Average Error: 6.4 → 6.7
Time: 13.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r277978 = 1.0;
        double r277979 = x;
        double r277980 = r277978 / r277979;
        double r277981 = y;
        double r277982 = z;
        double r277983 = r277982 * r277982;
        double r277984 = r277978 + r277983;
        double r277985 = r277981 * r277984;
        double r277986 = r277980 / r277985;
        return r277986;
}
↓
double f(double x, double y, double z) {
        double r277987 = 1.0;
        double r277988 = cbrt(r277987);
        double r277989 = r277988 * r277988;
        double r277990 = y;
        double r277991 = r277989 / r277990;
        double r277992 = x;
        double r277993 = r277988 / r277992;
        double r277994 = z;
        double r277995 = fma(r277994, r277994, r277987);
        double r277996 = r277993 / r277995;
        double r277997 = r277991 * r277996;
        return r277997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.8
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

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

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

  4. Applied add-cube-cbrt6.4

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

  5. Applied times-frac6.4

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

  6. Applied times-frac6.7

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

  7. Simplified6.7

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

  8. Simplified6.7

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

  9. Final simplification6.7

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

Reproduce

herbie shell --seed 2019208 +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)))))