Average Error: 0.0 → 0.0
Time: 5.2s
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 r6332090 = x;
        double r6332091 = y;
        double r6332092 = r6332090 * r6332091;
        double r6332093 = z;
        double r6332094 = t;
        double r6332095 = r6332093 * r6332094;
        double r6332096 = r6332092 + r6332095;
        return r6332096;
}
↓
double f(double x, double y, double z, double t) {
        double r6332097 = z;
        double r6332098 = t;
        double r6332099 = x;
        double r6332100 = y;
        double r6332101 = r6332099 * r6332100;
        double r6332102 = fma(r6332097, r6332098, r6332101);
        return r6332102;
}

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 2019179 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  (+ (* x y) (* z t)))