Average Error: 0.0 → 0.0
Time: 1.9s
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 r685879 = x;
        double r685880 = r685879 * r685879;
        double r685881 = 2.0;
        double r685882 = r685879 * r685881;
        double r685883 = y;
        double r685884 = r685882 * r685883;
        double r685885 = r685880 + r685884;
        double r685886 = r685883 * r685883;
        double r685887 = r685885 + r685886;
        return r685887;
}
↓
double f(double x, double y) {
        double r685888 = x;
        double r685889 = 2.0;
        double r685890 = r685888 * r685889;
        double r685891 = y;
        double r685892 = r685890 * r685891;
        double r685893 = fma(r685888, r685888, r685892);
        double r685894 = r685891 * r685891;
        double r685895 = r685893 + r685894;
        return r685895;
}

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 2020081 +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)))