Average Error: 0.0 → 0.0
Time: 7.2s
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 r436908 = x;
        double r436909 = r436908 * r436908;
        double r436910 = 2.0;
        double r436911 = r436908 * r436910;
        double r436912 = y;
        double r436913 = r436911 * r436912;
        double r436914 = r436909 + r436913;
        double r436915 = r436912 * r436912;
        double r436916 = r436914 + r436915;
        return r436916;
}
↓
double f(double x, double y) {
        double r436917 = x;
        double r436918 = 2.0;
        double r436919 = r436917 * r436918;
        double r436920 = y;
        double r436921 = r436919 * r436920;
        double r436922 = fma(r436917, r436917, r436921);
        double r436923 = r436920 * r436920;
        double r436924 = r436922 + r436923;
        return r436924;
}

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 2019323 +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)))