Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, \mathsf{fma}\left(2, x, y\right), x \cdot x\right)
double f(double x, double y) {
        double r25733020 = x;
        double r25733021 = r25733020 * r25733020;
        double r25733022 = 2.0;
        double r25733023 = r25733020 * r25733022;
        double r25733024 = y;
        double r25733025 = r25733023 * r25733024;
        double r25733026 = r25733021 + r25733025;
        double r25733027 = r25733024 * r25733024;
        double r25733028 = r25733026 + r25733027;
        return r25733028;
}
↓
double f(double x, double y) {
        double r25733029 = y;
        double r25733030 = 2.0;
        double r25733031 = x;
        double r25733032 = fma(r25733030, r25733031, r25733029);
        double r25733033 = r25733031 * r25733031;
        double r25733034 = fma(r25733029, r25733032, r25733033);
        return r25733034;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))