Average Error: 6.4 → 6.1
Time: 29.0s
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 r197375 = 1.0;
        double r197376 = x;
        double r197377 = r197375 / r197376;
        double r197378 = y;
        double r197379 = z;
        double r197380 = r197379 * r197379;
        double r197381 = r197375 + r197380;
        double r197382 = r197378 * r197381;
        double r197383 = r197377 / r197382;
        return r197383;
}
↓
double f(double x, double y, double z) {
        double r197384 = 1.0;
        double r197385 = sqrt(r197384);
        double r197386 = z;
        double r197387 = fma(r197386, r197386, r197384);
        double r197388 = sqrt(r197387);
        double r197389 = y;
        double r197390 = cbrt(r197389);
        double r197391 = r197390 * r197390;
        double r197392 = r197388 * r197391;
        double r197393 = r197385 / r197392;
        double r197394 = x;
        double r197395 = r197385 / r197394;
        double r197396 = r197395 / r197390;
        double r197397 = r197396 / r197388;
        double r197398 = r197393 * r197397;
        return r197398;
}

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)))))