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

Average Error: 6.1 → 6.4

Time: 18.2s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r391801 = 1.0;
        double r391802 = x;
        double r391803 = r391801 / r391802;
        double r391804 = y;
        double r391805 = z;
        double r391806 = r391805 * r391805;
        double r391807 = r391801 + r391806;
        double r391808 = r391804 * r391807;
        double r391809 = r391803 / r391808;
        return r391809;
}

↓

double f(double x, double y, double z) {
        double r391810 = 1.0;
        double r391811 = y;
        double r391812 = cbrt(r391811);
        double r391813 = r391812 * r391812;
        double r391814 = r391810 / r391813;
        double r391815 = 1.0;
        double r391816 = x;
        double r391817 = r391815 / r391816;
        double r391818 = cbrt(r391817);
        double r391819 = r391818 * r391818;
        double r391820 = r391818 / r391812;
        double r391821 = z;
        double r391822 = fma(r391821, r391821, r391815);
        double r391823 = r391820 / r391822;
        double r391824 = r391819 * r391823;
        double r391825 = r391814 * r391824;
        return r391825;
}

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 *-un-lft-identity 6.9

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

8. Applied times-frac 6.9

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

9. Applied times-frac 6.9

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

10. Applied times-frac 6.6

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

11. Simplified 6.6

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

13. Applied *-un-lft-identity 6.6

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

14. Applied *-un-lft-identity 6.6

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

15. Applied cbrt-prod 6.6

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

16. Applied add-cube-cbrt 6.7

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

17. Applied times-frac 6.7

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

18. Applied times-frac 6.4

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

19. Simplified 6.4

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

20. Final simplification 6.4

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

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)))))