\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Test:
NMSE problem 3.4.4
Bits:
128 bits
Bits error versus x
Time: 3.7 s
Input Error: 16.6
Output Error: 0.2
Log: ⚲
Profile: 🕒
\((e^{\log_* (1 + \sqrt{\frac{(e^{x \cdot 2} - 1)^*}{(e^{x} - 1)^*}})} - 1)^*\)
  1. Started with
    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    16.6
  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 expm1-log1p-u to get
    \[\color{red}{\sqrt{\frac{(e^{x \cdot 2} - 1)^*}{(e^{x} - 1)^*}}} \leadsto \color{blue}{(e^{\log_* (1 + \sqrt{\frac{(e^{x \cdot 2} - 1)^*}{(e^{x} - 1)^*}})} - 1)^*}\]
    0.2
  5. Removed slow pow expressions

Original test:


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