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

Average Error: 0.1 → 0.1

Time: 3.4s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)\]

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)}\]

3. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(3, y \cdot y, x \cdot x\right)\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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