Average Error: 0.0 → 0.0
Time: 7.5s
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 r444255 = x;
        double r444256 = r444255 * r444255;
        double r444257 = 2.0;
        double r444258 = r444255 * r444257;
        double r444259 = y;
        double r444260 = r444258 * r444259;
        double r444261 = r444256 + r444260;
        double r444262 = r444259 * r444259;
        double r444263 = r444261 + r444262;
        return r444263;
}
↓
double f(double x, double y) {
        double r444264 = x;
        double r444265 = 2.0;
        double r444266 = r444264 * r444265;
        double r444267 = y;
        double r444268 = r444266 * r444267;
        double r444269 = fma(r444264, r444264, r444268);
        double r444270 = r444267 * r444267;
        double r444271 = r444269 + r444270;
        return r444271;
}

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)))