Average Error: 6.4 → 6.2
Time: 14.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot y}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
1 \cdot \frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot y}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r245889 = 1.0;
        double r245890 = x;
        double r245891 = r245889 / r245890;
        double r245892 = y;
        double r245893 = z;
        double r245894 = r245893 * r245893;
        double r245895 = r245889 + r245894;
        double r245896 = r245892 * r245895;
        double r245897 = r245891 / r245896;
        return r245897;
}
↓
double f(double x, double y, double z) {
        double r245898 = 1.0;
        double r245899 = 1.0;
        double r245900 = x;
        double r245901 = r245899 / r245900;
        double r245902 = z;
        double r245903 = r245902 * r245902;
        double r245904 = r245898 + r245903;
        double r245905 = sqrt(r245904);
        double r245906 = y;
        double r245907 = r245905 * r245906;
        double r245908 = r245901 / r245907;
        double r245909 = r245908 / r245905;
        double r245910 = r245898 * r245909;
        return r245910;
}

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
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied *-un-lft-identity6.4

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

  4. Applied add-sqr-sqrt6.4

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

  5. Applied times-frac6.4

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

  6. Applied times-frac6.4

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

  7. Simplified6.4

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

  9. Applied add-sqr-sqrt6.4

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

  10. Applied div-inv6.4

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

  11. Applied times-frac6.4

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

  12. Applied associate-*r*6.0

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

  13. Simplified6.0

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

  15. Applied *-un-lft-identity6.0

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

  16. Applied div-inv6.0

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

  17. Applied times-frac6.0

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

  18. Applied associate-*l*6.0

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

  19. Simplified6.2

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

  20. Final simplification6.2

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

Reproduce

herbie shell --seed 2019326 
(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)))))