Average Error: 0.0 → 0.0
Time: 866.0ms
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)
double f(double x, double y) {
        double r786184 = x;
        double r786185 = r786184 * r786184;
        double r786186 = 2.0;
        double r786187 = r786184 * r786186;
        double r786188 = y;
        double r786189 = r786187 * r786188;
        double r786190 = r786185 + r786189;
        double r786191 = r786188 * r786188;
        double r786192 = r786190 + r786191;
        return r786192;
}
↓
double f(double x, double y) {
        double r786193 = y;
        double r786194 = 2.0;
        double r786195 = x;
        double r786196 = fma(r786193, r786194, r786195);
        double r786197 = r786193 * r786193;
        double r786198 = fma(r786196, r786195, r786197);
        return r786198;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2)))

  (+ (+ (* x x) (* (* x 2) y)) (* y y)))