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

Average Error: 0.1 → 0.1

Time: 5.5s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double x, double y, double z) {
	return fma(x, 0.5, (((1.0 - (z - (2.0 * log(cbrt(z))))) * y) + ((y * log(cbrt((cbrt(z) * cbrt(z))))) + (y * log(cbrt(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. Simplified 0.1

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

4. Applied distribute-lft-in 0.1

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

6. Applied add-cube-cbrt 0.1

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

7. Applied log-prod 0.1

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

8. Applied distribute-lft-in 0.1

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

9. Applied associate-+r+ 0.1

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

10. Simplified 0.1

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

12. Applied add-cube-cbrt 0.1

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

13. Applied cbrt-prod 0.1

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

14. Applied log-prod 0.1

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

15. Applied distribute-lft-in 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020057 +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)))))