Average Error: 0.0 → 0.0
Time: 1.7s
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 r121122 = x;
        double r121123 = y;
        double r121124 = r121122 * r121123;
        double r121125 = z;
        double r121126 = t;
        double r121127 = r121125 * r121126;
        double r121128 = r121124 + r121127;
        double r121129 = a;
        double r121130 = b;
        double r121131 = r121129 * r121130;
        double r121132 = r121128 + r121131;
        return r121132;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r121133 = b;
        double r121134 = a;
        double r121135 = x;
        double r121136 = y;
        double r121137 = z;
        double r121138 = t;
        double r121139 = r121137 * r121138;
        double r121140 = fma(r121135, r121136, r121139);
        double r121141 = fma(r121133, r121134, r121140);
        return r121141;
}

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 2020035 +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)))