Average Error: 0.0 → 0.0
Time: 926.0ms
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\mathsf{fma}\left(y, 2, x\right), x, y \cdot y\right)
double f(double x, double y) {
        double r583309 = x;
        double r583310 = r583309 * r583309;
        double r583311 = 2.0;
        double r583312 = r583309 * r583311;
        double r583313 = y;
        double r583314 = r583312 * r583313;
        double r583315 = r583310 + r583314;
        double r583316 = r583313 * r583313;
        double r583317 = r583315 + r583316;
        return r583317;
}
↓
double f(double x, double y) {
        double r583318 = y;
        double r583319 = 2.0;
        double r583320 = x;
        double r583321 = fma(r583318, r583319, r583320);
        double r583322 = r583318 * r583318;
        double r583323 = fma(r583321, r583320, r583322);
        return r583323;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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)))