\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 2.6 s
Input Error: 23.5
Output Error: 0
Log: ⚲
Profile: 🕒
\(a \cdot x\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    23.5
  2. Applied taylor to get
    \[e^{a \cdot x} - 1 \leadsto \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1\]
    24.8
  3. Taylor expanded around 0 to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1 \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1\]
    24.8
  4. Applied simplify to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a}\]
    14.0
  5. Applied taylor to get
    \[\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a \leadsto a \cdot x\]
    0
  6. Taylor expanded around 0 to get
    \[\color{red}{a \cdot x} \leadsto \color{blue}{a \cdot x}\]
    0

Original test:


(lambda ((a default) (x default))
  #:name "NMSE section 3.5"
  (- (exp (* a x)) 1)
  #:target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1)))