\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 4.8 s
Input Error: 18.0
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\frac{e^{x}}{{\left(\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x}\right)}^3}\)
  1. Started with
    \[\frac{e^{x}}{e^{x} - 1}\]
    18.0
  2. Applied taylor to get
    \[\frac{e^{x}}{e^{x} - 1} \leadsto \frac{e^{x}}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
    5.3
  3. Taylor expanded around 0 to get
    \[\frac{e^{x}}{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
    5.3
  4. Using strategy rm
    5.3
  5. Applied add-cube-cbrt to get
    \[\frac{e^{x}}{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \frac{e^{x}}{\color{blue}{{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}^3}}\]
    5.7
  6. Applied simplify to get
    \[\frac{e^{x}}{{\color{red}{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}}^3} \leadsto \frac{e^{x}}{{\color{blue}{\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\right)}}^3}\]
    0.5
  7. Applied simplify to get
    \[\frac{e^{x}}{{\left(\sqrt[3]{\color{red}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\right)}^3} \leadsto \frac{e^{x}}{{\left(\sqrt[3]{\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x}}\right)}^3}\]
    0.5

Original test:


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