\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 4.6 s
Input Error: 39.9
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\sqrt{1} \cdot \sqrt{e^{x} + 1}\)
  1. Started with
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    39.9
  2. Using strategy rm
    39.9
  3. Applied flip-- to get
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{\color{red}{e^{x} - 1}}} \leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}}}\]
    39.8
  4. Applied associate-/r/ to get
    \[\sqrt{\color{red}{\frac{e^{2 \cdot x} - 1}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}}} \leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{{\left(e^{x}\right)}^2 - {1}^2} \cdot \left(e^{x} + 1\right)}}\]
    39.8
  5. Applied sqrt-prod to get
    \[\color{red}{\sqrt{\frac{e^{2 \cdot x} - 1}{{\left(e^{x}\right)}^2 - {1}^2} \cdot \left(e^{x} + 1\right)}} \leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{{\left(e^{x}\right)}^2 - {1}^2}} \cdot \sqrt{e^{x} + 1}}\]
    39.8
  6. Applied simplify to get
    \[\color{red}{\sqrt{\frac{e^{2 \cdot x} - 1}{{\left(e^{x}\right)}^2 - {1}^2}}} \cdot \sqrt{e^{x} + 1} \leadsto \color{blue}{\sqrt{\frac{e^{x \cdot 2} - 1}{{\left(e^{x}\right)}^2 - 1}}} \cdot \sqrt{e^{x} + 1}\]
    39.8
  7. Applied simplify to get
    \[\sqrt{\color{red}{\frac{e^{x \cdot 2} - 1}{{\left(e^{x}\right)}^2 - 1}}} \cdot \sqrt{e^{x} + 1} \leadsto \sqrt{\color{blue}{1}} \cdot \sqrt{e^{x} + 1}\]
    0.0

Original test:


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