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

Average Error: 6.2 → 6.0

Time: 34.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r284325 = 1.0;
        double r284326 = x;
        double r284327 = r284325 / r284326;
        double r284328 = y;
        double r284329 = z;
        double r284330 = r284329 * r284329;
        double r284331 = r284325 + r284330;
        double r284332 = r284328 * r284331;
        double r284333 = r284327 / r284332;
        return r284333;
}

↓

double f(double x, double y, double z) {
        double r284334 = 1.0;
        double r284335 = cbrt(r284334);
        double r284336 = r284335 * r284335;
        double r284337 = x;
        double r284338 = cbrt(r284337);
        double r284339 = r284338 * r284338;
        double r284340 = r284336 / r284339;
        double r284341 = y;
        double r284342 = cbrt(r284341);
        double r284343 = r284342 * r284342;
        double r284344 = r284340 / r284343;
        double r284345 = r284335 / r284338;
        double r284346 = r284345 / r284342;
        double r284347 = z;
        double r284348 = fma(r284347, r284347, r284334);
        double r284349 = r284346 / r284348;
        double r284350 = r284344 * r284349;
        return r284350;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	5.5
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 6.2

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

2. Simplified 6.4

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

4. Applied *-un-lft-identity 6.4

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

5. Applied add-cube-cbrt 7.0

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

6. Applied add-cube-cbrt 7.2

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

7. Applied add-cube-cbrt 7.2

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

8. Applied times-frac 7.2

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

9. Applied times-frac 7.2

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

10. Applied times-frac 6.0

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

11. Simplified 6.0

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

12. Final simplification 6.0

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

Reproduce

herbie shell --seed 2019209 +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)))))