Average Error: 0.0 → 0.0
Time: 765.0ms
Precision: 64
\[x \cdot y + z \cdot t\]
↓
\[\mathsf{fma}\left(t, z, x \cdot y\right)\]
x \cdot y + z \cdot t
↓
\mathsf{fma}\left(t, z, x \cdot y\right)
double f(double x, double y, double z, double t) {
        double r146977 = x;
        double r146978 = y;
        double r146979 = r146977 * r146978;
        double r146980 = z;
        double r146981 = t;
        double r146982 = r146980 * r146981;
        double r146983 = r146979 + r146982;
        return r146983;
}
↓
double f(double x, double y, double z, double t) {
        double r146984 = t;
        double r146985 = z;
        double r146986 = x;
        double r146987 = y;
        double r146988 = r146986 * r146987;
        double r146989 = fma(r146984, r146985, r146988);
        return r146989;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.0

    \[x \cdot y + z \cdot t\]

  2. Simplified0.0

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{t \cdot z + x \cdot y}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))