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

Average Error: 6.6 → 5.9

Time: 25.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -9.4796465446858217 \cdot 10^{-131}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y \cdot x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r212848 = 1.0;
        double r212849 = x;
        double r212850 = r212848 / r212849;
        double r212851 = y;
        double r212852 = z;
        double r212853 = r212852 * r212852;
        double r212854 = r212848 + r212853;
        double r212855 = r212851 * r212854;
        double r212856 = r212850 / r212855;
        return r212856;
}

↓

double f(double x, double y, double z) {
        double r212857 = 1.0;
        double r212858 = x;
        double r212859 = r212857 / r212858;
        double r212860 = -9.479646544685822e-131;
        bool r212861 = r212859 <= r212860;
        double r212862 = 1.0;
        double r212863 = y;
        double r212864 = r212862 / r212863;
        double r212865 = r212857 * r212864;
        double r212866 = z;
        double r212867 = fma(r212866, r212866, r212857);
        double r212868 = r212867 * r212858;
        double r212869 = r212865 / r212868;
        double r212870 = sqrt(r212867);
        double r212871 = r212862 / r212870;
        double r212872 = r212863 * r212858;
        double r212873 = r212857 / r212872;
        double r212874 = r212873 / r212870;
        double r212875 = r212871 * r212874;
        double r212876 = r212861 ? r212869 : r212875;
        return r212876;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.6
Target	5.8
Herbie	5.9

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

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 x) < -9.479646544685822e-131

   1. Initial program 9.6

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

   2. Simplified 9.8

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

   4. Applied div-inv 9.8

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

   6. Applied associate-*l/ 9.8

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

   7. Applied associate-/l/ 7.8

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

   if -9.479646544685822e-131 < (/ 1.0 x)

   1. Initial program 5.1

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

   2. Simplified 4.8

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

   4. Applied div-inv 4.8

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

   6. Applied pow1 4.8

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

   7. Applied pow1 4.8

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

   8. Applied pow-prod-down 4.8

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

   9. Simplified 4.8

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

   11. Applied add-sqr-sqrt 4.8

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

   12. Applied *-un-lft-identity 4.8

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

   13. Applied *-un-lft-identity 4.8

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

   14. Applied *-un-lft-identity 4.8

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

   15. Applied times-frac 4.8

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

   16. Applied times-frac 4.8

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

   17. Applied unpow-prod-down 4.8

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

   18. Applied times-frac 4.8

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

   19. Simplified 4.8

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

   20. Simplified 4.9

       \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \color{blue}{\frac{\frac{1}{y \cdot x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -9.4796465446858217 \cdot 10^{-131}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y \cdot x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))