Average Error: 4.6 → 1.0
Time: 15.9s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-906485597919399}{9444732965739290427392}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-906485597919399}{9444732965739290427392}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)\right)\\

\end{array}
double f(double x) {
        double r34076 = 2.0;
        double r34077 = x;
        double r34078 = r34076 * r34077;
        double r34079 = exp(r34078);
        double r34080 = 1.0;
        double r34081 = r34079 - r34080;
        double r34082 = exp(r34077);
        double r34083 = r34082 - r34080;
        double r34084 = r34081 / r34083;
        double r34085 = sqrt(r34084);
        return r34085;
}
↓
double f(double x) {
        double r34086 = x;
        double r34087 = -906485597919399.0;
        double r34088 = 9.44473296573929e+21;
        double r34089 = r34087 / r34088;
        bool r34090 = r34086 <= r34089;
        double r34091 = 2.0;
        double r34092 = r34091 * r34086;
        double r34093 = exp(r34092);
        double r34094 = sqrt(r34093);
        double r34095 = 1.0;
        double r34096 = sqrt(r34095);
        double r34097 = r34094 + r34096;
        double r34098 = r34094 - r34096;
        double r34099 = r34097 * r34098;
        double r34100 = exp(r34086);
        double r34101 = r34100 - r34095;
        double r34102 = 3.0;
        double r34103 = pow(r34101, r34102);
        double r34104 = cbrt(r34103);
        double r34105 = r34099 / r34104;
        double r34106 = sqrt(r34105);
        double r34107 = r34095 / r34091;
        double r34108 = sqrt(r34091);
        double r34109 = r34086 / r34108;
        double r34110 = r34107 * r34109;
        double r34111 = 2.0;
        double r34112 = pow(r34086, r34111);
        double r34113 = r34112 / r34108;
        double r34114 = 4.0;
        double r34115 = r34095 / r34114;
        double r34116 = 8.0;
        double r34117 = r34095 / r34116;
        double r34118 = r34117 / r34091;
        double r34119 = r34115 - r34118;
        double r34120 = r34113 * r34119;
        double r34121 = r34108 + r34120;
        double r34122 = r34110 + r34121;
        double r34123 = r34090 ? r34106 : r34122;
        return r34123;
}

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 < -9.597789595615565e-08

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.0

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

    7. Applied add-cbrt-cube0.0

      \[\leadsto \sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}}\]

    8. Simplified0.0

      \[\leadsto \sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt[3]{\color{blue}{{\left(e^{x} - 1\right)}^{3}}}}}\]

    if -9.597789595615565e-08 < x

    1. Initial program 35.1

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

    2. Taylor expanded around 0 7.8

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

    3. Simplified7.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-906485597919399}{9444732965739290427392}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))