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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.284857974241761 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -4.284857974241761 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}}\\

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

\end{array}

double f(double x) {
        double r13584 = 2.0;
        double r13585 = x;
        double r13586 = r13584 * r13585;
        double r13587 = exp(r13586);
        double r13588 = 1.0;
        double r13589 = r13587 - r13588;
        double r13590 = exp(r13585);
        double r13591 = r13590 - r13588;
        double r13592 = r13589 / r13591;
        double r13593 = sqrt(r13592);
        return r13593;
}

↓

double f(double x) {
        double r13594 = x;
        double r13595 = -4.2848579742417606e-11;
        bool r13596 = r13594 <= r13595;
        double r13597 = 2.0;
        double r13598 = r13597 * r13594;
        double r13599 = exp(r13598);
        double r13600 = 1.0;
        double r13601 = r13599 - r13600;
        double r13602 = -r13600;
        double r13603 = r13594 + r13594;
        double r13604 = exp(r13603);
        double r13605 = fma(r13602, r13600, r13604);
        double r13606 = r13601 / r13605;
        double r13607 = exp(r13594);
        double r13608 = 3.0;
        double r13609 = pow(r13607, r13608);
        double r13610 = pow(r13600, r13608);
        double r13611 = r13609 + r13610;
        double r13612 = r13600 - r13607;
        double r13613 = 2.0;
        double r13614 = r13613 * r13594;
        double r13615 = exp(r13614);
        double r13616 = fma(r13600, r13612, r13615);
        double r13617 = r13611 / r13616;
        double r13618 = r13606 * r13617;
        double r13619 = sqrt(r13618);
        double r13620 = 0.5;
        double r13621 = pow(r13594, r13613);
        double r13622 = fma(r13600, r13594, r13597);
        double r13623 = fma(r13620, r13621, r13622);
        double r13624 = sqrt(r13623);
        double r13625 = r13596 ? r13619 : r13624;
        return r13625;
}

Derivation

Split input into 2 regimes
if x < -4.2848579742417606e-11
1. Initial program 0.4
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.2
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.2
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
6. Using strategy rm
7. Applied flip3-+0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]
8. Simplified0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}}}\]
if -4.2848579742417606e-11 < x
1. Initial program 34.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.6
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified7.6
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.284857974241761 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Error

Derivation

`if x < -4.2848579742417606e-11`

`if -4.2848579742417606e-11 < x`

Reproduce