Average Error: 39.2 → 0.0
Time: 2.9s
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 r16346 = x;
        double r16347 = 1.0;
        double r16348 = r16346 + r16347;
        double r16349 = r16348 * r16348;
        double r16350 = r16349 - r16347;
        return r16350;
}
↓
double f(double x) {
        double r16351 = x;
        double r16352 = 2.0;
        double r16353 = 2.0;
        double r16354 = pow(r16351, r16353);
        double r16355 = fma(r16351, r16352, r16354);
        return r16355;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.2

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

  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 2020059 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))