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 r925948 = x;
        double r925949 = r925948 * r925948;
        double r925950 = 2.0;
        double r925951 = r925948 * r925950;
        double r925952 = y;
        double r925953 = r925951 * r925952;
        double r925954 = r925949 + r925953;
        double r925955 = r925952 * r925952;
        double r925956 = r925954 + r925955;
        return r925956;
}
↓
double f(double x, double y) {
        double r925957 = x;
        double r925958 = 2.0;
        double r925959 = r925957 * r925958;
        double r925960 = y;
        double r925961 = r925959 * r925960;
        double r925962 = fma(r925957, r925957, r925961);
        double r925963 = r925960 * r925960;
        double r925964 = r925962 + r925963;
        return r925964;
}

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 2020034 +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)))