Average Error: 39.3 → 0.0
Time: 1.8s
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 r4405 = x;
        double r4406 = 1.0;
        double r4407 = r4405 + r4406;
        double r4408 = r4407 * r4407;
        double r4409 = r4408 - r4406;
        return r4409;
}
↓
double f(double x) {
        double r4410 = x;
        double r4411 = 2.0;
        double r4412 = 2.0;
        double r4413 = pow(r4410, r4412);
        double r4414 = fma(r4410, r4411, r4413);
        return r4414;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.3

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

  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 2020046 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))