Average Error: 0.0 → 0.0
Time: 9.2s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x + y, \left(x + y\right) \cdot y\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x + y, \left(x + y\right) \cdot y\right)
double f(double x, double y) {
        double r734462 = x;
        double r734463 = y;
        double r734464 = r734462 + r734463;
        double r734465 = r734464 * r734464;
        return r734465;
}
↓
double f(double x, double y) {
        double r734466 = x;
        double r734467 = y;
        double r734468 = r734466 + r734467;
        double r734469 = r734468 * r734467;
        double r734470 = fma(r734466, r734468, r734469);
        return r734470;
}

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. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(x + y\right)} + \left(x + y\right) \cdot y\]
  5. Using strategy rm

  6. Applied fma-def0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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