\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 4.5 s
Input Error: 16.5
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\sqrt{\sqrt[3]{{\left(\frac{(e^{2 \cdot x} - 1)^*}{(e^{x} - 1)^*}\right)}^3}}\)
  1. Started with
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    16.5
  2. Applied simplify to get
    \[\color{red}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \leadsto \color{blue}{\sqrt{\frac{(e^{x \cdot 2} - 1)^*}{(e^{x} - 1)^*}}}\]
    0.2
  3. Using strategy rm
    0.2
  4. Applied add-cbrt-cube to get
    \[\sqrt{\frac{(e^{x \cdot 2} - 1)^*}{\color{red}{(e^{x} - 1)^*}}} \leadsto \sqrt{\frac{(e^{x \cdot 2} - 1)^*}{\color{blue}{\sqrt[3]{{\left((e^{x} - 1)^*\right)}^3}}}}\]
    12.2
  5. Applied add-cbrt-cube to get
    \[\sqrt{\frac{\color{red}{(e^{x \cdot 2} - 1)^*}}{\sqrt[3]{{\left((e^{x} - 1)^*\right)}^3}}} \leadsto \sqrt{\frac{\color{blue}{\sqrt[3]{{\left((e^{x \cdot 2} - 1)^*\right)}^3}}}{\sqrt[3]{{\left((e^{x} - 1)^*\right)}^3}}}\]
    12.3
  6. Applied cbrt-undiv to get
    \[\sqrt{\color{red}{\frac{\sqrt[3]{{\left((e^{x \cdot 2} - 1)^*\right)}^3}}{\sqrt[3]{{\left((e^{x} - 1)^*\right)}^3}}}} \leadsto \sqrt{\color{blue}{\sqrt[3]{\frac{{\left((e^{x \cdot 2} - 1)^*\right)}^3}{{\left((e^{x} - 1)^*\right)}^3}}}}\]
    0.2
  7. Applied simplify to get
    \[\sqrt{\sqrt[3]{\color{red}{\frac{{\left((e^{x \cdot 2} - 1)^*\right)}^3}{{\left((e^{x} - 1)^*\right)}^3}}}} \leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(\frac{(e^{2 \cdot x} - 1)^*}{(e^{x} - 1)^*}\right)}^3}}}\]
    0.2

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.4.4"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))