Average Error: 0.0 → 0.0
Time: 1.0s
Precision: 64
\[x \cdot y + z \cdot t\]
↓
\[\mathsf{fma}\left(t, z, x \cdot y\right)\]
x \cdot y + z \cdot t
↓
\mathsf{fma}\left(t, z, x \cdot y\right)
double f(double x, double y, double z, double t) {
        double r93590 = x;
        double r93591 = y;
        double r93592 = r93590 * r93591;
        double r93593 = z;
        double r93594 = t;
        double r93595 = r93593 * r93594;
        double r93596 = r93592 + r93595;
        return r93596;
}
↓
double f(double x, double y, double z, double t) {
        double r93597 = t;
        double r93598 = z;
        double r93599 = x;
        double r93600 = y;
        double r93601 = r93599 * r93600;
        double r93602 = fma(r93597, r93598, r93601);
        return r93602;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.0

    \[x \cdot y + z \cdot t\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{t \cdot z + x \cdot y}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))