\[\left(e^{x} - 2\right) + e^{-x}\]
Test:
NMSE problem 3.3.7
Bits:
128 bits
Bits error versus x
Time: 8.1 s
Input Error: 14.6
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left(x + \left(\frac{1}{1920} \cdot {x}^{5} + {x}^3 \cdot \frac{1}{24}\right)\right) \cdot \left(x + \left(\frac{1}{1920} \cdot {x}^{5} + {x}^3 \cdot \frac{1}{24}\right)\right)\)
  1. Started with
    \[\left(e^{x} - 2\right) + e^{-x}\]
    14.6
  2. Using strategy rm
    14.6
  3. Applied add-sqr-sqrt to get
    \[\color{red}{\left(e^{x} - 2\right) + e^{-x}} \leadsto \color{blue}{{\left(\sqrt{\left(e^{x} - 2\right) + e^{-x}}\right)}^2}\]
    14.6
  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
    \[{\left(\frac{1}{1920} \cdot {x}^{5} + \left(\frac{1}{24} \cdot {x}^{3} + x\right)\right)}^2 \leadsto \left(x + \left(\frac{1}{1920} \cdot {x}^{5} + {x}^3 \cdot \frac{1}{24}\right)\right) \cdot \left(x + \left(\frac{1}{1920} \cdot {x}^{5} + {x}^3 \cdot \frac{1}{24}\right)\right)\]
    0.1
  7. Applied final simplification
  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)))))