Average Error: 0.0 → 0.0
Time: 8.1s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2 \cdot y, y \cdot y\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2 \cdot y, y \cdot y\right)\right)
double f(double x, double y) {
        double r27031663 = x;
        double r27031664 = y;
        double r27031665 = r27031663 + r27031664;
        double r27031666 = r27031665 * r27031665;
        return r27031666;
}
↓
double f(double x, double y) {
        double r27031667 = x;
        double r27031668 = 2.0;
        double r27031669 = y;
        double r27031670 = r27031668 * r27031669;
        double r27031671 = r27031669 * r27031669;
        double r27031672 = fma(r27031667, r27031670, r27031671);
        double r27031673 = fma(r27031667, r27031667, r27031672);
        return r27031673;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

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