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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\

\end{array}

double f(double x) {
        double r27468 = 2.0;
        double r27469 = x;
        double r27470 = r27468 * r27469;
        double r27471 = exp(r27470);
        double r27472 = 1.0;
        double r27473 = r27471 - r27472;
        double r27474 = exp(r27469);
        double r27475 = r27474 - r27472;
        double r27476 = r27473 / r27475;
        double r27477 = sqrt(r27476);
        return r27477;
}

↓

double f(double x) {
        double r27478 = x;
        double r27479 = -1.2886839166735242e-05;
        bool r27480 = r27478 <= r27479;
        double r27481 = 2.0;
        double r27482 = r27481 * r27478;
        double r27483 = exp(r27482);
        double r27484 = 1.0;
        double r27485 = r27483 - r27484;
        double r27486 = log1p(r27485);
        double r27487 = expm1(r27486);
        double r27488 = r27478 + r27478;
        double r27489 = exp(r27488);
        double r27490 = r27484 * r27484;
        double r27491 = r27489 - r27490;
        double r27492 = exp(r27478);
        double r27493 = r27492 + r27484;
        double r27494 = r27491 / r27493;
        double r27495 = r27487 / r27494;
        double r27496 = sqrt(r27495);
        double r27497 = 0.5;
        double r27498 = fma(r27497, r27478, r27484);
        double r27499 = fma(r27478, r27498, r27481);
        double r27500 = sqrt(r27499);
        double r27501 = r27480 ? r27496 : r27500;
        return r27501;
}

Derivation

Split input into 2 regimes
if x < -1.2886839166735242e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Simplified0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}}\]
5. Using strategy rm
6. Applied expm1-log1p-u0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\]
if -1.2886839166735242e-05 < x
1. Initial program 33.9
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.5
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.5
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{2 \cdot x} - 1\right)\right)}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x, 1\right), 2\right)}\\ \end{array}\]

could not determine a ground truth for program body (more)

Error

Derivation

`if x < -1.2886839166735242e-05`

`if -1.2886839166735242e-05 < x`

Reproduce