Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(y, y, x \cdot x\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(y, y, x \cdot x\right)\right)
double f(double x, double y) {
        double r29390638 = x;
        double r29390639 = y;
        double r29390640 = r29390638 + r29390639;
        double r29390641 = r29390640 * r29390640;
        return r29390641;
}
↓
double f(double x, double y) {
        double r29390642 = 2.0;
        double r29390643 = y;
        double r29390644 = x;
        double r29390645 = r29390643 * r29390644;
        double r29390646 = r29390644 * r29390644;
        double r29390647 = fma(r29390643, r29390643, r29390646);
        double r29390648 = fma(r29390642, r29390645, r29390647);
        return r29390648;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot \left(x + y\right)\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{y}^{2} + \left({x}^{2} + 2 \cdot \left(x \cdot y\right)\right)}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, x \cdot y, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(y, y, x \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2 (* y x))))

  (* (+ x y) (+ x y)))