\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 5.9 s
Input Error: 23.5
Output Error: 4.0
Log: ⚲
Profile: 🕒
\(\frac{-x \cdot a}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    23.5
  2. Applied taylor to get
    \[e^{a \cdot x} - 1 \leadsto \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1\]
    26.4
  3. Taylor expanded around 0 to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1 \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1\]
    26.4
  4. Applied simplify to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a}\]
    16.4
  5. Using strategy rm
    16.4
  6. Applied distribute-lft1-in to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right) \cdot \left(x \cdot a\right)}\]
    16.4
  7. Using strategy rm
    16.4
  8. Applied flip-+ to get
    \[\color{red}{\left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right)} \cdot \left(x \cdot a\right) \leadsto \color{blue}{\frac{{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}} \cdot \left(x \cdot a\right)\]
    16.4
  9. Applied associate-*l/ to get
    \[\color{red}{\frac{{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1} \cdot \left(x \cdot a\right)} \leadsto \color{blue}{\frac{\left({\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}}\]
    16.4
  10. Applied taylor to get
    \[\frac{\left({\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1} \leadsto \frac{\left(0 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}\]
    4.0
  11. Taylor expanded around 0 to get
    \[\frac{\left(\color{red}{0} - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1} \leadsto \frac{\left(\color{blue}{0} - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}\]
    4.0
  12. Applied simplify to get
    \[\frac{\left(0 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1} \leadsto \frac{-x \cdot a}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}\]
    4.0
  13. 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)))