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

Average Error: 0.1 → 0.1

Time: 25.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r262351 = x;
        double r262352 = 0.5;
        double r262353 = r262351 * r262352;
        double r262354 = y;
        double r262355 = 1.0;
        double r262356 = z;
        double r262357 = r262355 - r262356;
        double r262358 = log(r262356);
        double r262359 = r262357 + r262358;
        double r262360 = r262354 * r262359;
        double r262361 = r262353 + r262360;
        return r262361;
}

↓

double f(double x, double y, double z) {
        double r262362 = x;
        double r262363 = 0.5;
        double r262364 = r262362 * r262363;
        double r262365 = 2.0;
        double r262366 = z;
        double r262367 = cbrt(r262366);
        double r262368 = log(r262367);
        double r262369 = 1.0;
        double r262370 = r262369 - r262366;
        double r262371 = fma(r262365, r262368, r262370);
        double r262372 = y;
        double r262373 = r262371 * r262372;
        double r262374 = 0.6666666666666666;
        double r262375 = pow(r262366, r262374);
        double r262376 = pow(r262375, r262374);
        double r262377 = pow(r262367, r262374);
        double r262378 = r262376 * r262377;
        double r262379 = cbrt(r262378);
        double r262380 = cbrt(r262367);
        double r262381 = r262379 * r262380;
        double r262382 = log(r262381);
        double r262383 = r262382 * r262372;
        double r262384 = r262373 + r262383;
        double r262385 = r262364 + r262384;
        return r262385;
}

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. Simplified 0.1

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

6. Applied add-cube-cbrt 0.1

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

7. Applied log-prod 0.1

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

8. Applied distribute-rgt-in 0.1

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

9. Applied associate-+r+ 0.1

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

10. Simplified 0.1

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

12. Applied add-cube-cbrt 0.1

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

13. Applied cbrt-prod 0.1

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

14. Simplified 0.1

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

16. Applied add-cube-cbrt 0.1

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

17. Applied unpow-prod-down 0.1

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

18. Simplified 0.1

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

19. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019326 +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)))))