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

Average Error: 0.1 → 0.1

Time: 3.3s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))

↓

(FPCore (x y z) :precision binary64 (+ (* x y) (* 3.0 (* z 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. Simplified 0.1

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

4. Applied pow1_binary64 0.1

   \[\leadsto x \cdot y + z \cdot \left(z \cdot \color{blue}{{3}^{1}}\right)\]

5. Applied pow1_binary64 0.1

   \[\leadsto x \cdot y + z \cdot \left(\color{blue}{{z}^{1}} \cdot {3}^{1}\right)\]

6. Applied pow-prod-down_binary64 0.1

   \[\leadsto x \cdot y + z \cdot \color{blue}{{\left(z \cdot 3\right)}^{1}}\]

7. Applied pow1_binary64 0.1

   \[\leadsto x \cdot y + \color{blue}{{z}^{1}} \cdot {\left(z \cdot 3\right)}^{1}\]

8. Applied pow-prod-down_binary64 0.1

   \[\leadsto x \cdot y + \color{blue}{{\left(z \cdot \left(z \cdot 3\right)\right)}^{1}}\]

9. Simplified 0.1

   \[\leadsto x \cdot y + {\color{blue}{\left(3 \cdot \left(z \cdot z\right)\right)}}^{1}\]

10. Final simplification 0.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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