Average Error: 6.4 → 6.2
Time: 9.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{1}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{1}}} \cdot \left(\frac{\frac{\sqrt[3]{1}}{y}}{\sqrt{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{1}{x}}{\sqrt{\sqrt{1 + z \cdot z}}}\right)\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{1}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{1}}} \cdot \left(\frac{\frac{\sqrt[3]{1}}{y}}{\sqrt{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{1}{x}}{\sqrt{\sqrt{1 + z \cdot z}}}\right)
double f(double x, double y, double z) {
        double r342859 = 1.0;
        double r342860 = x;
        double r342861 = r342859 / r342860;
        double r342862 = y;
        double r342863 = z;
        double r342864 = r342863 * r342863;
        double r342865 = r342859 + r342864;
        double r342866 = r342862 * r342865;
        double r342867 = r342861 / r342866;
        return r342867;
}
↓
double f(double x, double y, double z) {
        double r342868 = 1.0;
        double r342869 = cbrt(r342868);
        double r342870 = z;
        double r342871 = r342870 * r342870;
        double r342872 = r342868 + r342871;
        double r342873 = sqrt(r342872);
        double r342874 = r342873 / r342869;
        double r342875 = r342869 / r342874;
        double r342876 = y;
        double r342877 = r342869 / r342876;
        double r342878 = sqrt(r342873);
        double r342879 = r342877 / r342878;
        double r342880 = 1.0;
        double r342881 = x;
        double r342882 = r342880 / r342881;
        double r342883 = r342882 / r342878;
        double r342884 = r342879 * r342883;
        double r342885 = r342875 * r342884;
        return r342885;
}

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

Original6.4
Target5.8
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 6.4

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

  3. Applied associate-/r*6.6

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

  4. Simplified6.6

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

  6. Applied add-sqr-sqrt6.6

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

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

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

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

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

  9. Applied add-cube-cbrt6.6

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

  10. Applied times-frac6.6

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

  11. Applied times-frac6.6

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

  12. Applied times-frac6.6

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

  13. Simplified6.6

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{1}}}} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{y}}{x}}{\sqrt{1 + z \cdot z}}\]
  14. Using strategy rm

  15. Applied add-sqr-sqrt6.6

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

  16. Applied sqrt-prod6.6

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

  17. Applied div-inv6.7

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

  18. Applied times-frac6.2

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

  19. Final simplification6.2

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

Reproduce

herbie shell --seed 2020062 
(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)))))