Average Error: 4.7 → 0.1
Time: 8.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.538252460152239631466209890241736190647 \cdot 10^{-15} \lor \neg \left(x \le 1.11731374600617465290906760712553421595 \cdot 10^{-16}\right):\\ \;\;\;\;\sqrt{\frac{e^{\left(2 \cdot \frac{1}{2}\right) \cdot x} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.538252460152239631466209890241736190647 \cdot 10^{-15} \lor \neg \left(x \le 1.11731374600617465290906760712553421595 \cdot 10^{-16}\right):\\
\;\;\;\;\sqrt{\frac{e^{\left(2 \cdot \frac{1}{2}\right) \cdot x} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \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 r19110 = 2.0;
        double r19111 = x;
        double r19112 = r19110 * r19111;
        double r19113 = exp(r19112);
        double r19114 = 1.0;
        double r19115 = r19113 - r19114;
        double r19116 = exp(r19111);
        double r19117 = r19116 - r19114;
        double r19118 = r19115 / r19117;
        double r19119 = sqrt(r19118);
        return r19119;
}
↓
double f(double x) {
        double r19120 = x;
        double r19121 = -2.5382524601522396e-15;
        bool r19122 = r19120 <= r19121;
        double r19123 = 1.1173137460061747e-16;
        bool r19124 = r19120 <= r19123;
        double r19125 = !r19124;
        bool r19126 = r19122 || r19125;
        double r19127 = 2.0;
        double r19128 = 0.5;
        double r19129 = r19127 * r19128;
        double r19130 = r19129 * r19120;
        double r19131 = exp(r19130);
        double r19132 = 1.0;
        double r19133 = sqrt(r19132);
        double r19134 = r19131 + r19133;
        double r19135 = exp(r19120);
        double r19136 = r19135 - r19132;
        double r19137 = exp(r19127);
        double r19138 = r19128 * r19120;
        double r19139 = pow(r19137, r19138);
        double r19140 = r19139 - r19133;
        double r19141 = r19136 / r19140;
        double r19142 = r19134 / r19141;
        double r19143 = sqrt(r19142);
        double r19144 = 0.5;
        double r19145 = sqrt(r19127);
        double r19146 = r19120 / r19145;
        double r19147 = r19144 * r19146;
        double r19148 = 2.0;
        double r19149 = pow(r19120, r19148);
        double r19150 = r19149 / r19145;
        double r19151 = 0.25;
        double r19152 = 0.125;
        double r19153 = r19152 / r19127;
        double r19154 = r19151 - r19153;
        double r19155 = r19150 * r19154;
        double r19156 = r19145 + r19155;
        double r19157 = r19147 + r19156;
        double r19158 = r19126 ? r19143 : r19157;
        return r19158;
}

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 < -2.5382524601522396e-15 or 1.1173137460061747e-16 < x

    1. Initial program 1.6

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

    3. Applied add-sqr-sqrt1.6

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

    4. Applied add-sqr-sqrt1.3

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

    5. Applied difference-of-squares0.6

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

    6. Applied associate-/l*0.6

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

    8. Applied add-log-exp0.6

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

    9. Applied exp-to-pow0.6

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

    10. Applied sqrt-pow10.1

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

    11. Simplified0.1

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

    13. Applied add-log-exp0.1

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

    14. Applied exp-to-pow0.1

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

    15. Applied sqrt-pow10.1

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

    16. Simplified0.1

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

    18. Applied pow-exp0.1

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

    19. Simplified0.1

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

    if -2.5382524601522396e-15 < x < 1.1173137460061747e-16

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.538252460152239631466209890241736190647 \cdot 10^{-15} \lor \neg \left(x \le 1.11731374600617465290906760712553421595 \cdot 10^{-16}\right):\\ \;\;\;\;\sqrt{\frac{e^{\left(2 \cdot \frac{1}{2}\right) \cdot x} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))