Average Error: 45.4 → 0
Time: 10.0s
Precision: 64
\[\sqrt{\left(x - 1\right) \cdot \left(x + 1\right) + 1}\]
\[\begin{array}{l} \mathbf{if}\;x \le 6.006836929901048229584203904238188713098 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x - 1\right) \cdot \left(x + 1\right) + 1}
\begin{array}{l}
\mathbf{if}\;x \le 6.006836929901048229584203904238188713098 \cdot 10^{-310}:\\
\;\;\;\;-x\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x) {
        double r54884532 = x;
        double r54884533 = 1.0;
        double r54884534 = r54884532 - r54884533;
        double r54884535 = r54884532 + r54884533;
        double r54884536 = r54884534 * r54884535;
        double r54884537 = r54884536 + r54884533;
        double r54884538 = sqrt(r54884537);
        return r54884538;
}

double f(double x) {
        double r54884539 = x;
        double r54884540 = 6.00683692990105e-310;
        bool r54884541 = r54884539 <= r54884540;
        double r54884542 = -r54884539;
        double r54884543 = r54884541 ? r54884542 : r54884539;
        return r54884543;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < 6.00683692990105e-310

    1. Initial program 45.6

      \[\sqrt{\left(x - 1\right) \cdot \left(x + 1\right) + 1}\]
    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-1 \cdot x}\]
    3. Simplified0

      \[\leadsto \color{blue}{-x}\]

    if 6.00683692990105e-310 < x

    1. Initial program 45.3

      \[\sqrt{\left(x - 1\right) \cdot \left(x + 1\right) + 1}\]
    2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{x}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.006836929901048229584203904238188713098 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt((x-1)*(x+1)+1)"
  (sqrt (+ (* (- x 1.0) (+ x 1.0)) 1.0)))