Average Error: 6.6 → 6.0
Time: 12.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt[3]{y}}}{x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt[3]{y}}}{x}
double f(double x, double y, double z) {
        double r190273 = 1.0;
        double r190274 = x;
        double r190275 = r190273 / r190274;
        double r190276 = y;
        double r190277 = z;
        double r190278 = r190277 * r190277;
        double r190279 = r190273 + r190278;
        double r190280 = r190276 * r190279;
        double r190281 = r190275 / r190280;
        return r190281;
}
↓
double f(double x, double y, double z) {
        double r190282 = 1.0;
        double r190283 = sqrt(r190282);
        double r190284 = z;
        double r190285 = fma(r190284, r190284, r190282);
        double r190286 = sqrt(r190285);
        double r190287 = r190283 / r190286;
        double r190288 = y;
        double r190289 = cbrt(r190288);
        double r190290 = r190289 * r190289;
        double r190291 = r190287 / r190290;
        double r190292 = r190287 / r190289;
        double r190293 = x;
        double r190294 = r190292 / r190293;
        double r190295 = r190291 * r190294;
        return r190295;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target6.0
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 6.6

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

  3. Applied *-un-lft-identity6.6

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

  4. Applied add-cube-cbrt6.6

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

  5. Applied times-frac6.6

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

  6. Applied associate-/l*6.9

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

  7. Simplified6.9

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

  9. Applied associate-/r*6.6

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

  10. Simplified6.6

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

  12. Applied associate-/r*6.5

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

  14. Applied *-un-lft-identity6.5

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

  15. Applied add-cube-cbrt7.1

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

  16. Applied add-sqr-sqrt7.1

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

  17. Applied add-sqr-sqrt7.1

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

  18. Applied times-frac7.1

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

  19. Applied times-frac7.1

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

  20. Applied times-frac6.0

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

  21. Simplified6.0

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

  22. Final simplification6.0

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

Reproduce

herbie shell --seed 2019304 +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)))))