Average Error: 4.2 → 0.7
Time: 10.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(e^{x + x} - 1 \cdot 1\right)}^{3}}} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(e^{x + x} - 1 \cdot 1\right)}^{3}}} \cdot \left(e^{x} + 1\right)}\\

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

\end{array}
double f(double x) {
        double r19897 = 2.0;
        double r19898 = x;
        double r19899 = r19897 * r19898;
        double r19900 = exp(r19899);
        double r19901 = 1.0;
        double r19902 = r19900 - r19901;
        double r19903 = exp(r19898);
        double r19904 = r19903 - r19901;
        double r19905 = r19902 / r19904;
        double r19906 = sqrt(r19905);
        return r19906;
}
↓
double f(double x) {
        double r19907 = x;
        double r19908 = -2.5679454952614643e-15;
        bool r19909 = r19907 <= r19908;
        double r19910 = 2.0;
        double r19911 = r19910 * r19907;
        double r19912 = exp(r19911);
        double r19913 = 1.0;
        double r19914 = r19912 - r19913;
        double r19915 = r19907 + r19907;
        double r19916 = exp(r19915);
        double r19917 = r19913 * r19913;
        double r19918 = r19916 - r19917;
        double r19919 = 3.0;
        double r19920 = pow(r19918, r19919);
        double r19921 = cbrt(r19920);
        double r19922 = r19914 / r19921;
        double r19923 = exp(r19907);
        double r19924 = r19923 + r19913;
        double r19925 = r19922 * r19924;
        double r19926 = sqrt(r19925);
        double r19927 = 0.25;
        double r19928 = 2.0;
        double r19929 = pow(r19907, r19928);
        double r19930 = sqrt(r19910);
        double r19931 = r19929 / r19930;
        double r19932 = r19927 * r19931;
        double r19933 = 0.5;
        double r19934 = r19907 / r19930;
        double r19935 = r19933 * r19934;
        double r19936 = r19930 + r19935;
        double r19937 = r19932 + r19936;
        double r19938 = 0.125;
        double r19939 = pow(r19930, r19919);
        double r19940 = r19929 / r19939;
        double r19941 = r19938 * r19940;
        double r19942 = r19937 - r19941;
        double r19943 = r19909 ? r19926 : r19942;
        return r19943;
}

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 < -2.5679454952614643e-15

    1. Initial program 0.7

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

    3. Applied flip--0.4

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

    4. Applied associate-/r/0.4

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

    5. Simplified0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Simplified0.0

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

    if -2.5679454952614643e-15 < x

    1. Initial program 38.5

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

    3. Applied flip--36.1

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

    4. Applied associate-/r/36.1

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

    5. Simplified29.1

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

    7. Applied add-cbrt-cube29.5

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

    8. Simplified29.5

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

    9. Taylor expanded around 0 7.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(e^{x + x} - 1 \cdot 1\right)}^{3}}} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))