Average Error: 4.6 → 1.1
Time: 7.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.19520557993145058 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x}}{\sqrt{e^{2 \cdot x}} - \sqrt{1}} - \frac{1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.19520557993145058 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x}}{\sqrt{e^{2 \cdot x}} - \sqrt{1}} - \frac{1}{\sqrt{e^{2 \cdot x}} - \sqrt{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 r24074 = 2.0;
        double r24075 = x;
        double r24076 = r24074 * r24075;
        double r24077 = exp(r24076);
        double r24078 = 1.0;
        double r24079 = r24077 - r24078;
        double r24080 = exp(r24075);
        double r24081 = r24080 - r24078;
        double r24082 = r24079 / r24081;
        double r24083 = sqrt(r24082);
        return r24083;
}
↓
double f(double x) {
        double r24084 = x;
        double r24085 = -7.195205579931451e-16;
        bool r24086 = r24084 <= r24085;
        double r24087 = 2.0;
        double r24088 = r24087 * r24084;
        double r24089 = exp(r24088);
        double r24090 = sqrt(r24089);
        double r24091 = 1.0;
        double r24092 = sqrt(r24091);
        double r24093 = r24090 + r24092;
        double r24094 = exp(r24084);
        double r24095 = r24090 - r24092;
        double r24096 = r24094 / r24095;
        double r24097 = r24091 / r24095;
        double r24098 = r24096 - r24097;
        double r24099 = r24093 / r24098;
        double r24100 = sqrt(r24099);
        double r24101 = 0.5;
        double r24102 = sqrt(r24087);
        double r24103 = r24084 / r24102;
        double r24104 = r24101 * r24103;
        double r24105 = 2.0;
        double r24106 = pow(r24084, r24105);
        double r24107 = r24106 / r24102;
        double r24108 = 0.25;
        double r24109 = 0.125;
        double r24110 = r24109 / r24087;
        double r24111 = r24108 - r24110;
        double r24112 = r24107 * r24111;
        double r24113 = r24102 + r24112;
        double r24114 = r24104 + r24113;
        double r24115 = r24086 ? r24100 : r24114;
        return r24115;
}

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 < -7.195205579931451e-16

    1. Initial program 0.7

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied add-sqr-sqrt0.6

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

    5. Applied difference-of-squares0.2

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

    6. Applied associate-/l*0.2

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

    8. Applied div-sub0.2

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

    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

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

  4. Final simplification1.1

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

Reproduce

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