Average Error: 38.7 → 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 r3351 = x;
        double r3352 = 1.0;
        double r3353 = r3351 + r3352;
        double r3354 = r3353 * r3353;
        double r3355 = r3354 - r3352;
        return r3355;
}
↓
double f(double x) {
        double r3356 = x;
        double r3357 = 2.0;
        double r3358 = 2.0;
        double r3359 = pow(r3356, r3358);
        double r3360 = fma(r3356, r3357, r3359);
        return r3360;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.7

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

  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 2019352 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))