\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 7.0 s
Input Error: 21.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    21.2
  2. Using strategy rm
    21.2
  3. Applied add-exp-log to get
    \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{e^{\log \left(e^{a \cdot x} - 1\right)}}\]
    23.0
  4. Applied taylor to get
    \[e^{\log \left(e^{a \cdot x} - 1\right)} \leadsto \frac{1}{2} \cdot \left(e^{\log x + \log a} \cdot \left(a \cdot x\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\log x + \log a} \cdot \left({a}^2 \cdot {x}^2\right)\right) + e^{\log x + \log a}\right)\]
    25.4
  5. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{2} \cdot \left(e^{\log x + \log a} \cdot \left(a \cdot x\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\log x + \log a} \cdot \left({a}^2 \cdot {x}^2\right)\right) + e^{\log x + \log a}\right)} \leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\log x + \log a} \cdot \left(a \cdot x\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\log x + \log a} \cdot \left({a}^2 \cdot {x}^2\right)\right) + e^{\log x + \log a}\right)}\]
    25.4
  6. Applied simplify to get
    \[\frac{1}{2} \cdot \left(e^{\log x + \log a} \cdot \left(a \cdot x\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\log x + \log a} \cdot \left({a}^2 \cdot {x}^2\right)\right) + e^{\log x + \log a}\right) \leadsto \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^2 + a \cdot x\right) + {\left(a \cdot x\right)}^2 \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right)\]
    0.2
  7. Applied final simplification
  8. Applied simplify to get
    \[\color{red}{\left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^2 + a \cdot x\right) + {\left(a \cdot x\right)}^2 \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right)} \leadsto \color{blue}{a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)}\]
    0.2

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)))