Average Error: 39.0 → 0.0
Time: 1.7s
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 r3232 = x;
        double r3233 = 1.0;
        double r3234 = r3232 + r3233;
        double r3235 = r3234 * r3234;
        double r3236 = r3235 - r3233;
        return r3236;
}
↓
double f(double x) {
        double r3237 = x;
        double r3238 = 2.0;
        double r3239 = 2.0;
        double r3240 = pow(r3237, r3239);
        double r3241 = fma(r3237, r3238, r3240);
        return r3241;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.0

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

  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 2020027 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))