Average Error: 6.3 → 5.9
Time: 13.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{1} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\sqrt[3]{1}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{1} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\sqrt[3]{1}}}
double f(double x, double y, double z) {
        double r306718 = 1.0;
        double r306719 = x;
        double r306720 = r306718 / r306719;
        double r306721 = y;
        double r306722 = z;
        double r306723 = r306722 * r306722;
        double r306724 = r306718 + r306723;
        double r306725 = r306721 * r306724;
        double r306726 = r306720 / r306725;
        return r306726;
}
↓
double f(double x, double y, double z) {
        double r306727 = 1.0;
        double r306728 = cbrt(r306727);
        double r306729 = x;
        double r306730 = r306728 / r306729;
        double r306731 = z;
        double r306732 = fma(r306731, r306731, r306727);
        double r306733 = sqrt(r306732);
        double r306734 = r306730 / r306733;
        double r306735 = y;
        double r306736 = r306734 / r306735;
        double r306737 = r306728 * r306736;
        double r306738 = r306733 / r306728;
        double r306739 = r306737 / r306738;
        return r306739;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.3
Target5.7
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Simplified6.3

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

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

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

  5. Applied add-sqr-sqrt6.3

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

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

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

  7. Applied add-cube-cbrt6.3

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

  8. Applied times-frac6.3

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

  9. Applied times-frac6.3

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

  10. Applied times-frac5.9

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

  11. Simplified5.9

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

  13. Applied associate-*l/5.9

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

  14. Final simplification5.9

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

Reproduce

herbie shell --seed 2020001 +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))) #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)))))