Average Error: 39.1 → 0.0
Time: 2.0s
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 r4955 = x;
        double r4956 = 1.0;
        double r4957 = r4955 + r4956;
        double r4958 = r4957 * r4957;
        double r4959 = r4958 - r4956;
        return r4959;
}
↓
double f(double x) {
        double r4960 = x;
        double r4961 = 2.0;
        double r4962 = 2.0;
        double r4963 = pow(r4960, r4962);
        double r4964 = fma(r4960, r4961, r4963);
        return r4964;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.1

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

  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 2019354 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))