\[\left(e^{x} - 2\right) + e^{-x}\]
Test:
NMSE problem 3.3.7
Bits:
128 bits
Bits error versus x
Time: 8.2 s
Input Error: 14.8
Output Error: 2.6
Log: ⚲
Profile: 🕒
\({\left(e^{\frac{1}{12}}\right)}^{\left(x \cdot x\right)} \cdot \frac{{x}^{2}}{{\left(e^{\frac{1}{1440}}\right)}^{\left({x}^{4}\right)}}\)
  1. Started with
    \[\left(e^{x} - 2\right) + e^{-x}\]
    14.8
  2. Using strategy rm
    14.8
  3. Applied add-exp-log to get
    \[\color{red}{\left(e^{x} - 2\right) + e^{-x}} \leadsto \color{blue}{e^{\log \left(\left(e^{x} - 2\right) + e^{-x}\right)}}\]
    14.8
  4. Applied taylor to get
    \[e^{\log \left(\left(e^{x} - 2\right) + e^{-x}\right)} \leadsto e^{\left(\frac{1}{12} \cdot {x}^2 + 2 \cdot \log x\right) - \frac{1}{1440} \cdot {x}^{4}}\]
    16.2
  5. Taylor expanded around 0 to get
    \[e^{\color{red}{\left(\frac{1}{12} \cdot {x}^2 + 2 \cdot \log x\right) - \frac{1}{1440} \cdot {x}^{4}}} \leadsto e^{\color{blue}{\left(\frac{1}{12} \cdot {x}^2 + 2 \cdot \log x\right) - \frac{1}{1440} \cdot {x}^{4}}}\]
    16.2
  6. Applied simplify to get
    \[e^{\left(\frac{1}{12} \cdot {x}^2 + 2 \cdot \log x\right) - \frac{1}{1440} \cdot {x}^{4}} \leadsto {\left(e^{\frac{1}{12}}\right)}^{\left(x \cdot x\right)} \cdot \frac{{x}^{2}}{{\left(e^{\frac{1}{1440}}\right)}^{\left({x}^{4}\right)}}\]
    2.6
  7. Applied final simplification

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.7"
  (+ (- (exp x) 2) (exp (- x)))
  #:target
  (* 4 (sqr (sinh (/ x 2)))))