\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 5.1 s
Input Error: 45.5
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{1 - e^{-x}} & \text{when } x \le -4.037641697290413 \cdot 10^{-05} \\ \frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right) & \text{otherwise} \end{cases}\)

    if x < -4.037641697290413e-05

    1. Started with
      \[\frac{e^{x}}{e^{x} - 1}\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied clear-num to get
      \[\color{red}{\frac{e^{x}}{e^{x} - 1}} \leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
      0.1
    4. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{e^{x} - 1}{e^{x}}}} \leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]
      0.1

    if -4.037641697290413e-05 < x

    1. Started with
      \[\frac{e^{x}}{e^{x} - 1}\]
      60.6
    2. Applied taylor to get
      \[\frac{e^{x}}{e^{x} - 1} \leadsto \frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\]
      0.1
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)} \leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
      0.1
  1. Removed slow pow expressions

Original test:


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