Average Error: 4.3 → 0.8
Time: 5.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.1802985128712644 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.1802985128712644 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double x) {
        double r12922 = 2.0;
        double r12923 = x;
        double r12924 = r12922 * r12923;
        double r12925 = exp(r12924);
        double r12926 = 1.0;
        double r12927 = r12925 - r12926;
        double r12928 = exp(r12923);
        double r12929 = r12928 - r12926;
        double r12930 = r12927 / r12929;
        double r12931 = sqrt(r12930);
        return r12931;
}
↓
double f(double x) {
        double r12932 = x;
        double r12933 = -3.1802985128712644e-06;
        bool r12934 = r12932 <= r12933;
        double r12935 = 2.0;
        double r12936 = r12935 * r12932;
        double r12937 = exp(r12936);
        double r12938 = sqrt(r12937);
        double r12939 = 1.0;
        double r12940 = sqrt(r12939);
        double r12941 = r12938 + r12940;
        double r12942 = 1.0;
        double r12943 = r12941 / r12942;
        double r12944 = sqrt(r12943);
        double r12945 = r12938 - r12940;
        double r12946 = exp(r12932);
        double r12947 = r12946 - r12939;
        double r12948 = r12945 / r12947;
        double r12949 = sqrt(r12948);
        double r12950 = r12944 * r12949;
        double r12951 = 0.5;
        double r12952 = sqrt(r12935);
        double r12953 = r12932 / r12952;
        double r12954 = r12951 * r12953;
        double r12955 = 2.0;
        double r12956 = pow(r12932, r12955);
        double r12957 = r12956 / r12952;
        double r12958 = 0.25;
        double r12959 = 0.125;
        double r12960 = r12959 / r12935;
        double r12961 = r12958 - r12960;
        double r12962 = r12957 * r12961;
        double r12963 = r12952 + r12962;
        double r12964 = r12954 + r12963;
        double r12965 = r12934 ? r12950 : r12964;
        return r12965;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -3.1802985128712644e-06

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.0

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

    8. Applied sqrt-prod0.0

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

    if -3.1802985128712644e-06 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 6.8

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

    3. Simplified6.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.1802985128712644 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))