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

Average Error: 0.1 → 0.1

Time: 3.8s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((double) (((double) (x * 0.5)) + ((double) (y * ((double) (((double) (1.0 - z)) + ((double) log(z))))))));
}

↓

double code(double x, double y, double z) {
	return ((double) (((double) (((double) (y * ((double) (((double) (1.0 - z)) + ((double) log(((double) pow(z, 0.6666666666666666)))))))) + ((double) (x * 0.5)))) + ((double) (y * ((double) log(((double) cbrt(z))))))));
}

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

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

7. Applied add-cube-cbrt 0.1

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

8. Applied log-prod 0.1

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

9. Applied distribute-lft-in 0.1

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

10. Applied associate-+r+ 0.1

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

11. Simplified 0.1

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

13. Applied pow1/3 0.1

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

14. Applied pow1/3 0.1

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

15. Applied pow-prod-up 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020168 
(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 (+ (- 1.0 z) (log z)))))