Average Error: 0.0 → 0.0
Time: 10.3s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(a, b, \mathsf{fma}\left(x, y, z \cdot t\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r143901 = x;
        double r143902 = y;
        double r143903 = r143901 * r143902;
        double r143904 = z;
        double r143905 = t;
        double r143906 = r143904 * r143905;
        double r143907 = r143903 + r143906;
        double r143908 = a;
        double r143909 = b;
        double r143910 = r143908 * r143909;
        double r143911 = r143907 + r143910;
        return r143911;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r143912 = a;
        double r143913 = b;
        double r143914 = x;
        double r143915 = y;
        double r143916 = z;
        double r143917 = t;
        double r143918 = r143916 * r143917;
        double r143919 = fma(r143914, r143915, r143918);
        double r143920 = fma(r143912, r143913, r143919);
        return r143920;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +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)))