Average Error: 6.6 → 6.3
Time: 17.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{\sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{x}}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right)\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{\sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{x}}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right)
double f(double x, double y, double z) {
        double r297767 = 1.0;
        double r297768 = x;
        double r297769 = r297767 / r297768;
        double r297770 = y;
        double r297771 = z;
        double r297772 = r297771 * r297771;
        double r297773 = r297767 + r297772;
        double r297774 = r297770 * r297773;
        double r297775 = r297769 / r297774;
        return r297775;
}
↓
double f(double x, double y, double z) {
        double r297776 = 1.0;
        double r297777 = x;
        double r297778 = r297776 / r297777;
        double r297779 = cbrt(r297778);
        double r297780 = r297779 * r297779;
        double r297781 = z;
        double r297782 = fma(r297781, r297781, r297776);
        double r297783 = sqrt(r297782);
        double r297784 = r297780 / r297783;
        double r297785 = sqrt(r297776);
        double r297786 = cbrt(r297777);
        double r297787 = r297786 * r297786;
        double r297788 = r297785 / r297787;
        double r297789 = cbrt(r297788);
        double r297790 = r297785 / r297786;
        double r297791 = cbrt(r297790);
        double r297792 = y;
        double r297793 = r297791 / r297792;
        double r297794 = r297793 / r297783;
        double r297795 = r297789 * r297794;
        double r297796 = r297784 * r297795;
        return r297796;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target6.0
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  2. Simplified6.7

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{\mathsf{fma}\left(z, z, 1\right)}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt6.8

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

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

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

  6. Applied add-cube-cbrt7.3

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

  7. Applied times-frac7.3

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

  8. Applied times-frac6.3

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

  9. Simplified6.3

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

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

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

  12. Applied sqrt-prod6.3

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

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

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

  14. Applied add-cube-cbrt6.3

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

  15. Applied add-sqr-sqrt6.3

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

  16. Applied times-frac6.3

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

  17. Applied cbrt-prod6.3

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

  18. Applied times-frac6.3

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

  19. Applied times-frac6.3

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

  20. Simplified6.3

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

  21. Final simplification6.3

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))