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

Average Error: 0.1 → 0.1

Time: 5.1s

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 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 ((x * 0.5) + (y * ((fma(2.0, log(cbrt(sqrt(z))), (1.0 - z)) + log(cbrt(sqrt(z)))) + log(sqrt(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 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 2020078 +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)))))