Average Error: 0.0 → 0.0
Time: 13.5s
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 r27403228 = x;
        double r27403229 = r27403228 * r27403228;
        double r27403230 = 2.0;
        double r27403231 = r27403228 * r27403230;
        double r27403232 = y;
        double r27403233 = r27403231 * r27403232;
        double r27403234 = r27403229 + r27403233;
        double r27403235 = r27403232 * r27403232;
        double r27403236 = r27403234 + r27403235;
        return r27403236;
}
↓
double f(double x, double y) {
        double r27403237 = x;
        double r27403238 = y;
        double r27403239 = 2.0;
        double r27403240 = fma(r27403238, r27403239, r27403237);
        double r27403241 = r27403238 * r27403238;
        double r27403242 = fma(r27403237, r27403240, r27403241);
        return r27403242;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))