\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 6.9 s
Input Error: 49.4
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + 0\)
  1. Started with
    \[e^{a \cdot x} - 1\]
    49.4
  2. Using strategy rm
    49.4
  3. Applied add-cube-cbrt to get
    \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}^3}\]
    49.4
  4. Applied taylor to get
    \[{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\right)}^3\]
    28.5
  5. Taylor expanded around 0 to get
    \[{\left(\sqrt[3]{\color{red}{\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}}\right)}^3\]
    28.5
  6. Applied simplify to get
    \[\color{red}{{\left(\sqrt[3]{\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\right)}^3} \leadsto \color{blue}{\left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + \left({x}^3 \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot \left(a \cdot a\right)}\]
    25.8
  7. Applied taylor to get
    \[\left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + \left({x}^3 \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot \left(a \cdot a\right) \leadsto \left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + 0\]
    0.2
  8. Taylor expanded around 0 to get
    \[\left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + \color{red}{0} \leadsto \left(x \cdot a + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\right) + \color{blue}{0}\]
    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)))