Average Error: 0.0 → 0.0
Time: 1.4s
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 r773725 = x;
        double r773726 = r773725 * r773725;
        double r773727 = 2.0;
        double r773728 = r773725 * r773727;
        double r773729 = y;
        double r773730 = r773728 * r773729;
        double r773731 = r773726 + r773730;
        double r773732 = r773729 * r773729;
        double r773733 = r773731 + r773732;
        return r773733;
}
↓
double f(double x, double y) {
        double r773734 = x;
        double r773735 = 2.0;
        double r773736 = r773734 * r773735;
        double r773737 = y;
        double r773738 = r773736 * r773737;
        double r773739 = fma(r773734, r773734, r773738);
        double r773740 = r773737 * r773737;
        double r773741 = r773739 + r773740;
        return r773741;
}

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 2020020 +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)))