Average Error: 0.0 → 0.0
Time: 3.3s
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 r371189 = x;
        double r371190 = r371189 * r371189;
        double r371191 = 2.0;
        double r371192 = r371189 * r371191;
        double r371193 = y;
        double r371194 = r371192 * r371193;
        double r371195 = r371190 + r371194;
        double r371196 = r371193 * r371193;
        double r371197 = r371195 + r371196;
        return r371197;
}
↓
double f(double x, double y) {
        double r371198 = x;
        double r371199 = y;
        double r371200 = 2.0;
        double r371201 = fma(r371199, r371200, r371198);
        double r371202 = r371199 * r371199;
        double r371203 = fma(r371198, r371201, r371202);
        return r371203;
}

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 2019306 +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)))