Average Error: 0.2 → 0.0
Time: 5.3s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.56286823819809495:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 \cdot 1 - x\right) - 1} \cdot \left(1 - \sqrt{x + 1}\right)\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 5.56286823819809495:\\
\;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\

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

\end{array}
double f(double x) {
        double r135271 = x;
        double r135272 = 1.0;
        double r135273 = r135271 + r135272;
        double r135274 = sqrt(r135273);
        double r135275 = r135272 + r135274;
        double r135276 = r135271 / r135275;
        return r135276;
}
↓
double f(double x) {
        double r135277 = x;
        double r135278 = 5.562868238198095;
        bool r135279 = r135277 <= r135278;
        double r135280 = 1.0;
        double r135281 = 3.0;
        double r135282 = pow(r135280, r135281);
        double r135283 = r135277 + r135280;
        double r135284 = sqrt(r135283);
        double r135285 = pow(r135284, r135281);
        double r135286 = r135282 + r135285;
        double r135287 = r135277 / r135286;
        double r135288 = r135280 * r135280;
        double r135289 = r135284 * r135284;
        double r135290 = r135280 * r135284;
        double r135291 = r135289 - r135290;
        double r135292 = r135288 + r135291;
        double r135293 = r135287 * r135292;
        double r135294 = r135288 - r135277;
        double r135295 = r135294 - r135280;
        double r135296 = r135277 / r135295;
        double r135297 = r135280 - r135284;
        double r135298 = r135296 * r135297;
        double r135299 = r135279 ? r135293 : r135298;
        return r135299;
}

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 < 5.562868238198095

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

      \[\leadsto \frac{x}{\color{blue}{\frac{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)}}}\]

    4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)}\]

    if 5.562868238198095 < 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. Applied associate-/r/0.6

      \[\leadsto \color{blue}{\frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(1 - \sqrt{x + 1}\right)}\]

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.56286823819809495:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 \cdot 1 - x\right) - 1} \cdot \left(1 - \sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))