Average Error: 0.0 → 0.0
Time: 3.7s
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 r32920281 = x;
        double r32920282 = r32920281 * r32920281;
        double r32920283 = 2.0;
        double r32920284 = r32920281 * r32920283;
        double r32920285 = y;
        double r32920286 = r32920284 * r32920285;
        double r32920287 = r32920282 + r32920286;
        double r32920288 = r32920285 * r32920285;
        double r32920289 = r32920287 + r32920288;
        return r32920289;
}
↓
double f(double x, double y) {
        double r32920290 = y;
        double r32920291 = 2.0;
        double r32920292 = x;
        double r32920293 = fma(r32920291, r32920292, r32920290);
        double r32920294 = r32920292 * r32920292;
        double r32920295 = fma(r32920290, r32920293, r32920294);
        return r32920295;
}

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 2019172 +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)))