Average Error: 0.0 → 0.0
Time: 7.2s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)
double f(double x, double y) {
        double r428154 = x;
        double r428155 = r428154 * r428154;
        double r428156 = 2.0;
        double r428157 = r428154 * r428156;
        double r428158 = y;
        double r428159 = r428157 * r428158;
        double r428160 = r428155 + r428159;
        double r428161 = r428158 * r428158;
        double r428162 = r428160 + r428161;
        return r428162;
}
↓
double f(double x, double y) {
        double r428163 = y;
        double r428164 = 2.0;
        double r428165 = x;
        double r428166 = fma(r428164, r428165, r428163);
        double r428167 = r428165 * r428165;
        double r428168 = fma(r428163, r428166, r428167);
        return r428168;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))