Average Error: 4.5 → 0.4
Time: 12.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.608732958076172508184858989110921356769 \cdot 10^{-5} \lor \neg \left(x \le 1.324600673656064518905561606948045724394 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.608732958076172508184858989110921356769 \cdot 10^{-5} \lor \neg \left(x \le 1.324600673656064518905561606948045724394 \cdot 10^{-14}\right):\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r39248 = 2.0;
        double r39249 = x;
        double r39250 = r39248 * r39249;
        double r39251 = exp(r39250);
        double r39252 = 1.0;
        double r39253 = r39251 - r39252;
        double r39254 = exp(r39249);
        double r39255 = r39254 - r39252;
        double r39256 = r39253 / r39255;
        double r39257 = sqrt(r39256);
        return r39257;
}
↓
double f(double x) {
        double r39258 = x;
        double r39259 = -3.6087329580761725e-05;
        bool r39260 = r39258 <= r39259;
        double r39261 = 1.3246006736560645e-14;
        bool r39262 = r39258 <= r39261;
        double r39263 = !r39262;
        bool r39264 = r39260 || r39263;
        double r39265 = 2.0;
        double r39266 = r39265 * r39258;
        double r39267 = exp(r39266);
        double r39268 = sqrt(r39267);
        double r39269 = 1.0;
        double r39270 = sqrt(r39269);
        double r39271 = r39268 + r39270;
        double r39272 = 3.0;
        double r39273 = pow(r39271, r39272);
        double r39274 = cbrt(r39273);
        double r39275 = exp(r39258);
        double r39276 = r39275 - r39269;
        double r39277 = r39268 - r39270;
        double r39278 = r39276 / r39277;
        double r39279 = r39274 / r39278;
        double r39280 = sqrt(r39279);
        double r39281 = 0.5;
        double r39282 = r39281 * r39258;
        double r39283 = r39269 + r39282;
        double r39284 = r39258 * r39283;
        double r39285 = r39284 + r39265;
        double r39286 = sqrt(r39285);
        double r39287 = r39264 ? r39280 : r39286;
        return r39287;
}

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 < -3.6087329580761725e-05 or 1.3246006736560645e-14 < x

    1. Initial program 0.8

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.8

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{e^{x} - 1}}\]

    4. Applied add-sqr-sqrt0.8

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{e^{x} - 1}}\]

    5. Applied difference-of-squares0.3

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]

    6. Applied associate-/l*0.3

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube0.4

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt[3]{\left(\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)\right) \cdot \left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\]

    9. Simplified0.4

      \[\leadsto \sqrt{\frac{\sqrt[3]{\color{blue}{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\]

    if -3.6087329580761725e-05 < x < 1.3246006736560645e-14

    1. Initial program 45.2

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified0.1

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.608732958076172508184858989110921356769 \cdot 10^{-5} \lor \neg \left(x \le 1.324600673656064518905561606948045724394 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))