\[\left(e^{x} - 2\right) + e^{-x}\]
Test:
NMSE problem 3.3.7
Bits:
128 bits
Bits error versus x
Time: 7.4 s
Input Error: 14.5
Output Error: 0.1
Log: ⚲
Profile: 🕒
\({\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}^2\)
  1. Started with
    \[\left(e^{x} - 2\right) + e^{-x}\]
    14.5
  2. Applied taylor to get
    \[\left(e^{x} - 2\right) + e^{-x} \leadsto {x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)\]
    2.7
  3. Taylor expanded around 0 to get
    \[\color{red}{{x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)} \leadsto \color{blue}{{x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]
    2.7
  4. Using strategy rm
    2.7
  5. Applied add-sqr-sqrt to get
    \[\color{red}{{x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)} \leadsto \color{blue}{{\left(\sqrt{{x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\right)}^2}\]
    2.7
  6. Applied taylor to get
    \[{\left(\sqrt{{x}^2 + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\right)}^2 \leadsto {\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}^2\]
    0.1
  7. 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
  8. Removed slow pow expressions

Original test:


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