Average Error: 0.0 → 0.0
Time: 1.7s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)
double f(double x, double y) {
        double r669392 = x;
        double r669393 = r669392 * r669392;
        double r669394 = 2.0;
        double r669395 = r669392 * r669394;
        double r669396 = y;
        double r669397 = r669395 * r669396;
        double r669398 = r669393 + r669397;
        double r669399 = r669396 * r669396;
        double r669400 = r669398 + r669399;
        return r669400;
}
↓
double f(double x, double y) {
        double r669401 = x;
        double r669402 = y;
        double r669403 = 2.0;
        double r669404 = fma(r669402, r669403, r669401);
        double r669405 = r669402 * r669402;
        double r669406 = fma(r669401, r669404, r669405);
        return r669406;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +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)))