Average Error: 6.4 → 6.2
Time: 8.3s
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 r294702 = 1.0;
        double r294703 = x;
        double r294704 = r294702 / r294703;
        double r294705 = y;
        double r294706 = z;
        double r294707 = r294706 * r294706;
        double r294708 = r294702 + r294707;
        double r294709 = r294705 * r294708;
        double r294710 = r294704 / r294709;
        return r294710;
}
↓
double f(double x, double y, double z) {
        double r294711 = 1.0;
        double r294712 = cbrt(r294711);
        double r294713 = z;
        double r294714 = r294713 * r294713;
        double r294715 = r294711 + r294714;
        double r294716 = sqrt(r294715);
        double r294717 = r294716 / r294712;
        double r294718 = r294712 / r294717;
        double r294719 = y;
        double r294720 = r294712 / r294719;
        double r294721 = sqrt(r294716);
        double r294722 = r294720 / r294721;
        double r294723 = 1.0;
        double r294724 = x;
        double r294725 = r294723 / r294724;
        double r294726 = r294725 / r294721;
        double r294727 = r294722 * r294726;
        double r294728 = r294718 * r294727;
        return r294728;
}

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

Original6.4
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)))))