Average Error: 0.0 → 0.0
Time: 7.2s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(x, x, y \cdot y\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(x, x, y \cdot y\right)\right)
double f(double x, double y) {
        double r22224759 = x;
        double r22224760 = y;
        double r22224761 = r22224759 + r22224760;
        double r22224762 = r22224761 * r22224761;
        return r22224762;
}
↓
double f(double x, double y) {
        double r22224763 = 2.0;
        double r22224764 = y;
        double r22224765 = x;
        double r22224766 = r22224764 * r22224765;
        double r22224767 = r22224764 * r22224764;
        double r22224768 = fma(r22224765, r22224765, r22224767);
        double r22224769 = fma(r22224763, r22224766, r22224768);
        return r22224769;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))