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 r172629 = x;
        double r172630 = y;
        double r172631 = r172629 * r172630;
        double r172632 = z;
        double r172633 = t;
        double r172634 = r172632 * r172633;
        double r172635 = r172631 + r172634;
        return r172635;
}
↓
double f(double x, double y, double z, double t) {
        double r172636 = t;
        double r172637 = z;
        double r172638 = x;
        double r172639 = y;
        double r172640 = r172638 * r172639;
        double r172641 = fma(r172636, r172637, r172640);
        return r172641;
}

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. Simplified0.0

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020020 +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)))