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

Average Error: 0.1 → 0.1

Time: 3.8s

Precision: binary64

Math

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

↓

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

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) (x * 0.5)) + ((double) (((double) (y * ((double) (((double) (1.0 - z)) + ((double) log(((double) sqrt(z)))))))) + ((double) (((double) (y * ((double) log(((double) sqrt(((double) (((double) cbrt(z)) * ((double) cbrt(z)))))))))) + ((double) (y * ((double) log(((double) sqrt(((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 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 distribute-lft-in 0.1

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

9. Applied add-cube-cbrt 0.1

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

10. Applied sqrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-lft-in 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020161 
(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)))))