Average Error: 6.5 → 6.0
Time: 33.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r18001855 = 1.0;
        double r18001856 = x;
        double r18001857 = r18001855 / r18001856;
        double r18001858 = y;
        double r18001859 = z;
        double r18001860 = r18001859 * r18001859;
        double r18001861 = r18001855 + r18001860;
        double r18001862 = r18001858 * r18001861;
        double r18001863 = r18001857 / r18001862;
        return r18001863;
}
↓
double f(double x, double y, double z) {
        double r18001864 = 1.0;
        double r18001865 = x;
        double r18001866 = r18001864 / r18001865;
        double r18001867 = z;
        double r18001868 = fma(r18001867, r18001867, r18001864);
        double r18001869 = sqrt(r18001868);
        double r18001870 = r18001866 / r18001869;
        double r18001871 = 1.0;
        double r18001872 = y;
        double r18001873 = r18001871 / r18001872;
        double r18001874 = r18001873 / r18001869;
        double r18001875 = r18001870 * r18001874;
        return r18001875;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.7
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.7

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

  4. Applied add-sqr-sqrt6.7

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

  5. Applied div-inv6.7

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

  6. Applied times-frac6.0

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

  7. Final simplification6.0

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

Reproduce

herbie shell --seed 2019174 +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)))))