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

Average Error: 6.6 → 6.0

Time: 13.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r213980 = 1.0;
        double r213981 = x;
        double r213982 = r213980 / r213981;
        double r213983 = y;
        double r213984 = z;
        double r213985 = r213984 * r213984;
        double r213986 = r213980 + r213985;
        double r213987 = r213983 * r213986;
        double r213988 = r213982 / r213987;
        return r213988;
}

↓

double f(double x, double y, double z) {
        double r213989 = 1.0;
        double r213990 = cbrt(r213989);
        double r213991 = r213990 * r213990;
        double r213992 = y;
        double r213993 = r213991 / r213992;
        double r213994 = z;
        double r213995 = fma(r213994, r213994, r213989);
        double r213996 = sqrt(r213995);
        double r213997 = r213993 / r213996;
        double r213998 = x;
        double r213999 = r213990 / r213998;
        double r214000 = r213999 / r213996;
        double r214001 = r213997 * r214000;
        return r214001;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.6
Target	5.8
Herbie	6.0

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

Derivation

1. Initial program 6.6

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

3. Applied *-un-lft-identity 6.6

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

4. Applied add-cube-cbrt 6.6

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

5. Applied times-frac 6.6

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

6. Applied times-frac 6.4

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

7. Simplified 6.4

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

8. Simplified 6.4

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

10. Applied add-sqr-sqrt 6.4

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

11. Applied *-un-lft-identity 6.4

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

12. Applied *-un-lft-identity 6.4

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

13. Applied cbrt-prod 6.4

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

14. Applied times-frac 6.4

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

15. Applied times-frac 6.4

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

16. Applied associate-*r* 6.0

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

17. Simplified 6.0

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

18. Final simplification 6.0

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

Reproduce

herbie shell --seed 2019235 +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))) -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)))))