Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, z \cdot t\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r109603 = x;
        double r109604 = y;
        double r109605 = r109603 * r109604;
        double r109606 = z;
        double r109607 = t;
        double r109608 = r109606 * r109607;
        double r109609 = r109605 + r109608;
        double r109610 = a;
        double r109611 = b;
        double r109612 = r109610 * r109611;
        double r109613 = r109609 + r109612;
        return r109613;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r109614 = a;
        double r109615 = b;
        double r109616 = x;
        double r109617 = y;
        double r109618 = z;
        double r109619 = t;
        double r109620 = r109618 * r109619;
        double r109621 = fma(r109616, r109617, r109620);
        double r109622 = fma(r109614, r109615, r109621);
        return r109622;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))