1.0 + (z * z)))) * (cbrt(x) * cbrt(x)))) * ((sqrt(1.0) / cbrt(x)) / cbrt((1.0 + (z * z)))));
}

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

Original6.3
Target5.6
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

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

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

  4. Applied add-sqr-sqrt6.3

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

  5. Applied times-frac6.3

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

  6. Applied times-frac6.3

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

  7. Simplified6.3

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

  9. Applied add-cube-cbrt6.5

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

  10. Applied add-cube-cbrt6.9

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

  11. Applied *-un-lft-identity6.9

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

  12. Applied sqrt-prod6.9

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

  13. Applied times-frac6.9

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

  14. Applied times-frac6.9

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

  15. Applied associate-*r*5.8

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

  16. Simplified5.9

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

  17. Final simplification5.9

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

Reproduce

herbie shell --seed 2020058 
(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)))))