\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 4.7 s
Input Error: 53.1
Output Error: 0.9
Log: ⚲
Profile: 🕒
\(\frac{1}{\frac{1}{a \cdot x} + \left(\left(a \cdot x\right) \cdot \frac{1}{12} - \frac{1}{2}\right)}\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    53.1
  2. Using strategy rm
    53.1
  3. Applied flip-- to get
    \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1}}\]
    53.1
  4. Using strategy rm
    53.1
  5. Applied clear-num to get
    \[\color{red}{\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1}} \leadsto \color{blue}{\frac{1}{\frac{e^{a \cdot x} + 1}{{\left(e^{a \cdot x}\right)}^2 - {1}^2}}}\]
    53.1
  6. Applied simplify to get
    \[\frac{1}{\color{red}{\frac{e^{a \cdot x} + 1}{{\left(e^{a \cdot x}\right)}^2 - {1}^2}}} \leadsto \frac{1}{\color{blue}{\frac{1}{e^{x \cdot a} - 1}}}\]
    53.1
  7. Applied taylor to get
    \[\frac{1}{\frac{1}{e^{x \cdot a} - 1}} \leadsto \frac{1}{\left(\frac{1}{a \cdot x} + \frac{1}{12} \cdot \left(a \cdot x\right)\right) - \frac{1}{2}}\]
    0.9
  8. Taylor expanded around 0 to get
    \[\frac{1}{\color{red}{\left(\frac{1}{a \cdot x} + \frac{1}{12} \cdot \left(a \cdot x\right)\right) - \frac{1}{2}}} \leadsto \frac{1}{\color{blue}{\left(\frac{1}{a \cdot x} + \frac{1}{12} \cdot \left(a \cdot x\right)\right) - \frac{1}{2}}}\]
    0.9
  9. Applied simplify to get
    \[\frac{1}{\left(\frac{1}{a \cdot x} + \frac{1}{12} \cdot \left(a \cdot x\right)\right) - \frac{1}{2}} \leadsto \frac{1}{\frac{1}{a \cdot x} + \left(\left(a \cdot x\right) \cdot \frac{1}{12} - \frac{1}{2}\right)}\]
    0.9
  10. Applied final simplification

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)))