Average Error: 0.0 → 0.0
Time: 9.4s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, x\right), y \cdot y\right)
double f(double x, double y) {
        double r818092 = x;
        double r818093 = r818092 * r818092;
        double r818094 = 2.0;
        double r818095 = r818092 * r818094;
        double r818096 = y;
        double r818097 = r818095 * r818096;
        double r818098 = r818093 + r818097;
        double r818099 = r818096 * r818096;
        double r818100 = r818098 + r818099;
        return r818100;
}
↓
double f(double x, double y) {
        double r818101 = x;
        double r818102 = y;
        double r818103 = 2.0;
        double r818104 = fma(r818102, r818103, r818101);
        double r818105 = r818102 * r818102;
        double r818106 = fma(r818101, r818104, r818105);
        return r818106;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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)))