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

Average Error: 6.7 → 6.2

Time: 30.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r492390 = 1.0;
        double r492391 = x;
        double r492392 = r492390 / r492391;
        double r492393 = y;
        double r492394 = z;
        double r492395 = r492394 * r492394;
        double r492396 = r492390 + r492395;
        double r492397 = r492393 * r492396;
        double r492398 = r492392 / r492397;
        return r492398;
}

↓

double f(double x, double y, double z) {
        double r492399 = 1.0;
        double r492400 = cbrt(r492399);
        double r492401 = r492400 * r492400;
        double r492402 = x;
        double r492403 = cbrt(r492402);
        double r492404 = r492403 * r492403;
        double r492405 = r492401 / r492404;
        double r492406 = z;
        double r492407 = r492406 * r492406;
        double r492408 = r492399 + r492407;
        double r492409 = sqrt(r492408);
        double r492410 = r492405 / r492409;
        double r492411 = sqrt(r492399);
        double r492412 = cbrt(r492411);
        double r492413 = cbrt(r492404);
        double r492414 = r492412 / r492413;
        double r492415 = cbrt(r492408);
        double r492416 = fabs(r492415);
        double r492417 = r492414 / r492416;
        double r492418 = cbrt(r492403);
        double r492419 = r492412 / r492418;
        double r492420 = y;
        double r492421 = r492419 / r492420;
        double r492422 = sqrt(r492415);
        double r492423 = r492421 / r492422;
        double r492424 = r492417 * r492423;
        double r492425 = r492410 * r492424;
        return r492425;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	6.1
Herbie	6.2

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

Derivation

1. Initial program 6.7

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

3. Applied associate-/r* 6.9

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

5. Applied add-sqr-sqrt 6.9

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

6. Applied *-un-lft-identity 6.9

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

7. Applied add-cube-cbrt 7.5

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

8. Applied add-cube-cbrt 7.5

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

9. Applied times-frac 7.5

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

10. Applied times-frac 7.5

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

11. Applied times-frac 6.3

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

12. Simplified 6.3

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

14. Applied add-cube-cbrt 6.3

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

15. Applied sqrt-prod 6.3

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

16. Applied *-un-lft-identity 6.3

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

17. Applied add-cube-cbrt 6.4

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

18. Applied cbrt-prod 6.4

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

19. Applied add-sqr-sqrt 6.4

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

20. Applied cbrt-prod 6.4

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

21. Applied times-frac 6.4

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

22. Applied times-frac 6.4

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

23. Applied times-frac 6.2

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

24. Simplified 6.2

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

25. Final simplification 6.2

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

Reproduce

herbie shell --seed 2020046 
(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)))))