Average Error: 6.4 → 6.1
Time: 27.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} \cdot \frac{\frac{\frac{\sqrt{1}}{x}}{\sqrt[3]{y}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} \cdot \frac{\frac{\frac{\sqrt{1}}{x}}{\sqrt[3]{y}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r337324 = 1.0;
        double r337325 = x;
        double r337326 = r337324 / r337325;
        double r337327 = y;
        double r337328 = z;
        double r337329 = r337328 * r337328;
        double r337330 = r337324 + r337329;
        double r337331 = r337327 * r337330;
        double r337332 = r337326 / r337331;
        return r337332;
}
↓
double f(double x, double y, double z) {
        double r337333 = 1.0;
        double r337334 = sqrt(r337333);
        double r337335 = z;
        double r337336 = fma(r337335, r337335, r337333);
        double r337337 = sqrt(r337336);
        double r337338 = y;
        double r337339 = cbrt(r337338);
        double r337340 = r337339 * r337339;
        double r337341 = r337337 * r337340;
        double r337342 = r337334 / r337341;
        double r337343 = x;
        double r337344 = r337334 / r337343;
        double r337345 = r337344 / r337339;
        double r337346 = r337345 / r337337;
        double r337347 = r337342 * r337346;
        return r337347;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 6.4

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

  2. Simplified6.6

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

  4. Applied add-sqr-sqrt6.6

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

  5. Applied add-cube-cbrt7.2

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

  6. Applied *-un-lft-identity7.2

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

  7. Applied add-sqr-sqrt7.2

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

  8. Applied times-frac7.2

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

  9. Applied times-frac7.2

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

  10. Applied times-frac6.1

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

  11. Simplified6.1

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

  12. Final simplification6.1

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

Reproduce

herbie shell --seed 2019326 +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)))))