Average Error: 0.0 → 0.0
Time: 1.4s
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 r108523 = x;
        double r108524 = y;
        double r108525 = r108523 * r108524;
        double r108526 = z;
        double r108527 = t;
        double r108528 = r108526 * r108527;
        double r108529 = r108525 + r108528;
        double r108530 = a;
        double r108531 = b;
        double r108532 = r108530 * r108531;
        double r108533 = r108529 + r108532;
        return r108533;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r108534 = t;
        double r108535 = z;
        double r108536 = a;
        double r108537 = b;
        double r108538 = x;
        double r108539 = y;
        double r108540 = r108538 * r108539;
        double r108541 = fma(r108536, r108537, r108540);
        double r108542 = fma(r108534, r108535, r108541);
        return r108542;
}

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

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

  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 2020065 +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)))