Average Error: 0.0 → 0.0
Time: 939.0ms
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 r760421 = x;
        double r760422 = r760421 * r760421;
        double r760423 = 2.0;
        double r760424 = r760421 * r760423;
        double r760425 = y;
        double r760426 = r760424 * r760425;
        double r760427 = r760422 + r760426;
        double r760428 = r760425 * r760425;
        double r760429 = r760427 + r760428;
        return r760429;
}
↓
double f(double x, double y) {
        double r760430 = y;
        double r760431 = 2.0;
        double r760432 = x;
        double r760433 = fma(r760430, r760431, r760432);
        double r760434 = r760430 * r760430;
        double r760435 = fma(r760433, r760432, r760434);
        return r760435;
}

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 2020056 +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)))