Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r200257 = x;
        double r200258 = y;
        double r200259 = r200257 * r200258;
        double r200260 = z;
        double r200261 = t;
        double r200262 = r200260 * r200261;
        double r200263 = r200259 + r200262;
        double r200264 = a;
        double r200265 = b;
        double r200266 = r200264 * r200265;
        double r200267 = r200263 + r200266;
        return r200267;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r200268 = t;
        double r200269 = z;
        double r200270 = a;
        double r200271 = b;
        double r200272 = x;
        double r200273 = y;
        double r200274 = r200272 * r200273;
        double r200275 = fma(r200270, r200271, r200274);
        double r200276 = fma(r200268, r200269, r200275);
        return r200276;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{t \cdot z + \left(a \cdot b + x \cdot y\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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)))