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

Average Error: 6.1 → 6.4

Time: 21.0s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r303310 = 1.0;
        double r303311 = x;
        double r303312 = r303310 / r303311;
        double r303313 = y;
        double r303314 = z;
        double r303315 = r303314 * r303314;
        double r303316 = r303310 + r303315;
        double r303317 = r303313 * r303316;
        double r303318 = r303312 / r303317;
        return r303318;
}

↓

double f(double x, double y, double z) {
        double r303319 = 1.0;
        double r303320 = cbrt(r303319);
        double r303321 = r303320 * r303320;
        double r303322 = y;
        double r303323 = cbrt(r303322);
        double r303324 = r303323 * r303323;
        double r303325 = r303321 / r303324;
        double r303326 = x;
        double r303327 = r303320 / r303326;
        double r303328 = cbrt(r303327);
        double r303329 = r303328 * r303328;
        double r303330 = r303328 / r303323;
        double r303331 = z;
        double r303332 = fma(r303331, r303331, r303319);
        double r303333 = r303330 / r303332;
        double r303334 = r303329 * r303333;
        double r303335 = r303325 * r303334;
        return r303335;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	5.5
Herbie	6.4

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

Derivation

1. Initial program 6.1

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

2. Simplified 6.3

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

4. Applied *-un-lft-identity 6.3

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

5. Applied add-cube-cbrt 6.9

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

6. Applied *-un-lft-identity 6.9

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

7. Applied add-cube-cbrt 6.9

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

8. Applied times-frac 6.9

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

9. Applied times-frac 6.9

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

10. Applied times-frac 6.6

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

11. Simplified 6.6

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

13. Applied *-un-lft-identity 6.6

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

14. Applied *-un-lft-identity 6.6

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

15. Applied cbrt-prod 6.6

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

16. Applied add-cube-cbrt 6.7

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

17. Applied times-frac 6.7

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

18. Applied times-frac 6.4

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

19. Simplified 6.4

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

20. Final simplification 6.4

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

Reproduce

herbie shell --seed 2020047 +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))) #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)))))