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

Average Error: 6.7 → 6.3

Time: 17.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r232470 = 1.0;
        double r232471 = x;
        double r232472 = r232470 / r232471;
        double r232473 = y;
        double r232474 = z;
        double r232475 = r232474 * r232474;
        double r232476 = r232470 + r232475;
        double r232477 = r232473 * r232476;
        double r232478 = r232472 / r232477;
        return r232478;
}

↓

double f(double x, double y, double z) {
        double r232479 = 1.0;
        double r232480 = cbrt(r232479);
        double r232481 = r232480 * r232480;
        double r232482 = x;
        double r232483 = cbrt(r232482);
        double r232484 = r232483 * r232483;
        double r232485 = r232481 / r232484;
        double r232486 = y;
        double r232487 = cbrt(r232486);
        double r232488 = r232487 * r232487;
        double r232489 = r232485 / r232488;
        double r232490 = r232480 / r232483;
        double r232491 = r232490 / r232487;
        double r232492 = z;
        double r232493 = r232492 * r232492;
        double r232494 = r232479 + r232493;
        double r232495 = r232491 / r232494;
        double r232496 = r232489 * r232495;
        return r232496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	6.1
Herbie	6.3

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

Derivation

1. Initial program 6.7

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

3. Applied add-cube-cbrt 7.3

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

4. Applied times-frac 7.1

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

6. Applied associate-*r/ 7.4

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

7. Simplified 6.8

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

9. Applied *-un-lft-identity 6.8

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

10. Applied add-cube-cbrt 7.4

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

11. Applied add-cube-cbrt 7.6

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

12. Applied add-cube-cbrt 7.6

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

13. Applied times-frac 7.6

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

14. Applied times-frac 7.6

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

15. Applied times-frac 6.3

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

16. Simplified 6.3

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

17. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))