\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 3.8 s
Input Error: 18.2
Output Error: 1.8
Log: ⚲
Profile: 🕒
\(\frac{e^{x}}{\log_* (1 + (e^{(e^{x} - 1)^*} - 1)^*)}\)
  1. Started with
    \[\frac{e^{x}}{e^{x} - 1}\]
    18.2
  2. Applied simplify to get
    \[\color{red}{\frac{e^{x}}{e^{x} - 1}} \leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]
    1.8
  3. Using strategy rm
    1.8
  4. Applied log1p-expm1-u to get
    \[\frac{e^{x}}{\color{red}{(e^{x} - 1)^*}} \leadsto \frac{e^{x}}{\color{blue}{\log_* (1 + (e^{(e^{x} - 1)^*} - 1)^*)}}\]
    1.8

Original test:


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