Average Error: 4.5 → 0.9
Time: 6.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.8282416705026107 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.8282416705026107 \cdot 10^{-14}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{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 r14059 = 2.0;
        double r14060 = x;
        double r14061 = r14059 * r14060;
        double r14062 = exp(r14061);
        double r14063 = 1.0;
        double r14064 = r14062 - r14063;
        double r14065 = exp(r14060);
        double r14066 = r14065 - r14063;
        double r14067 = r14064 / r14066;
        double r14068 = sqrt(r14067);
        return r14068;
}
↓
double f(double x) {
        double r14069 = x;
        double r14070 = -1.8282416705026107e-14;
        bool r14071 = r14069 <= r14070;
        double r14072 = 2.0;
        double r14073 = r14072 * r14069;
        double r14074 = exp(r14073);
        double r14075 = sqrt(r14074);
        double r14076 = 1.0;
        double r14077 = sqrt(r14076);
        double r14078 = r14075 + r14077;
        double r14079 = 3.0;
        double r14080 = pow(r14078, r14079);
        double r14081 = cbrt(r14080);
        double r14082 = 1.0;
        double r14083 = r14081 / r14082;
        double r14084 = sqrt(r14083);
        double r14085 = r14075 - r14077;
        double r14086 = exp(r14069);
        double r14087 = r14086 - r14076;
        double r14088 = r14085 / r14087;
        double r14089 = sqrt(r14088);
        double r14090 = r14084 * r14089;
        double r14091 = 0.5;
        double r14092 = sqrt(r14072);
        double r14093 = r14069 / r14092;
        double r14094 = r14091 * r14093;
        double r14095 = 2.0;
        double r14096 = pow(r14069, r14095);
        double r14097 = r14096 / r14092;
        double r14098 = 0.25;
        double r14099 = 0.125;
        double r14100 = r14099 / r14072;
        double r14101 = r14098 - r14100;
        double r14102 = r14097 * r14101;
        double r14103 = r14092 + r14102;
        double r14104 = r14094 + r14103;
        double r14105 = r14071 ? r14090 : r14104;
        return r14105;
}

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 < -1.8282416705026107e-14

    1. Initial program 0.6

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

    3. Applied *-un-lft-identity0.6

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

    4. Applied add-sqr-sqrt0.6

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

    5. Applied add-sqr-sqrt0.5

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

    6. Applied difference-of-squares0.2

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

    7. Applied times-frac0.2

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

    8. Applied sqrt-prod0.2

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

    10. Applied add-cbrt-cube0.2

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

    11. Simplified0.2

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

    if -1.8282416705026107e-14 < x

    1. Initial program 38.4

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

    2. Taylor expanded around 0 6.6

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

    3. Simplified6.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8282416705026107 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)}^{3}}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))