\[e^{x} - 1\]
Test:
NMSE example 3.7
Bits:
128 bits
Bits error versus x
Time: 4.1 s
Input Error: 26.2
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x\)
  1. Started with
    \[e^{x} - 1\]
    26.2
  2. Applied taylor to get
    \[e^{x} - 1 \leadsto \frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)\]
    0.2
  3. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)} \leadsto \color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
    0.2
  4. Applied simplify to get
    \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)} \leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\]
    0.1
  5. Applied simplify to get
    \[\color{red}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)} + x \leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2} + x\]
    0.1

Original test:


(lambda ((x default))
  #:name "NMSE example 3.7"
  (- (exp x) 1)
  #:target
  (* x (+ (+ 1 (/ x 2)) (/ (sqr x) 6))))