\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 6.4 s
Input Error: 60.3
Output Error: 15.1
Log: ⚲
Profile: 🕒
\(\sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}\)
  1. Started with
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    60.3
  2. Applied taylor to get
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}\]
    15.1
  3. Taylor expanded around 0 to get
    \[\sqrt{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}} \leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}}\]
    15.1

Original test:


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