Average Error: 0.0 → 0.0
Time: 2.9s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r137345 = x;
        double r137346 = y;
        double r137347 = r137345 * r137346;
        double r137348 = z;
        double r137349 = t;
        double r137350 = r137348 * r137349;
        double r137351 = r137347 + r137350;
        double r137352 = a;
        double r137353 = b;
        double r137354 = r137352 * r137353;
        double r137355 = r137351 + r137354;
        return r137355;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r137356 = t;
        double r137357 = z;
        double r137358 = a;
        double r137359 = b;
        double r137360 = x;
        double r137361 = y;
        double r137362 = r137360 * r137361;
        double r137363 = fma(r137358, r137359, r137362);
        double r137364 = fma(r137356, r137357, r137363);
        return r137364;
}

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +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)))