\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 10.5 s
Input Error: 60.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right) \cdot \frac{{x}^2}{\sqrt{2}}\)
  1. Started with
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    60.2
  2. Applied taylor to get
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}\]
    0.2
  3. Taylor expanded around 0 to get
    \[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}}\]
    0.2
  4. Applied simplify to get
    \[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}\]
    0.2
  5. Applied simplify to get
    \[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{red}{\frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{\left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right) \cdot \frac{{x}^2}{\sqrt{2}}}\]
    0.2

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.4.4"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))