Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y
double f(double x, double y) {
        double r627623 = x;
        double r627624 = r627623 * r627623;
        double r627625 = 2.0;
        double r627626 = r627623 * r627625;
        double r627627 = y;
        double r627628 = r627626 * r627627;
        double r627629 = r627624 + r627628;
        double r627630 = r627627 * r627627;
        double r627631 = r627629 + r627630;
        return r627631;
}
↓
double f(double x, double y) {
        double r627632 = x;
        double r627633 = 2.0;
        double r627634 = r627632 * r627633;
        double r627635 = y;
        double r627636 = r627634 * r627635;
        double r627637 = fma(r627632, r627632, r627636);
        double r627638 = r627635 * r627635;
        double r627639 = r627637 + r627638;
        return r627639;
}

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. Using strategy rm

  3. Applied fma-def0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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)))