Average Error: 0.0 → 0.0
Time: 2.4s
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 r116215 = x;
        double r116216 = y;
        double r116217 = r116215 * r116216;
        double r116218 = z;
        double r116219 = t;
        double r116220 = r116218 * r116219;
        double r116221 = r116217 + r116220;
        double r116222 = a;
        double r116223 = b;
        double r116224 = r116222 * r116223;
        double r116225 = r116221 + r116224;
        return r116225;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r116226 = b;
        double r116227 = a;
        double r116228 = x;
        double r116229 = y;
        double r116230 = z;
        double r116231 = t;
        double r116232 = r116230 * r116231;
        double r116233 = fma(r116228, r116229, r116232);
        double r116234 = fma(r116226, r116227, r116233);
        return r116234;
}

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 2020020 +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)))