\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 1.8 s
Input Error: 27.8
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\)
  1. Started with
    \[\frac{e^{x}}{e^{x} - 1}\]
    27.8
  2. Applied taylor to get
    \[\frac{e^{x}}{e^{x} - 1} \leadsto \frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\]
    0.2
  3. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)} \leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
    0.2

Original test:


(lambda ((x default))
  #:name "NMSE section 3.11"
  (/ (exp x) (- (exp x) 1))
  #:target
  (/ 1 (- 1 (exp (- x)))))