Average Error: 0.0 → 0.0
Time: 6.0s
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 r413269 = x;
        double r413270 = r413269 * r413269;
        double r413271 = 2.0;
        double r413272 = r413269 * r413271;
        double r413273 = y;
        double r413274 = r413272 * r413273;
        double r413275 = r413270 + r413274;
        double r413276 = r413273 * r413273;
        double r413277 = r413275 + r413276;
        return r413277;
}
↓
double f(double x, double y) {
        double r413278 = x;
        double r413279 = 2.0;
        double r413280 = r413278 * r413279;
        double r413281 = y;
        double r413282 = r413280 * r413281;
        double r413283 = fma(r413278, r413278, r413282);
        double r413284 = r413281 * r413281;
        double r413285 = r413283 + r413284;
        return r413285;
}

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 2019305 +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)))