Average Error: 0.0 → 0.0
Time: 10.6s
Precision: 64
\[x \cdot y + z \cdot t\]
↓
\[\mathsf{fma}\left(z, t, x \cdot y\right)\]
x \cdot y + z \cdot t
↓
\mathsf{fma}\left(z, t, x \cdot y\right)
double f(double x, double y, double z, double t) {
        double r5357073 = x;
        double r5357074 = y;
        double r5357075 = r5357073 * r5357074;
        double r5357076 = z;
        double r5357077 = t;
        double r5357078 = r5357076 * r5357077;
        double r5357079 = r5357075 + r5357078;
        return r5357079;
}
↓
double f(double x, double y, double z, double t) {
        double r5357080 = z;
        double r5357081 = t;
        double r5357082 = x;
        double r5357083 = y;
        double r5357084 = r5357082 * r5357083;
        double r5357085 = fma(r5357080, r5357081, r5357084);
        return r5357085;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  (+ (* x y) (* z t)))