Average Error: 6.3 → 6.2
Time: 15.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{1}{y}}{\left|\sqrt[3]{1 + z \cdot z}\right|}}{\sqrt{\sqrt[3]{1 + z \cdot z}} \cdot \left(\sqrt{1 + z \cdot z} \cdot \sqrt[3]{x}\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{1}{y}}{\left|\sqrt[3]{1 + z \cdot z}\right|}}{\sqrt{\sqrt[3]{1 + z \cdot z}} \cdot \left(\sqrt{1 + z \cdot z} \cdot \sqrt[3]{x}\right)}
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) (1.0 / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) (((double) (1.0 / y)) / ((double) fabs(((double) cbrt(((double) (1.0 + ((double) (z * z)))))))))))) / ((double) (((double) sqrt(((double) cbrt(((double) (1.0 + ((double) (z * z)))))))) * ((double) (((double) sqrt(((double) (1.0 + ((double) (z * z)))))) * ((double) cbrt(x))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original6.3
Target5.7
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied add-sqr-sqrt6.3

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

  4. Applied associate-*r*6.3

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

  5. Applied div-inv6.3

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

  6. Applied times-frac6.2

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

  8. Applied add-cube-cbrt6.8

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

  9. Applied associate-/r*6.8

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

  10. Applied associate-/l/6.8

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

  11. Applied add-cube-cbrt6.8

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

  12. Applied sqrt-prod6.8

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

  13. Applied associate-*r*6.8

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

  14. Applied associate-/r*6.6

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

  15. Applied frac-times6.2

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

  16. Simplified6.2

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

  17. Final simplification6.2

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

Reproduce

herbie shell --seed 2020113 
(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)))))