Average Error: 6.7 → 6.3
Time: 19.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r250573 = 1.0;
        double r250574 = x;
        double r250575 = r250573 / r250574;
        double r250576 = y;
        double r250577 = z;
        double r250578 = r250577 * r250577;
        double r250579 = r250573 + r250578;
        double r250580 = r250576 * r250579;
        double r250581 = r250575 / r250580;
        return r250581;
}
↓
double f(double x, double y, double z) {
        double r250582 = 1.0;
        double r250583 = x;
        double r250584 = cbrt(r250583);
        double r250585 = r250584 * r250584;
        double r250586 = r250582 / r250585;
        double r250587 = y;
        double r250588 = cbrt(r250587);
        double r250589 = r250588 * r250588;
        double r250590 = r250586 / r250589;
        double r250591 = 1.0;
        double r250592 = r250591 / r250584;
        double r250593 = r250592 / r250588;
        double r250594 = z;
        double r250595 = fma(r250594, r250594, r250591);
        double r250596 = r250593 / r250595;
        double r250597 = r250590 * r250596;
        return r250597;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.7
Target6.1
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 6.7

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

  2. Simplified6.8

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

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

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

  5. Applied add-cube-cbrt7.4

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

  6. Applied add-cube-cbrt7.6

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

  7. Applied *-un-lft-identity7.6

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

  8. Applied times-frac7.6

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

  9. Applied times-frac7.6

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

  10. Applied times-frac6.3

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

  11. Simplified6.3

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

  12. Final simplification6.3

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

Reproduce

herbie shell --seed 2019323 +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)))))