Average Error: 38.3 → 0.0
Time: 2.4s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(x, 2, {x}^{2}\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, 2, {x}^{2}\right)
double f(double x) {
        double r64 = x;
        double r65 = 1.0;
        double r66 = r64 + r65;
        double r67 = r66 * r66;
        double r68 = r67 - r65;
        return r68;
}
↓
double f(double x) {
        double r69 = x;
        double r70 = 2.0;
        double r71 = 2.0;
        double r72 = pow(r69, r71);
        double r73 = fma(r69, r70, r72);
        return r73;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.3

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + 2 \cdot x}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))