System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Average Error: 0.1 → 0.1

Time: 20.9s

Precision: 64

Math

\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

↓

\[x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\sqrt[3]{{\left({z}^{\frac{2}{3}}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{z}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot y\right)\]

C

double f(double x, double y, double z) {
        double r329803 = x;
        double r329804 = 0.5;
        double r329805 = r329803 * r329804;
        double r329806 = y;
        double r329807 = 1.0;
        double r329808 = z;
        double r329809 = r329807 - r329808;
        double r329810 = log(r329808);
        double r329811 = r329809 + r329810;
        double r329812 = r329806 * r329811;
        double r329813 = r329805 + r329812;
        return r329813;
}

↓

double f(double x, double y, double z) {
        double r329814 = x;
        double r329815 = 0.5;
        double r329816 = r329814 * r329815;
        double r329817 = y;
        double r329818 = 2.0;
        double r329819 = z;
        double r329820 = cbrt(r329819);
        double r329821 = log(r329820);
        double r329822 = r329818 * r329821;
        double r329823 = 1.0;
        double r329824 = r329823 - r329819;
        double r329825 = r329822 + r329824;
        double r329826 = r329817 * r329825;
        double r329827 = 0.6666666666666666;
        double r329828 = pow(r329819, r329827);
        double r329829 = pow(r329828, r329827);
        double r329830 = cbrt(r329828);
        double r329831 = r329829 * r329830;
        double r329832 = cbrt(r329831);
        double r329833 = cbrt(r329820);
        double r329834 = r329832 * r329833;
        double r329835 = log(r329834);
        double r329836 = r329835 * r329817;
        double r329837 = r329826 + r329836;
        double r329838 = r329816 + r329837;
        return r329838;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

1. Initial program 0.1

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
2. Using strategy rm

3. Applied distribute-lft-in 0.1

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)}\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.1

   \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + y \cdot \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\]

6. Applied log-prod 0.1

   \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + y \cdot \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\]

7. Applied distribute-rgt-in 0.1

   \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y + \log \left(\sqrt[3]{z}\right) \cdot y\right)}\right)\]

8. Applied associate-+r+ 0.1

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(\left(y \cdot \left(1 - z\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y\right) + \log \left(\sqrt[3]{z}\right) \cdot y\right)}\]

9. Simplified 0.1

   \[\leadsto x \cdot 0.5 + \left(\color{blue}{y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right)} + \log \left(\sqrt[3]{z}\right) \cdot y\right)\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right) \cdot y\right)\]

12. Applied cbrt-prod 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right)} \cdot y\right)\]

13. Simplified 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\color{blue}{\sqrt[3]{{z}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot y\right)\]
14. Using strategy rm

15. Applied add-cube-cbrt 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{{z}^{\frac{2}{3}}} \cdot \sqrt[3]{{z}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{{z}^{\frac{2}{3}}}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot y\right)\]

16. Simplified 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\sqrt[3]{\color{blue}{{\left({z}^{\frac{2}{3}}\right)}^{\frac{2}{3}}} \cdot \sqrt[3]{{z}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot y\right)\]

17. Final simplification 0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(1 - z\right)\right) + \log \left(\sqrt[3]{{\left({z}^{\frac{2}{3}}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{z}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot y\right)\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))