Average Error: 0.0 → 0.0
Time: 6.1s
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 r426427 = x;
        double r426428 = r426427 * r426427;
        double r426429 = 2.0;
        double r426430 = r426427 * r426429;
        double r426431 = y;
        double r426432 = r426430 * r426431;
        double r426433 = r426428 + r426432;
        double r426434 = r426431 * r426431;
        double r426435 = r426433 + r426434;
        return r426435;
}
↓
double f(double x, double y) {
        double r426436 = x;
        double r426437 = y;
        double r426438 = 2.0;
        double r426439 = fma(r426437, r426438, r426436);
        double r426440 = r426437 * r426437;
        double r426441 = fma(r426436, r426439, r426440);
        return r426441;
}

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 2019325 +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)))