Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x + y, x \cdot y\right) + y \cdot y\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x + y, x \cdot y\right) + y \cdot y
double f(double x, double y) {
        double r930040 = x;
        double r930041 = y;
        double r930042 = r930040 + r930041;
        double r930043 = r930042 * r930042;
        return r930043;
}
↓
double f(double x, double y) {
        double r930044 = x;
        double r930045 = y;
        double r930046 = r930044 + r930045;
        double r930047 = r930044 * r930045;
        double r930048 = fma(r930044, r930046, r930047);
        double r930049 = r930045 * r930045;
        double r930050 = r930048 + r930049;
        return r930050;
}

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. Simplified0.0

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

  7. Applied distribute-lft-in0.0

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

  8. Applied associate-+r+0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 +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)))