Average Error: 6.6 → 6.5
Time: 8.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r170920 = 1.0;
        double r170921 = x;
        double r170922 = r170920 / r170921;
        double r170923 = y;
        double r170924 = z;
        double r170925 = r170924 * r170924;
        double r170926 = r170920 + r170925;
        double r170927 = r170923 * r170926;
        double r170928 = r170922 / r170927;
        return r170928;
}
↓
double f(double x, double y, double z) {
        double r170929 = 1.0;
        double r170930 = y;
        double r170931 = r170929 / r170930;
        double r170932 = 1.0;
        double r170933 = x;
        double r170934 = r170932 / r170933;
        double r170935 = z;
        double r170936 = fma(r170935, r170935, r170929);
        double r170937 = r170934 / r170936;
        double r170938 = r170931 * r170937;
        return r170938;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target5.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied div-inv6.6

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

  4. Applied times-frac6.5

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

  5. Simplified6.5

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

  6. Final simplification6.5

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

Reproduce

herbie shell --seed 2019212 +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)))))