\[\frac{e^{x}}{e^{x} - 1}\]
Test:
NMSE section 3.11
Bits:
128 bits
Bits error versus x
Time: 3.5 s
Input Error: 18.0
Output Error: 2.1
Log: ⚲
Profile: 🕒
\({\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{(e^{x} - 1)^*}}\right)}^3\)
  1. Started with
    \[\frac{e^{x}}{e^{x} - 1}\]
    18.0
  2. Applied simplify to get
    \[\color{red}{\frac{e^{x}}{e^{x} - 1}} \leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]
    1.6
  3. Using strategy rm
    1.6
  4. Applied add-cube-cbrt to get
    \[\frac{e^{x}}{\color{red}{(e^{x} - 1)^*}} \leadsto \frac{e^{x}}{\color{blue}{{\left(\sqrt[3]{(e^{x} - 1)^*}\right)}^3}}\]
    2.0
  5. Applied add-cube-cbrt to get
    \[\frac{\color{red}{e^{x}}}{{\left(\sqrt[3]{(e^{x} - 1)^*}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{e^{x}}\right)}^3}}{{\left(\sqrt[3]{(e^{x} - 1)^*}\right)}^3}\]
    2.1
  6. Applied cube-undiv to get
    \[\color{red}{\frac{{\left(\sqrt[3]{e^{x}}\right)}^3}{{\left(\sqrt[3]{(e^{x} - 1)^*}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{(e^{x} - 1)^*}}\right)}^3}\]
    2.1

Original test:


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