\[\left(e^{x} - 2\right) + e^{-x}\]
Test:
NMSE problem 3.3.7
Bits:
128 bits
Bits error versus x
Time: 6.2 s
Input Error: 14.4
Output Error: 0.2
Log: ⚲
Profile: 🕒
\((\frac{1}{1920} * \left({x}^{5}\right) + \left((\left({x}^3\right) * \frac{1}{24} + x)_*\right))_* \cdot (\frac{1}{1920} * \left({x}^{5}\right) + \left((\left({x}^3\right) * \frac{1}{24} + x)_*\right))_*\)
  1. Started with
    \[\left(e^{x} - 2\right) + e^{-x}\]
    14.4
  2. Using strategy rm
    14.4
  3. Applied add-sqr-sqrt to get
    \[\color{red}{\left(e^{x} - 2\right) + e^{-x}} \leadsto \color{blue}{{\left(\sqrt{\left(e^{x} - 2\right) + e^{-x}}\right)}^2}\]
    14.4
  4. Applied taylor to get
    \[{\left(\sqrt{\left(e^{x} - 2\right) + e^{-x}}\right)}^2 \leadsto {\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}^2\]
    0.1
  5. Taylor expanded around 0 to get
    \[{\color{red}{\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}}^2 \leadsto {\color{blue}{\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}}^2\]
    0.1
  6. Applied simplify to get
    \[{\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}^2 \leadsto (\frac{1}{1920} * \left({x}^{5}\right) + \left((\left({x}^3\right) * \frac{1}{24} + x)_*\right))_* \cdot (\frac{1}{1920} * \left({x}^{5}\right) + \left((\left({x}^3\right) * \frac{1}{24} + x)_*\right))_*\]
    0.2
  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)))))