\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 26.1 s
Input Error: 45.1
Output Error: 7.5
Log: ⚲
\(\begin{cases} \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} & \text{when } x \le -8.867720861083589 \cdot 10^{-13} \\ \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)} & \text{otherwise} \end{cases}\)

    if x < -8.867720861083589e-13

    1. Started with
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
      0.6

    if -8.867720861083589e-13 < x

    1. Started with
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
      60.7
    2. Applied taylor to get
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}\]
      9.9
    3. Taylor expanded around 0 to get
      \[\sqrt{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}} \leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}}\]
      9.9
  1. Removed slow pow expressions

Original test:


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