Average Error: 38.9 → 0.0
Time: 2.2s
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 r4944 = x;
        double r4945 = 1.0;
        double r4946 = r4944 + r4945;
        double r4947 = r4946 * r4946;
        double r4948 = r4947 - r4945;
        return r4948;
}
↓
double f(double x) {
        double r4949 = x;
        double r4950 = 2.0;
        double r4951 = 2.0;
        double r4952 = pow(r4949, r4951);
        double r4953 = fma(r4949, r4950, r4952);
        return r4953;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.9

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

  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 2020100 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))