Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x + y, y, \left(x + y\right) \cdot x\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x + y, y, \left(x + y\right) \cdot x\right)
double f(double x, double y) {
        double r1404249 = x;
        double r1404250 = y;
        double r1404251 = r1404249 + r1404250;
        double r1404252 = r1404251 * r1404251;
        return r1404252;
}
↓
double f(double x, double y) {
        double r1404253 = x;
        double r1404254 = y;
        double r1404255 = r1404253 + r1404254;
        double r1404256 = r1404255 * r1404253;
        double r1404257 = fma(r1404255, r1404254, r1404256);
        return r1404257;
}

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. Simplified0.0

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

  4. Applied distribute-lft-in0.0

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

  5. Simplified0.0

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

  7. Applied fma-def0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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)))