Average Error: 6.6 → 6.2
Time: 4.1m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\left(\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\left(\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r147516677 = 1.0;
        double r147516678 = x;
        double r147516679 = r147516677 / r147516678;
        double r147516680 = y;
        double r147516681 = z;
        double r147516682 = r147516681 * r147516681;
        double r147516683 = r147516677 + r147516682;
        double r147516684 = r147516680 * r147516683;
        double r147516685 = r147516679 / r147516684;
        return r147516685;
}
↓
double f(double x, double y, double z) {
        double r147516686 = 1.0;
        double r147516687 = x;
        double r147516688 = r147516686 / r147516687;
        double r147516689 = cbrt(r147516688);
        double r147516690 = y;
        double r147516691 = cbrt(r147516690);
        double r147516692 = r147516689 / r147516691;
        double r147516693 = r147516692 * r147516692;
        double r147516694 = z;
        double r147516695 = fma(r147516694, r147516694, r147516686);
        double r147516696 = r147516692 / r147516695;
        double r147516697 = r147516693 * r147516696;
        return r147516697;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target6.0
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  2. Simplified6.9

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

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

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

  5. Applied add-cube-cbrt7.5

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

  6. Applied add-cube-cbrt7.6

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

  7. Applied times-frac7.6

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

  8. Applied times-frac6.2

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

  9. Simplified6.2

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

  10. Final simplification6.2

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))