\[e^{x} - 1\]
Test:
NMSE example 3.7
Bits:
128 bits
Bits error versus x
Time: 9.8 s
Input Error: 59.4
Output Error: 0.1
Log: ⚲
\(x + {x}^2 \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\)
  1. Started with
    \[e^{x} - 1\]
    59.4
  2. Applied taylor to get
    \[e^{x} - 1 \leadsto \frac{1}{2} \cdot {x}^2 + \left(x + \frac{1}{6} \cdot {x}^{3}\right)\]
    0.1
  3. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(x + \frac{1}{6} \cdot {x}^{3}\right)} \leadsto \color{blue}{\frac{1}{2} \cdot {x}^2 + \left(x + \frac{1}{6} \cdot {x}^{3}\right)}\]
    0.1
  4. Applied simplify to get
    \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(x + \frac{1}{6} \cdot {x}^{3}\right)} \leadsto \color{blue}{x + {x}^2 \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\]
    0.1
  5. Removed slow pow expressions

Original test:


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