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

Average Error: 15.7 → 1.5

Time: 11.5s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r264079 = x;
        double r264080 = y;
        double r264081 = r264079 * r264080;
        double r264082 = z;
        double r264083 = r264082 * r264082;
        double r264084 = 1.0;
        double r264085 = r264082 + r264084;
        double r264086 = r264083 * r264085;
        double r264087 = r264081 / r264086;
        return r264087;
}

↓

double f(double x, double y, double z) {
        double r264088 = x;
        double r264089 = cbrt(r264088);
        double r264090 = z;
        double r264091 = r264089 / r264090;
        double r264092 = r264091 * r264089;
        double r264093 = y;
        double r264094 = cbrt(r264091);
        double r264095 = 3.0;
        double r264096 = pow(r264094, r264095);
        double r264097 = r264093 * r264096;
        double r264098 = 1.0;
        double r264099 = r264090 + r264098;
        double r264100 = r264097 / r264099;
        double r264101 = r264092 * r264100;
        double r264102 = cbrt(r264101);
        double r264103 = r264102 * r264102;
        double r264104 = r264103 * r264102;
        return r264104;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.7
Target	4.4
Herbie	1.5

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

Derivation

1. Initial program 15.7

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

3. Applied times-frac 11.8

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

5. Applied add-cube-cbrt 12.1

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

6. Applied times-frac 7.0

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

7. Applied associate-*l* 1.3

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

9. Applied add-cube-cbrt 1.5

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

10. Applied associate-*l* 1.5

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

12. Applied add-cube-cbrt 1.6

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

13. Simplified 1.6

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

14. Simplified 1.5

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

15. Final simplification 1.5

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

Reproduce

herbie shell --seed 2020047 +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))))