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 r27005 = 2.0;
        double r27006 = x;
        double r27007 = r27005 * r27006;
        double r27008 = exp(r27007);
        double r27009 = 1.0;
        double r27010 = r27008 - r27009;
        double r27011 = exp(r27006);
        double r27012 = r27011 - r27009;
        double r27013 = r27010 / r27012;
        double r27014 = sqrt(r27013);
        return r27014;
}
↓
double f(double x) {
        double r27015 = x;
        double r27016 = -7.195205579931451e-16;
        bool r27017 = r27015 <= r27016;
        double r27018 = 2.0;
        double r27019 = r27018 * r27015;
        double r27020 = exp(r27019);
        double r27021 = 1.0;
        double r27022 = r27020 - r27021;
        double r27023 = -r27021;
        double r27024 = r27015 + r27015;
        double r27025 = exp(r27024);
        double r27026 = fma(r27023, r27021, r27025);
        double r27027 = exp(r27015);
        double r27028 = r27027 + r27021;
        double r27029 = r27026 / r27028;
        double r27030 = r27022 / r27029;
        double r27031 = sqrt(r27030);
        double r27032 = 0.5;
        double r27033 = sqrt(r27018);
        double r27034 = r27015 / r27033;
        double r27035 = r27032 * r27034;
        double r27036 = 2.0;
        double r27037 = pow(r27015, r27036);
        double r27038 = r27037 / r27033;
        double r27039 = 0.25;
        double r27040 = 0.125;
        double r27041 = r27040 / r27018;
        double r27042 = r27039 - r27041;
        double r27043 = r27038 * r27042;
        double r27044 = r27033 + r27043;
        double r27045 = log(r27044);
        double r27046 = log1p(r27045);
        double r27047 = expm1(r27046);
        double r27048 = exp(r27047);
        double r27049 = r27035 + r27048;
        double r27050 = r27017 ? r27031 : r27049;
        return r27050;
}

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))))