Average Error: 0.0 → 0.0
Time: 1.1s
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 r134968 = x;
        double r134969 = y;
        double r134970 = r134968 * r134969;
        double r134971 = z;
        double r134972 = t;
        double r134973 = r134971 * r134972;
        double r134974 = r134970 + r134973;
        return r134974;
}
↓
double f(double x, double y, double z, double t) {
        double r134975 = t;
        double r134976 = z;
        double r134977 = x;
        double r134978 = y;
        double r134979 = r134977 * r134978;
        double r134980 = fma(r134975, r134976, r134979);
        return r134980;
}

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 2020057 +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)))