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

Average Error: 0.1 → 0.1

Time: 13.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r378919 = x;
        double r378920 = 0.5;
        double r378921 = r378919 * r378920;
        double r378922 = y;
        double r378923 = 1.0;
        double r378924 = z;
        double r378925 = r378923 - r378924;
        double r378926 = log(r378924);
        double r378927 = r378925 + r378926;
        double r378928 = r378922 * r378927;
        double r378929 = r378921 + r378928;
        return r378929;
}

↓

double f(double x, double y, double z) {
        double r378930 = x;
        double r378931 = 1.0;
        double r378932 = 2.0;
        double r378933 = r378931 / r378932;
        double r378934 = r378930 * r378933;
        double r378935 = z;
        double r378936 = r378931 - r378935;
        double r378937 = sqrt(r378935);
        double r378938 = log(r378937);
        double r378939 = r378936 + r378938;
        double r378940 = y;
        double r378941 = r378939 * r378940;
        double r378942 = r378934 + r378941;
        double r378943 = 3.0;
        double r378944 = cbrt(r378937);
        double r378945 = log(r378944);
        double r378946 = r378940 * r378945;
        double r378947 = r378943 * r378946;
        double r378948 = r378942 + r378947;
        return r378948;
}

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. Simplified 0.1

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

4. Applied distribute-lft-in 0.1

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

6. Applied add-sqr-sqrt 0.1

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

7. Applied log-prod 0.1

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

8. Applied distribute-lft-in 0.1

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

9. Applied associate-+r+ 0.1

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

10. Simplified 0.1

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

12. Applied add-cube-cbrt 0.1

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

13. Applied log-prod 0.1

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

14. Applied distribute-lft-in 0.1

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

15. Applied associate-+r+ 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019303 
(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)))))