\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 8.0 s
Input Error: 52.2
Output Error: 10.3
Log: ⚲
Profile: 🕒
\(\frac{\left(1 + x \cdot a\right) \cdot \left(x \cdot \left(a \cdot 2\right)\right)}{1 + e^{x \cdot a}}\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    52.2
  2. Using strategy rm
    52.2
  3. Applied flip-- to get
    \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1}}\]
    52.2
  4. Applied taylor to get
    \[\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1} \leadsto \frac{\left(2 \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + 2 \cdot \left(a \cdot x\right)\right)\right) - {1}^2}{e^{a \cdot x} + 1}\]
    45.5
  5. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(2 \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + 2 \cdot \left(a \cdot x\right)\right)\right)} - {1}^2}{e^{a \cdot x} + 1} \leadsto \frac{\color{blue}{\left(2 \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + 2 \cdot \left(a \cdot x\right)\right)\right)} - {1}^2}{e^{a \cdot x} + 1}\]
    45.5
  6. Applied simplify to get
    \[\color{red}{\frac{\left(2 \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + 2 \cdot \left(a \cdot x\right)\right)\right) - {1}^2}{e^{a \cdot x} + 1}} \leadsto \color{blue}{\frac{2 \cdot \left(x \cdot a + \left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{1 + e^{x \cdot a}}}\]
    10.3
  7. Applied simplify to get
    \[\frac{\color{red}{2 \cdot \left(x \cdot a + \left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}}{1 + e^{x \cdot a}} \leadsto \frac{\color{blue}{\left(1 + x \cdot a\right) \cdot \left(x \cdot \left(a \cdot 2\right)\right)}}{1 + e^{x \cdot a}}\]
    10.3

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)))