Average Error: 0.0 → 0.0
Time: 2.3s
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 r659226 = x;
        double r659227 = r659226 * r659226;
        double r659228 = 2.0;
        double r659229 = r659226 * r659228;
        double r659230 = y;
        double r659231 = r659229 * r659230;
        double r659232 = r659227 + r659231;
        double r659233 = r659230 * r659230;
        double r659234 = r659232 + r659233;
        return r659234;
}
↓
double f(double x, double y) {
        double r659235 = x;
        double r659236 = 2.0;
        double r659237 = r659235 * r659236;
        double r659238 = y;
        double r659239 = r659237 * r659238;
        double r659240 = fma(r659235, r659235, r659239);
        double r659241 = r659238 * r659238;
        double r659242 = r659240 + r659241;
        return r659242;
}

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 2020036 +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)))