Average Error: 0.0 → 0.0
Time: 7.4s
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 r82767 = x;
        double r82768 = y;
        double r82769 = r82767 * r82768;
        double r82770 = z;
        double r82771 = t;
        double r82772 = r82770 * r82771;
        double r82773 = r82769 + r82772;
        double r82774 = a;
        double r82775 = b;
        double r82776 = r82774 * r82775;
        double r82777 = r82773 + r82776;
        return r82777;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r82778 = t;
        double r82779 = z;
        double r82780 = a;
        double r82781 = b;
        double r82782 = x;
        double r82783 = y;
        double r82784 = r82782 * r82783;
        double r82785 = fma(r82780, r82781, r82784);
        double r82786 = fma(r82778, r82779, r82785);
        return r82786;
}

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 2019198 +o rules:numerics
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  (+ (+ (* x y) (* z t)) (* a b)))