Average Error: 0.0 → 0.0
Time: 2.9s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)
double f(double x, double y) {
        double r22868223 = x;
        double r22868224 = r22868223 * r22868223;
        double r22868225 = 2.0;
        double r22868226 = r22868223 * r22868225;
        double r22868227 = y;
        double r22868228 = r22868226 * r22868227;
        double r22868229 = r22868224 + r22868228;
        double r22868230 = r22868227 * r22868227;
        double r22868231 = r22868229 + r22868230;
        return r22868231;
}
↓
double f(double x, double y) {
        double r22868232 = y;
        double r22868233 = 2.0;
        double r22868234 = x;
        double r22868235 = fma(r22868233, r22868234, r22868232);
        double r22868236 = r22868234 * r22868234;
        double r22868237 = fma(r22868232, r22868235, r22868236);
        return r22868237;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))