Linear.Quaternion:$c/ from linear-1.19.1.3, A

Average Error: 0.1 → 0.1

Time: 2.9s

Precision: 64

Math

\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]

↓

\[x \cdot y + \left(3 \cdot z\right) \cdot z\]

C

double code(double x, double y, double z) {
	return ((((x * y) + (z * z)) + (z * z)) + (z * z));
}

↓

double code(double x, double y, double z) {
	return ((x * y) + ((3.0 * z) * z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(3 \cdot z\right) \cdot z + y \cdot x\]

Derivation

1. Initial program 0.1

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
2. Using strategy rm

3. Applied associate-+l+ 0.1

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

4. Simplified 0.1

   \[\leadsto \left(x \cdot y + \color{blue}{z \cdot \left(z + z\right)}\right) + z \cdot z\]
5. Using strategy rm

6. Applied associate-+l+ 0.1

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

7. Simplified 0.1

   \[\leadsto x \cdot y + \color{blue}{\left(3 \cdot z\right) \cdot z}\]

8. Final simplification 0.1

   \[\leadsto x \cdot y + \left(3 \cdot z\right) \cdot z\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))