Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 15.2 → 3.8

Time: 18.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 455039715215592194048:\\ \;\;\;\;\frac{\frac{1}{z}}{z + 1} \cdot \left(\frac{x}{z} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{\frac{\sqrt[3]{x}}{z}}{\sqrt[3]{z + 1} \cdot \frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r287849 = x;
        double r287850 = y;
        double r287851 = r287849 * r287850;
        double r287852 = z;
        double r287853 = r287852 * r287852;
        double r287854 = 1.0;
        double r287855 = r287852 + r287854;
        double r287856 = r287853 * r287855;
        double r287857 = r287851 / r287856;
        return r287857;
}

↓

double f(double x, double y, double z) {
        double r287858 = x;
        double r287859 = 4.550397152155922e+20;
        bool r287860 = r287858 <= r287859;
        double r287861 = 1.0;
        double r287862 = z;
        double r287863 = r287861 / r287862;
        double r287864 = 1.0;
        double r287865 = r287862 + r287864;
        double r287866 = r287863 / r287865;
        double r287867 = r287858 / r287862;
        double r287868 = y;
        double r287869 = r287867 * r287868;
        double r287870 = r287866 * r287869;
        double r287871 = cbrt(r287858);
        double r287872 = r287871 * r287871;
        double r287873 = cbrt(r287865);
        double r287874 = r287873 * r287873;
        double r287875 = r287872 / r287874;
        double r287876 = r287871 / r287862;
        double r287877 = r287862 / r287868;
        double r287878 = r287873 * r287877;
        double r287879 = r287876 / r287878;
        double r287880 = r287875 * r287879;
        double r287881 = r287860 ? r287870 : r287880;
        return r287881;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.2
Target	4.1
Herbie	3.8

\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < 4.550397152155922e+20

   1. Initial program 14.0

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

   2. Simplified 3.5

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

   4. Applied *-un-lft-identity 3.5

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

   5. Applied div-inv 3.5

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

   6. Applied times-frac 4.6

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

   7. Applied associate-*r* 4.4

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

   8. Simplified 4.0

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

   if 4.550397152155922e+20 < x

   1. Initial program 20.3

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

   2. Simplified 3.5

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

   4. Applied *-un-lft-identity 3.5

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

   5. Applied div-inv 3.6

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

   6. Applied times-frac 8.0

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

   7. Applied associate-*r* 8.0

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

   8. Simplified 9.8

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

   10. Applied associate-*r/ 5.4

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

   11. Simplified 5.3

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

   13. Applied associate-/l* 4.1

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

   15. Applied add-cube-cbrt 4.5

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

   16. Applied *-un-lft-identity 4.5

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

   17. Applied *-un-lft-identity 4.5

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

   18. Applied times-frac 4.5

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

   19. Applied *-un-lft-identity 4.5

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

   20. Applied add-cube-cbrt 4.7

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

   21. Applied times-frac 4.7

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

   22. Applied times-frac 2.4

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

   23. Applied times-frac 2.6

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

   24. Simplified 2.6

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

   25. Simplified 2.8

       \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{x}}{z}}{\sqrt[3]{z + 1} \cdot \frac{z}{y}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 455039715215592194048:\\ \;\;\;\;\frac{\frac{1}{z}}{z + 1} \cdot \left(\frac{x}{z} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{\frac{\sqrt[3]{x}}{z}}{\sqrt[3]{z + 1} \cdot \frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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