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

Average Error: 0.1 → 0.1

Time: 14.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r276983 = x;
        double r276984 = 0.5;
        double r276985 = r276983 * r276984;
        double r276986 = y;
        double r276987 = 1.0;
        double r276988 = z;
        double r276989 = r276987 - r276988;
        double r276990 = log(r276988);
        double r276991 = r276989 + r276990;
        double r276992 = r276986 * r276991;
        double r276993 = r276985 + r276992;
        return r276993;
}

↓

double f(double x, double y, double z) {
        double r276994 = 0.5;
        double r276995 = x;
        double r276996 = r276994 * r276995;
        double r276997 = y;
        double r276998 = 1.0;
        double r276999 = z;
        double r277000 = r276998 - r276999;
        double r277001 = sqrt(r276999);
        double r277002 = log(r277001);
        double r277003 = r277000 + r277002;
        double r277004 = 2.0;
        double r277005 = cbrt(r277001);
        double r277006 = log(r277005);
        double r277007 = r277004 * r277006;
        double r277008 = r277003 + r277007;
        double r277009 = r276997 * r277008;
        double r277010 = r276996 + r277009;
        double r277011 = r277006 * r276997;
        double r277012 = r277010 + r277011;
        return r277012;
}

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. Applied associate-+r+ 0.1

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

5. Simplified 0.1

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

7. Applied add-sqr-sqrt 0.1

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

8. Applied log-prod 0.1

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

9. Applied distribute-lft-in 0.1

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

10. Applied associate-+r+ 0.1

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

11. Simplified 0.1

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

13. Applied add-cube-cbrt 0.1

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

14. Applied log-prod 0.1

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

15. Applied distribute-rgt-in 0.1

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

16. Applied associate-+r+ 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

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)))))