Average Error: 0.0 → 0.0
Time: 894.0ms
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)
double f(double x, double y) {
        double r598245 = x;
        double r598246 = r598245 * r598245;
        double r598247 = 2.0;
        double r598248 = r598245 * r598247;
        double r598249 = y;
        double r598250 = r598248 * r598249;
        double r598251 = r598246 + r598250;
        double r598252 = r598249 * r598249;
        double r598253 = r598251 + r598252;
        return r598253;
}
↓
double f(double x, double y) {
        double r598254 = y;
        double r598255 = 2.0;
        double r598256 = x;
        double r598257 = fma(r598254, r598255, r598256);
        double r598258 = r598254 * r598254;
        double r598259 = fma(r598257, r598256, r598258);
        return r598259;
}

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

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

  3. Final simplification0.0

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

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)))