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

Average Error: 0.1 → 0.1

Time: 5.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r315900 = x;
        double r315901 = 0.5;
        double r315902 = r315900 * r315901;
        double r315903 = y;
        double r315904 = 1.0;
        double r315905 = z;
        double r315906 = r315904 - r315905;
        double r315907 = log(r315905);
        double r315908 = r315906 + r315907;
        double r315909 = r315903 * r315908;
        double r315910 = r315902 + r315909;
        return r315910;
}

↓

double f(double x, double y, double z) {
        double r315911 = x;
        double r315912 = 0.5;
        double r315913 = r315911 * r315912;
        double r315914 = y;
        double r315915 = 2.0;
        double r315916 = z;
        double r315917 = sqrt(r315916);
        double r315918 = cbrt(r315917);
        double r315919 = log(r315918);
        double r315920 = 1.0;
        double r315921 = r315920 - r315916;
        double r315922 = fma(r315915, r315919, r315921);
        double r315923 = r315922 + r315919;
        double r315924 = log(r315917);
        double r315925 = r315923 + r315924;
        double r315926 = r315914 * r315925;
        double r315927 = r315913 + r315926;
        return r315927;
}

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 add-sqr-sqrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied associate-+r+ 0.1

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

7. Applied add-cube-cbrt 0.1

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

8. Applied log-prod 0.1

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

9. Applied associate-+r+ 0.1

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

10. Simplified 0.1

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

11. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(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)))))