Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(b, a, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(b, a, \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 r102195 = x;
        double r102196 = y;
        double r102197 = r102195 * r102196;
        double r102198 = z;
        double r102199 = t;
        double r102200 = r102198 * r102199;
        double r102201 = r102197 + r102200;
        double r102202 = a;
        double r102203 = b;
        double r102204 = r102202 * r102203;
        double r102205 = r102201 + r102204;
        return r102205;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r102206 = b;
        double r102207 = a;
        double r102208 = x;
        double r102209 = y;
        double r102210 = z;
        double r102211 = t;
        double r102212 = r102210 * r102211;
        double r102213 = fma(r102208, r102209, r102212);
        double r102214 = fma(r102206, r102207, r102213);
        return r102214;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))