Average Error: 38.9 → 0.0
Time: 1.5s
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 r3182 = x;
        double r3183 = 1.0;
        double r3184 = r3182 + r3183;
        double r3185 = r3184 * r3184;
        double r3186 = r3185 - r3183;
        return r3186;
}
↓
double f(double x) {
        double r3187 = x;
        double r3188 = 2.0;
        double r3189 = 2.0;
        double r3190 = pow(r3187, r3189);
        double r3191 = fma(r3187, r3188, r3190);
        return r3191;
}

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 2020056 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))