Average Error: 6.4 → 6.3
Time: 11.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{1}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{1}}} \cdot \frac{\frac{\sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z} \cdot x}\]
\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 \frac{\frac{\sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z} \cdot x}
double f(double x, double y, double z) {
        double r392718 = 1.0;
        double r392719 = x;
        double r392720 = r392718 / r392719;
        double r392721 = y;
        double r392722 = z;
        double r392723 = r392722 * r392722;
        double r392724 = r392718 + r392723;
        double r392725 = r392721 * r392724;
        double r392726 = r392720 / r392725;
        return r392726;
}
↓
double f(double x, double y, double z) {
        double r392727 = 1.0;
        double r392728 = cbrt(r392727);
        double r392729 = z;
        double r392730 = r392729 * r392729;
        double r392731 = r392727 + r392730;
        double r392732 = sqrt(r392731);
        double r392733 = r392732 / r392728;
        double r392734 = r392728 / r392733;
        double r392735 = y;
        double r392736 = r392728 / r392735;
        double r392737 = x;
        double r392738 = r392732 * r392737;
        double r392739 = r392736 / r392738;
        double r392740 = r392734 * r392739;
        return r392740;
}

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

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*6.5

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

  4. Simplified6.5

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

  6. Applied add-sqr-sqrt6.5

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

  7. Applied *-un-lft-identity6.5

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

  8. Applied *-un-lft-identity6.5

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

  9. Applied add-cube-cbrt6.5

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

  10. Applied times-frac6.5

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

  11. Applied times-frac6.5

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

  12. Applied times-frac6.5

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

  13. Simplified6.5

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

  15. Applied associate-/l/6.3

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

  16. Final simplification6.3

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

Reproduce

herbie shell --seed 2020083 
(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)))))