Average Error: 0.0 → 0.0
Time: 6.8s
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 r201207 = x;
        double r201208 = y;
        double r201209 = r201207 * r201208;
        double r201210 = z;
        double r201211 = t;
        double r201212 = r201210 * r201211;
        double r201213 = r201209 + r201212;
        double r201214 = a;
        double r201215 = b;
        double r201216 = r201214 * r201215;
        double r201217 = r201213 + r201216;
        return r201217;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r201218 = t;
        double r201219 = z;
        double r201220 = a;
        double r201221 = b;
        double r201222 = x;
        double r201223 = y;
        double r201224 = r201222 * r201223;
        double r201225 = fma(r201220, r201221, r201224);
        double r201226 = fma(r201218, r201219, r201225);
        return r201226;
}

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

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

  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 2019326 +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)))