Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b\]
↓
\[\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\]
\left(x \cdot y + z \cdot t\right) + a \cdot b
↓
\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, x \cdot y\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r143229 = x;
        double r143230 = y;
        double r143231 = r143229 * r143230;
        double r143232 = z;
        double r143233 = t;
        double r143234 = r143232 * r143233;
        double r143235 = r143231 + r143234;
        double r143236 = a;
        double r143237 = b;
        double r143238 = r143236 * r143237;
        double r143239 = r143235 + r143238;
        return r143239;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r143240 = b;
        double r143241 = a;
        double r143242 = t;
        double r143243 = z;
        double r143244 = x;
        double r143245 = y;
        double r143246 = r143244 * r143245;
        double r143247 = fma(r143242, r143243, r143246);
        double r143248 = fma(r143240, r143241, r143247);
        return r143248;
}

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

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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)))