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

Average Error: 6.4 → 5.8

Time: 6.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r372817 = 1.0;
        double r372818 = x;
        double r372819 = r372817 / r372818;
        double r372820 = y;
        double r372821 = z;
        double r372822 = r372821 * r372821;
        double r372823 = r372817 + r372822;
        double r372824 = r372820 * r372823;
        double r372825 = r372819 / r372824;
        return r372825;
}

↓

double f(double x, double y, double z) {
        double r372826 = 1.0;
        double r372827 = cbrt(r372826);
        double r372828 = x;
        double r372829 = r372826 / r372828;
        double r372830 = cbrt(r372829);
        double r372831 = r372827 * r372830;
        double r372832 = z;
        double r372833 = fma(r372832, r372832, r372826);
        double r372834 = cbrt(r372833);
        double r372835 = r372834 * r372834;
        double r372836 = cbrt(r372828);
        double r372837 = r372836 * r372836;
        double r372838 = cbrt(r372837);
        double r372839 = r372835 * r372838;
        double r372840 = cbrt(r372836);
        double r372841 = r372839 * r372840;
        double r372842 = r372831 / r372841;
        double r372843 = y;
        double r372844 = r372830 / r372834;
        double r372845 = r372843 / r372844;
        double r372846 = r372842 / r372845;
        return r372846;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	5.6
Herbie	5.8

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Initial program 6.4

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

2. Simplified 6.3

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

4. Applied add-cube-cbrt 6.5

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

5. Applied add-cube-cbrt 6.9

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

6. Applied times-frac 6.9

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

7. Applied associate-/l* 5.8

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

9. Applied cbrt-div 5.7

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

10. Applied associate-*l/ 5.7

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

11. Applied associate-/l/ 5.7

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

13. Applied add-cube-cbrt 5.8

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

14. Applied cbrt-prod 5.8

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

15. Applied associate-*r* 5.8

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

16. Final simplification 5.8

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

Reproduce

herbie shell --seed 2020060 +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)))))