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

Average Error: 6.1 → 6.4

Time: 15.7s

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 r419395 = 1.0;
        double r419396 = x;
        double r419397 = r419395 / r419396;
        double r419398 = y;
        double r419399 = z;
        double r419400 = r419399 * r419399;
        double r419401 = r419395 + r419400;
        double r419402 = r419398 * r419401;
        double r419403 = r419397 / r419402;
        return r419403;
}

↓

double f(double x, double y, double z) {
        double r419404 = 1.0;
        double r419405 = y;
        double r419406 = cbrt(r419405);
        double r419407 = r419406 * r419406;
        double r419408 = r419404 / r419407;
        double r419409 = 1.0;
        double r419410 = x;
        double r419411 = r419409 / r419410;
        double r419412 = cbrt(r419411);
        double r419413 = r419412 * r419412;
        double r419414 = r419412 / r419406;
        double r419415 = z;
        double r419416 = fma(r419415, r419415, r419404);
        double r419417 = r419414 / r419416;
        double r419418 = r419413 * r419417;
        double r419419 = r419408 * r419418;
        return r419419;
}

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 div-inv 6.9

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

7. Applied times-frac 6.9

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

8. Applied times-frac 6.6

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

9. Simplified 6.6

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

11. Applied *-un-lft-identity 6.6

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

12. Applied *-un-lft-identity 6.6

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

13. Applied cbrt-prod 6.6

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

14. Applied add-cube-cbrt 6.7

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

15. Applied times-frac 6.7

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

16. Applied times-frac 6.4

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

17. Simplified 6.4

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

18. 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)))))