Average Error: 0.0 → 0.0
Time: 9.1s
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 r416974 = x;
        double r416975 = r416974 * r416974;
        double r416976 = 2.0;
        double r416977 = r416974 * r416976;
        double r416978 = y;
        double r416979 = r416977 * r416978;
        double r416980 = r416975 + r416979;
        double r416981 = r416978 * r416978;
        double r416982 = r416980 + r416981;
        return r416982;
}
↓
double f(double x, double y) {
        double r416983 = x;
        double r416984 = 2.0;
        double r416985 = r416983 * r416984;
        double r416986 = y;
        double r416987 = r416985 * r416986;
        double r416988 = fma(r416983, r416983, r416987);
        double r416989 = r416986 * r416986;
        double r416990 = r416988 + r416989;
        return r416990;
}

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 2019323 +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)))