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

Average Error: 15.0 → 1.9

Time: 12.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.544292488324377 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z + 1} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{z + 1}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{\frac{\sqrt[3]{x}}{\sqrt{z + 1}}}{\sqrt[3]{z}} \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r414663 = x;
        double r414664 = y;
        double r414665 = r414663 * r414664;
        double r414666 = z;
        double r414667 = r414666 * r414666;
        double r414668 = 1.0;
        double r414669 = r414666 + r414668;
        double r414670 = r414667 * r414669;
        double r414671 = r414665 / r414670;
        return r414671;
}

↓

double f(double x, double y, double z) {
        double r414672 = z;
        double r414673 = -1.544292488324377e-131;
        bool r414674 = r414672 <= r414673;
        double r414675 = x;
        double r414676 = cbrt(r414675);
        double r414677 = r414676 * r414676;
        double r414678 = 1.0;
        double r414679 = r414672 + r414678;
        double r414680 = r414677 / r414679;
        double r414681 = r414676 / r414672;
        double r414682 = y;
        double r414683 = r414682 / r414672;
        double r414684 = r414681 * r414683;
        double r414685 = r414680 * r414684;
        double r414686 = sqrt(r414679);
        double r414687 = r414677 / r414686;
        double r414688 = cbrt(r414672);
        double r414689 = r414688 * r414688;
        double r414690 = r414687 / r414689;
        double r414691 = r414676 / r414686;
        double r414692 = r414691 / r414688;
        double r414693 = r414692 * r414682;
        double r414694 = r414693 / r414672;
        double r414695 = r414690 * r414694;
        double r414696 = r414674 ? r414685 : r414695;
        return r414696;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.0
Target	4.1
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.544292488324377e-131

   1. Initial program 10.1

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

   3. Applied associate-/l* 8.4

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

   4. Simplified 6.5

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

   6. Applied add-cube-cbrt 6.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(z + 1\right) \cdot \frac{{z}^{2}}{y}}\]

   7. Applied times-frac 5.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z + 1} \cdot \frac{\sqrt[3]{x}}{\frac{{z}^{2}}{y}}}\]

   8. Simplified 2.0

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

   if -1.544292488324377e-131 < z

   1. Initial program 18.8

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

   3. Applied associate-/l* 17.0

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

   4. Simplified 15.6

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

   6. Applied add-cube-cbrt 15.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(z + 1\right) \cdot \frac{{z}^{2}}{y}}\]

   7. Applied times-frac 14.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z + 1} \cdot \frac{\sqrt[3]{x}}{\frac{{z}^{2}}{y}}}\]

   8. Simplified 4.9

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

   10. Applied associate-*r* 4.1

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

   11. Simplified 4.1

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

   13. Applied add-cube-cbrt 4.2

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

   14. Applied add-sqr-sqrt 4.2

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

   15. Applied unpow3 4.2

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

   16. Applied times-frac 4.2

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

   17. Applied times-frac 4.2

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

   18. Applied associate-*l* 2.2

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

   19. Simplified 1.8

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.544292488324377 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z + 1} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{z + 1}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{\frac{\sqrt[3]{x}}{\sqrt{z + 1}}}{\sqrt[3]{z}} \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))