Average Error: 6.4 → 5.9
Time: 5.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\frac{\mathsf{fma}\left(z, z, 1\right)}{1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\frac{\mathsf{fma}\left(z, z, 1\right)}{1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)}
double f(double x, double y, double z) {
        double r277706 = 1.0;
        double r277707 = x;
        double r277708 = r277706 / r277707;
        double r277709 = y;
        double r277710 = z;
        double r277711 = r277710 * r277710;
        double r277712 = r277706 + r277711;
        double r277713 = r277709 * r277712;
        double r277714 = r277708 / r277713;
        return r277714;
}
↓
double f(double x, double y, double z) {
        double r277715 = 1.0;
        double r277716 = y;
        double r277717 = cbrt(r277716);
        double r277718 = r277717 * r277717;
        double r277719 = r277715 / r277718;
        double r277720 = x;
        double r277721 = cbrt(r277720);
        double r277722 = r277721 * r277721;
        double r277723 = r277719 / r277722;
        double r277724 = z;
        double r277725 = 1.0;
        double r277726 = fma(r277724, r277724, r277725);
        double r277727 = r277726 / r277725;
        double r277728 = r277721 * r277717;
        double r277729 = r277727 * r277728;
        double r277730 = r277723 / r277729;
        return r277730;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.9
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied associate-/r*6.4

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

  4. Simplified6.4

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

  6. Applied add-cube-cbrt7.0

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

  7. Applied add-cube-cbrt7.1

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

  8. Applied *-un-lft-identity7.1

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

  9. Applied times-frac7.1

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

  10. Applied times-frac7.1

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

  11. Applied associate-/l*5.9

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

  12. Simplified5.9

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

  13. Final simplification5.9

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

Reproduce

herbie shell --seed 2020033 +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)))))