Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.5 → 6.3

Time: 16.8s

Precision: 64

Math

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

↓

\[\frac{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\frac{\sqrt{\frac{{\left(\sqrt[3]{1}\right)}^{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y}}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}\right)\]

C

double f(double x, double y, double z) {
        double r244463 = 1.0;
        double r244464 = x;
        double r244465 = r244463 / r244464;
        double r244466 = y;
        double r244467 = z;
        double r244468 = r244467 * r244467;
        double r244469 = r244463 + r244468;
        double r244470 = r244466 * r244469;
        double r244471 = r244465 / r244470;
        return r244471;
}

↓

double f(double x, double y, double z) {
        double r244472 = 1.0;
        double r244473 = cbrt(r244472);
        double r244474 = r244473 * r244473;
        double r244475 = x;
        double r244476 = cbrt(r244475);
        double r244477 = r244476 * r244476;
        double r244478 = r244474 / r244477;
        double r244479 = sqrt(r244478);
        double r244480 = y;
        double r244481 = cbrt(r244480);
        double r244482 = r244481 * r244481;
        double r244483 = r244479 / r244482;
        double r244484 = 2.0;
        double r244485 = pow(r244473, r244484);
        double r244486 = r244485 / r244477;
        double r244487 = sqrt(r244486);
        double r244488 = r244487 / r244481;
        double r244489 = z;
        double r244490 = r244489 * r244489;
        double r244491 = r244472 + r244490;
        double r244492 = sqrt(r244491);
        double r244493 = r244488 / r244492;
        double r244494 = r244473 / r244476;
        double r244495 = r244494 / r244492;
        double r244496 = r244493 * r244495;
        double r244497 = r244483 * r244496;
        return r244497;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	5.9
Herbie	6.3

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

Derivation

1. Initial program 6.5

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

3. Applied add-cube-cbrt 7.1

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

4. Applied add-cube-cbrt 7.1

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

5. Applied times-frac 7.1

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

6. Applied times-frac 6.8

   \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{1 + z \cdot z}}\]
7. Using strategy rm

8. Applied add-cube-cbrt 7.0

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

9. Applied add-sqr-sqrt 7.0

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

10. Applied times-frac 7.0

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

11. Applied associate-*l* 6.3

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

13. Applied add-sqr-sqrt 6.3

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

14. Applied *-un-lft-identity 6.3

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

15. Applied cbrt-prod 6.3

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

16. Applied *-un-lft-identity 6.3

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

17. Applied cbrt-prod 6.3

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

18. Applied times-frac 6.3

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

19. Applied times-frac 6.3

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

20. Applied associate-*r* 6.3

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

21. Simplified 6.3

    \[\leadsto \frac{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\color{blue}{\frac{\frac{\sqrt{\frac{{\left(\sqrt[3]{1}\right)}^{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y}}}{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}\right)\]

22. Final simplification 6.3

    \[\leadsto \frac{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\frac{\sqrt{\frac{{\left(\sqrt[3]{1}\right)}^{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y}}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}\right)\]

Reproduce

herbie shell --seed 2019305 
(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)))))