Average Error: 6.4 → 5.9
Time: 4.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x}
double f(double x, double y, double z) {
        double r274915 = 1.0;
        double r274916 = x;
        double r274917 = r274915 / r274916;
        double r274918 = y;
        double r274919 = z;
        double r274920 = r274919 * r274919;
        double r274921 = r274915 + r274920;
        double r274922 = r274918 * r274921;
        double r274923 = r274917 / r274922;
        return r274923;
}
↓
double f(double x, double y, double z) {
        double r274924 = 1.0;
        double r274925 = z;
        double r274926 = fma(r274925, r274925, r274924);
        double r274927 = sqrt(r274926);
        double r274928 = r274924 / r274927;
        double r274929 = y;
        double r274930 = r274928 / r274929;
        double r274931 = x;
        double r274932 = r274927 * r274931;
        double r274933 = r274930 / r274932;
        return r274933;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.3

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

  4. Applied add-sqr-sqrt6.3

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

  5. Applied div-inv6.3

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

  6. Applied times-frac6.3

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

  7. Applied associate-/l*6.1

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

  8. Simplified6.1

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

  10. Applied associate-/r*5.9

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

  11. Final simplification5.9

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

Reproduce

herbie shell --seed 2020056 +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)))))