Average Error: 0.2 → 0.0
Time: 10.6s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.4582407169354019704776703747484134510159:\\ \;\;\;\;\frac{x}{1 + \log \left(e^{\sqrt{x + 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1 \cdot 1 - \left(x + 1\right)}{1 - \sqrt{x + 1}}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.4582407169354019704776703747484134510159:\\
\;\;\;\;\frac{x}{1 + \log \left(e^{\sqrt{x + 1}}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{1 \cdot 1 - \left(x + 1\right)}{1 - \sqrt{x + 1}}}\\

\end{array}
double f(double x) {
        double r11596426 = x;
        double r11596427 = 1.0;
        double r11596428 = r11596426 + r11596427;
        double r11596429 = sqrt(r11596428);
        double r11596430 = r11596427 + r11596429;
        double r11596431 = r11596426 / r11596430;
        return r11596431;
}
↓
double f(double x) {
        double r11596432 = x;
        double r11596433 = 0.45824071693540197;
        bool r11596434 = r11596432 <= r11596433;
        double r11596435 = 1.0;
        double r11596436 = r11596432 + r11596435;
        double r11596437 = sqrt(r11596436);
        double r11596438 = exp(r11596437);
        double r11596439 = log(r11596438);
        double r11596440 = r11596435 + r11596439;
        double r11596441 = r11596432 / r11596440;
        double r11596442 = r11596435 * r11596435;
        double r11596443 = r11596442 - r11596436;
        double r11596444 = r11596435 - r11596437;
        double r11596445 = r11596443 / r11596444;
        double r11596446 = r11596432 / r11596445;
        double r11596447 = r11596434 ? r11596441 : r11596446;
        return r11596447;
}

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 < 0.45824071693540197

    1. Initial program 0.0

      \[\frac{x}{1 + \sqrt{x + 1}}\]
    2. Using strategy rm

    3. Applied add-log-exp0.0

      \[\leadsto \frac{x}{1 + \color{blue}{\log \left(e^{\sqrt{x + 1}}\right)}}\]

    if 0.45824071693540197 < x

    1. Initial program 0.5

      \[\frac{x}{1 + \sqrt{x + 1}}\]
    2. Using strategy rm

    3. Applied flip-+0.5

      \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 - \sqrt{x + 1}}}}\]

    4. Simplified0.0

      \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot 1 - \left(x + 1\right)}}{1 - \sqrt{x + 1}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.4582407169354019704776703747484134510159:\\ \;\;\;\;\frac{x}{1 + \log \left(e^{\sqrt{x + 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1 \cdot 1 - \left(x + 1\right)}{1 - \sqrt{x + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))