Average Error: 38.8 → 0.0
Time: 4.4s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[x \cdot x + x \cdot 2\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
x \cdot x + x \cdot 2
double f(double x) {
        double r10568 = x;
        double r10569 = 1.0;
        double r10570 = r10568 + r10569;
        double r10571 = r10570 * r10570;
        double r10572 = r10571 - r10569;
        return r10572;
}
↓
double f(double x) {
        double r10573 = x;
        double r10574 = r10573 * r10573;
        double r10575 = 2.0;
        double r10576 = r10573 * r10575;
        double r10577 = r10574 + r10576;
        return r10577;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.8

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

  2. Simplified38.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(1 + x, 1 + x, -1\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(x + 2\right)}\]
  5. Using strategy rm

  6. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{x \cdot x + x \cdot 2}\]

  7. Final simplification0.0

    \[\leadsto x \cdot x + x \cdot 2\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))