\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 6.6 s
Input Error: 18.3
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(e^{x} \cdot \frac{1}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x}\)
  1. Started with
    \[\frac{e^{x}}{e^{x} - 1}\]
    18.3
  2. Applied taylor to get
    \[\frac{e^{x}}{e^{x} - 1} \leadsto \frac{e^{x}}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
    5.1
  3. Taylor expanded around 0 to get
    \[\frac{e^{x}}{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
    5.1
  4. Using strategy rm
    5.1
  5. Applied div-inv to get
    \[\color{red}{\frac{e^{x}}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \color{blue}{e^{x} \cdot \frac{1}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
    5.1
  6. Applied simplify to get
    \[e^{x} \cdot \color{red}{\frac{1}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto e^{x} \cdot \color{blue}{\frac{1}{x + \left(x \cdot x\right) \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}}\]
    0.1
  7. Applied simplify to get
    \[e^{x} \cdot \frac{1}{\color{red}{x + \left(x \cdot x\right) \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}} \leadsto e^{x} \cdot \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x}}\]
    0.1

Original test:


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