Average Error: 0.0 → 0.0
Time: 24.2s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x \cdot y, 2, \mathsf{fma}\left(y, y, x \cdot x\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x \cdot y, 2, \mathsf{fma}\left(y, y, x \cdot x\right)\right)
double f(double x, double y) {
        double r28336670 = x;
        double r28336671 = y;
        double r28336672 = r28336670 + r28336671;
        double r28336673 = r28336672 * r28336672;
        return r28336673;
}
↓
double f(double x, double y) {
        double r28336674 = x;
        double r28336675 = y;
        double r28336676 = r28336674 * r28336675;
        double r28336677 = 2.0;
        double r28336678 = r28336674 * r28336674;
        double r28336679 = fma(r28336675, r28336675, r28336678);
        double r28336680 = fma(r28336676, r28336677, r28336679);
        return r28336680;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +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)))