Average Error: 6.1 → 6.0
Time: 10.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{1 + z \cdot z} \cdot x}}{y}\right) \cdot \frac{1}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{1 + z \cdot z} \cdot x}}{y}\right) \cdot \frac{1}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r484769 = 1.0;
        double r484770 = x;
        double r484771 = r484769 / r484770;
        double r484772 = y;
        double r484773 = z;
        double r484774 = r484773 * r484773;
        double r484775 = r484769 + r484774;
        double r484776 = r484772 * r484775;
        double r484777 = r484771 / r484776;
        return r484777;
}
↓
double f(double x, double y, double z) {
        double r484778 = 1.0;
        double r484779 = cbrt(r484778);
        double r484780 = r484779 * r484779;
        double r484781 = z;
        double r484782 = r484781 * r484781;
        double r484783 = r484778 + r484782;
        double r484784 = sqrt(r484783);
        double r484785 = x;
        double r484786 = r484784 * r484785;
        double r484787 = r484779 / r484786;
        double r484788 = y;
        double r484789 = r484787 / r484788;
        double r484790 = r484780 * r484789;
        double r484791 = 1.0;
        double r484792 = r484791 / r484784;
        double r484793 = r484790 * r484792;
        return r484793;
}

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

Original6.1
Target5.5
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied *-un-lft-identity6.1

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

  4. Applied add-cube-cbrt6.1

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

  5. Applied times-frac6.1

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

  6. Applied times-frac6.4

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

  7. Simplified6.4

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

  9. Applied add-sqr-sqrt6.4

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

  10. Applied associate-/r*6.4

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

  12. Applied div-inv6.4

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

  13. Applied associate-*r*6.0

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

  15. Applied div-inv6.0

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

  16. Applied associate-*l*6.0

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

  17. Simplified6.0

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

  18. Final simplification6.0

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

Reproduce

herbie shell --seed 2020047 
(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)))))