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

Average Error: 0.1 → 0.1

Time: 3.4s

Precision: 64

Math

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

↓

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

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) + fma((1.0 - z), y, (log(z) * y)));
}

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-rgt-in 0.1

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

4. Applied associate-+r+ 0.1

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

5. Simplified 0.1

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

7. Applied fma-udef 0.1

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

8. Applied associate-+l+ 0.1

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

9. Simplified 0.1

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

10. Final simplification 0.1

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

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)))))