Average Error: 6.7 → 6.2
Time: 27.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{1 + z \cdot z}} \cdot \left(\frac{\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\left|\sqrt[3]{1 + z \cdot z}\right|} \cdot \frac{\frac{\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{\sqrt[3]{x}}}}{y}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}\right)\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{1 + z \cdot z}} \cdot \left(\frac{\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\left|\sqrt[3]{1 + z \cdot z}\right|} \cdot \frac{\frac{\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{\sqrt[3]{x}}}}{y}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}\right)
double f(double x, double y, double z) {
        double r456652 = 1.0;
        double r456653 = x;
        double r456654 = r456652 / r456653;
        double r456655 = y;
        double r456656 = z;
        double r456657 = r456656 * r456656;
        double r456658 = r456652 + r456657;
        double r456659 = r456655 * r456658;
        double r456660 = r456654 / r456659;
        return r456660;
}
↓
double f(double x, double y, double z) {
        double r456661 = 1.0;
        double r456662 = sqrt(r456661);
        double r456663 = x;
        double r456664 = cbrt(r456663);
        double r456665 = r456664 * r456664;
        double r456666 = r456662 / r456665;
        double r456667 = z;
        double r456668 = r456667 * r456667;
        double r456669 = r456661 + r456668;
        double r456670 = sqrt(r456669);
        double r456671 = r456666 / r456670;
        double r456672 = sqrt(r456662);
        double r456673 = cbrt(r456665);
        double r456674 = r456672 / r456673;
        double r456675 = cbrt(r456669);
        double r456676 = fabs(r456675);
        double r456677 = r456674 / r456676;
        double r456678 = cbrt(r456664);
        double r456679 = r456672 / r456678;
        double r456680 = y;
        double r456681 = r456679 / r456680;
        double r456682 = sqrt(r456675);
        double r456683 = r456681 / r456682;
        double r456684 = r456677 * r456683;
        double r456685 = r456671 * r456684;
        return r456685;
}

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

Original6.7
Target6.1
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 6.7

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

  3. Applied associate-/r*6.9

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

  5. Applied add-sqr-sqrt6.9

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

  6. Applied *-un-lft-identity6.9

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

  7. Applied add-cube-cbrt7.5

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

  8. Applied add-sqr-sqrt7.5

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

  9. Applied times-frac7.5

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

  10. Applied times-frac7.5

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

  11. Applied times-frac6.3

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

  12. Simplified6.3

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

  14. Applied add-cube-cbrt6.3

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

  15. Applied sqrt-prod6.3

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

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

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

  17. Applied add-cube-cbrt6.4

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

  18. Applied cbrt-prod6.4

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

  19. Applied add-sqr-sqrt6.4

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

  20. Applied sqrt-prod6.4

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

  21. Applied times-frac6.4

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

  22. Applied times-frac6.4

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

  23. Applied times-frac6.2

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

  24. Simplified6.2

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

  25. Final simplification6.2

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

Reproduce

herbie shell --seed 2020046 
(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)))))