Average Error: 0.2 → 0.0
Time: 21.2s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.76990929119128312565564679437277867402 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)} \cdot \left(1 \cdot 1\right) + \frac{\left(\sqrt{1 + x} \cdot \left(\sqrt{1 + x} - 1\right)\right) \cdot x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{1 + x} + 1}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.76990929119128312565564679437277867402 \cdot 10^{-9}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)} \cdot \left(1 \cdot 1\right) + \frac{\left(\sqrt{1 + x} \cdot \left(\sqrt{1 + x} - 1\right)\right) \cdot x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{1 + x} + 1}\\

\end{array}
double f(double x) {
        double r6132150 = x;
        double r6132151 = 1.0;
        double r6132152 = r6132150 + r6132151;
        double r6132153 = sqrt(r6132152);
        double r6132154 = r6132151 + r6132153;
        double r6132155 = r6132150 / r6132154;
        return r6132155;
}
↓
double f(double x) {
        double r6132156 = x;
        double r6132157 = 2.769909291191283e-09;
        bool r6132158 = r6132156 <= r6132157;
        double r6132159 = 1.0;
        double r6132160 = r6132159 + r6132156;
        double r6132161 = sqrt(r6132160);
        double r6132162 = r6132159 * r6132159;
        double r6132163 = r6132159 * r6132162;
        double r6132164 = fma(r6132160, r6132161, r6132163);
        double r6132165 = r6132156 / r6132164;
        double r6132166 = r6132165 * r6132162;
        double r6132167 = r6132161 - r6132159;
        double r6132168 = r6132161 * r6132167;
        double r6132169 = r6132168 * r6132156;
        double r6132170 = r6132169 / r6132164;
        double r6132171 = r6132166 + r6132170;
        double r6132172 = sqrt(r6132156);
        double r6132173 = r6132161 + r6132159;
        double r6132174 = r6132172 / r6132173;
        double r6132175 = r6132172 * r6132174;
        double r6132176 = r6132158 ? r6132171 : r6132175;
        return r6132176;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 2.769909291191283e-09

    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

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, \left(1 \cdot 1\right) \cdot 1\right)}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
    6. Using strategy rm

    7. Applied distribute-lft-in0.0

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

    8. Simplified0.0

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

    if 2.769909291191283e-09 < x

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied times-frac0.1

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

    6. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.76990929119128312565564679437277867402 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)} \cdot \left(1 \cdot 1\right) + \frac{\left(\sqrt{1 + x} \cdot \left(\sqrt{1 + x} - 1\right)\right) \cdot x}{\mathsf{fma}\left(1 + x, \sqrt{1 + x}, 1 \cdot \left(1 \cdot 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{1 + x} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))