Average Error: 4.6 → 0.9
Time: 7.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.19520557993145058 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.19520557993145058 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)\right)}\\

\end{array}
double f(double x) {
        double r27009 = 2.0;
        double r27010 = x;
        double r27011 = r27009 * r27010;
        double r27012 = exp(r27011);
        double r27013 = 1.0;
        double r27014 = r27012 - r27013;
        double r27015 = exp(r27010);
        double r27016 = r27015 - r27013;
        double r27017 = r27014 / r27016;
        double r27018 = sqrt(r27017);
        return r27018;
}
↓
double f(double x) {
        double r27019 = x;
        double r27020 = -7.195205579931451e-16;
        bool r27021 = r27019 <= r27020;
        double r27022 = 2.0;
        double r27023 = r27022 * r27019;
        double r27024 = exp(r27023);
        double r27025 = 1.0;
        double r27026 = r27024 - r27025;
        double r27027 = -r27025;
        double r27028 = r27019 + r27019;
        double r27029 = exp(r27028);
        double r27030 = fma(r27027, r27025, r27029);
        double r27031 = exp(r27019);
        double r27032 = r27031 + r27025;
        double r27033 = r27030 / r27032;
        double r27034 = r27026 / r27033;
        double r27035 = sqrt(r27034);
        double r27036 = 0.5;
        double r27037 = sqrt(r27022);
        double r27038 = r27019 / r27037;
        double r27039 = r27036 * r27038;
        double r27040 = 2.0;
        double r27041 = pow(r27019, r27040);
        double r27042 = r27041 / r27037;
        double r27043 = 0.25;
        double r27044 = 0.125;
        double r27045 = r27044 / r27022;
        double r27046 = r27043 - r27045;
        double r27047 = r27042 * r27046;
        double r27048 = r27037 + r27047;
        double r27049 = log(r27048);
        double r27050 = log1p(r27049);
        double r27051 = expm1(r27050);
        double r27052 = exp(r27051);
        double r27053 = r27039 + r27052;
        double r27054 = r27021 ? r27035 : r27053;
        return r27054;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -7.195205579931451e-16

    1. Initial program 0.7

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

    3. Applied flip--0.5

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

    4. Simplified0.0

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

    if -7.195205579931451e-16 < x

    1. Initial program 37.5

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

    2. Taylor expanded around 0 8.9

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    3. Simplified8.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
    4. Using strategy rm

    5. Applied add-exp-log8.9

      \[\leadsto 0.5 \cdot \frac{x}{\sqrt{2}} + \color{blue}{e^{\log \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}}\]
    6. Using strategy rm

    7. Applied expm1-log1p-u8.9

      \[\leadsto 0.5 \cdot \frac{x}{\sqrt{2}} + e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.19520557993145058 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))