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

Average Error: 0.1 → 0.1

Time: 11.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r28286002 = x;
        double r28286003 = y;
        double r28286004 = r28286002 * r28286003;
        double r28286005 = z;
        double r28286006 = r28286005 * r28286005;
        double r28286007 = r28286004 + r28286006;
        double r28286008 = r28286007 + r28286006;
        double r28286009 = r28286008 + r28286006;
        return r28286009;
}

↓

double f(double x, double y, double z) {
        double r28286010 = y;
        double r28286011 = x;
        double r28286012 = 3.0;
        double r28286013 = z;
        double r28286014 = r28286013 * r28286013;
        double r28286015 = r28286012 * r28286014;
        double r28286016 = fma(r28286010, r28286011, r28286015);
        return r28286016;
}

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

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"

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

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