Average Error: 6.6 → 6.2
Time: 20.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}}}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}}}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r208229 = 1.0;
        double r208230 = x;
        double r208231 = r208229 / r208230;
        double r208232 = y;
        double r208233 = z;
        double r208234 = r208233 * r208233;
        double r208235 = r208229 + r208234;
        double r208236 = r208232 * r208235;
        double r208237 = r208231 / r208236;
        return r208237;
}
↓
double f(double x, double y, double z) {
        double r208238 = 1.0;
        double r208239 = sqrt(r208238);
        double r208240 = y;
        double r208241 = cbrt(r208240);
        double r208242 = r208241 * r208241;
        double r208243 = r208239 / r208242;
        double r208244 = z;
        double r208245 = r208244 * r208244;
        double r208246 = r208238 + r208245;
        double r208247 = sqrt(r208246);
        double r208248 = sqrt(r208247);
        double r208249 = r208243 / r208248;
        double r208250 = r208239 / r208241;
        double r208251 = r208250 / r208248;
        double r208252 = x;
        double r208253 = r208251 / r208252;
        double r208254 = r208253 / r208247;
        double r208255 = r208249 * r208254;
        return r208255;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.6
Target6.0
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied add-sqr-sqrt6.6

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

  4. Applied associate-*r*6.6

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}}\]
  5. Using strategy rm

  6. Applied div-inv6.6

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

  7. Applied times-frac6.4

    \[\leadsto \color{blue}{\frac{1}{y \cdot \sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}\]
  8. Using strategy rm

  9. Applied associate-*r/6.3

    \[\leadsto \color{blue}{\frac{\frac{1}{y \cdot \sqrt{1 + z \cdot z}} \cdot \frac{1}{x}}{\sqrt{1 + z \cdot z}}}\]

  10. Simplified6.2

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x}}}{\sqrt{1 + z \cdot z}}\]
  11. Using strategy rm

  12. Applied *-un-lft-identity6.2

    \[\leadsto \frac{\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x}}{\color{blue}{1 \cdot \sqrt{1 + z \cdot z}}}\]

  13. Applied *-un-lft-identity6.2

    \[\leadsto \frac{\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{\color{blue}{1 \cdot x}}}{1 \cdot \sqrt{1 + z \cdot z}}\]

  14. Applied add-sqr-sqrt6.2

    \[\leadsto \frac{\frac{\frac{\frac{1}{y}}{\sqrt{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}}{1 \cdot x}}{1 \cdot \sqrt{1 + z \cdot z}}\]

  15. Applied sqrt-prod6.2

    \[\leadsto \frac{\frac{\frac{\frac{1}{y}}{\color{blue}{\sqrt{\sqrt{1 + z \cdot z}} \cdot \sqrt{\sqrt{1 + z \cdot z}}}}}{1 \cdot x}}{1 \cdot \sqrt{1 + z \cdot z}}\]

  16. Applied add-cube-cbrt6.8

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

  17. Applied add-sqr-sqrt6.8

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

  18. Applied times-frac6.8

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

  19. Applied times-frac6.8

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

  20. Applied times-frac6.4

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

  21. Applied times-frac6.2

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

  22. Simplified6.2

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

  23. Final simplification6.2

    \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{y}}}{\sqrt{\sqrt{1 + z \cdot z}}}}{x}}{\sqrt{1 + z \cdot z}}\]

Reproduce

herbie shell --seed 2019304 
(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)))))