Average Error: 4.6 → 0.1
Time: 28.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.813313662631737005199832057922293415686 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{elif}\;x \le 1.578024241221897350305036072937170388286 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.5 + 1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.813313662631737005199832057922293415686 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\

\mathbf{elif}\;x \le 1.578024241221897350305036072937170388286 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.5 + 1, 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\

\end{array}
double f(double x) {
        double r1125610 = 2.0;
        double r1125611 = x;
        double r1125612 = r1125610 * r1125611;
        double r1125613 = exp(r1125612);
        double r1125614 = 1.0;
        double r1125615 = r1125613 - r1125614;
        double r1125616 = exp(r1125611);
        double r1125617 = r1125616 - r1125614;
        double r1125618 = r1125615 / r1125617;
        double r1125619 = sqrt(r1125618);
        return r1125619;
}
↓
double f(double x) {
        double r1125620 = x;
        double r1125621 = -9.813313662631737e-06;
        bool r1125622 = r1125620 <= r1125621;
        double r1125623 = 1.0;
        double r1125624 = sqrt(r1125623);
        double r1125625 = 2.0;
        double r1125626 = r1125625 * r1125620;
        double r1125627 = exp(r1125626);
        double r1125628 = sqrt(r1125627);
        double r1125629 = r1125624 + r1125628;
        double r1125630 = exp(r1125620);
        double r1125631 = r1125630 - r1125623;
        double r1125632 = exp(r1125625);
        double r1125633 = 2.0;
        double r1125634 = r1125620 / r1125633;
        double r1125635 = pow(r1125632, r1125634);
        double r1125636 = r1125635 - r1125624;
        double r1125637 = r1125631 / r1125636;
        double r1125638 = r1125629 / r1125637;
        double r1125639 = sqrt(r1125638);
        double r1125640 = 1.5780242412218974e-06;
        bool r1125641 = r1125620 <= r1125640;
        double r1125642 = 0.5;
        double r1125643 = r1125620 * r1125642;
        double r1125644 = r1125643 + r1125623;
        double r1125645 = fma(r1125620, r1125644, r1125625);
        double r1125646 = sqrt(r1125645);
        double r1125647 = r1125641 ? r1125646 : r1125639;
        double r1125648 = r1125622 ? r1125639 : r1125647;
        return r1125648;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -9.813313662631737e-06 or 1.5780242412218974e-06 < x

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied difference-of-squares0.1

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

    6. Applied associate-/l*0.1

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

    8. Applied add-log-exp0.1

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

    9. Applied exp-to-pow0.1

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

    10. Applied sqrt-pow10.1

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

    if -9.813313662631737e-06 < x < 1.5780242412218974e-06

    1. Initial program 41.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.813313662631737005199832057922293415686 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{elif}\;x \le 1.578024241221897350305036072937170388286 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x \cdot 0.5 + 1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))