\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 7.5 s
Input Error: 45.3
Output Error: 7.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}} & \text{when } x \le -8.61646249982272 \cdot 10^{-12} \\ \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)} & \text{otherwise} \end{cases}\)

    if x < -8.61646249982272e-12

    1. Started with
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
      0.4
    2. Using strategy rm
      0.4
    3. Applied flip-- to get
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{\color{red}{e^{x} - 1}}} \leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}}}\]
      0.3

    if -8.61646249982272e-12 < x

    1. Started with
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
      60.5
    2. Applied taylor to get
      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}\]
      10.1
    3. Taylor expanded around 0 to get
      \[\sqrt{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}} \leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)}}\]
      10.1
  1. Removed slow pow expressions

Original test:


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