Average Error: 39.4 → 0.0
Time: 2.0s
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 r4749 = x;
        double r4750 = 1.0;
        double r4751 = r4749 + r4750;
        double r4752 = r4751 * r4751;
        double r4753 = r4752 - r4750;
        return r4753;
}
↓
double f(double x) {
        double r4754 = x;
        double r4755 = 2.0;
        double r4756 = 2.0;
        double r4757 = pow(r4754, r4756);
        double r4758 = fma(r4754, r4755, r4757);
        return r4758;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.4

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

  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 2019356 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))