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

Average Error: 6.4 → 6.0

Time: 10.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r302626 = 1.0;
        double r302627 = x;
        double r302628 = r302626 / r302627;
        double r302629 = y;
        double r302630 = z;
        double r302631 = r302630 * r302630;
        double r302632 = r302626 + r302631;
        double r302633 = r302629 * r302632;
        double r302634 = r302628 / r302633;
        return r302634;
}

↓

double f(double x, double y, double z) {
        double r302635 = 1.0;
        double r302636 = cbrt(r302635);
        double r302637 = y;
        double r302638 = r302636 / r302637;
        double r302639 = cbrt(r302638);
        double r302640 = r302639 * r302639;
        double r302641 = x;
        double r302642 = cbrt(r302641);
        double r302643 = r302642 * r302642;
        double r302644 = r302640 / r302643;
        double r302645 = r302636 * r302644;
        double r302646 = z;
        double r302647 = r302646 * r302646;
        double r302648 = r302635 + r302647;
        double r302649 = sqrt(r302648);
        double r302650 = r302649 / r302636;
        double r302651 = r302645 / r302650;
        double r302652 = r302639 / r302642;
        double r302653 = r302652 / r302649;
        double r302654 = r302651 * r302653;
        return r302654;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	5.8
Herbie	6.0

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Initial program 6.4

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

3. Applied associate-/r* 6.4

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

4. Simplified 6.4

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

6. Applied add-sqr-sqrt 6.4

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

7. Applied *-un-lft-identity 6.4

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

8. Applied *-un-lft-identity 6.4

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

9. Applied add-cube-cbrt 6.4

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

10. Applied times-frac 6.4

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

11. Applied times-frac 6.4

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

12. Applied times-frac 6.4

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

13. Simplified 6.4

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

15. Applied *-un-lft-identity 6.4

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

16. Applied sqrt-prod 6.4

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

17. Applied add-cube-cbrt 7.0

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

18. Applied add-cube-cbrt 7.1

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

19. Applied times-frac 7.1

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

20. Applied times-frac 6.3

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

21. Applied associate-*r* 6.0

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

22. Simplified 6.0

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

23. Final simplification 6.0

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

Reproduce

herbie shell --seed 2020034 
(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)))))