\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 6.9 s
Input Error: 33.2
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1} & \text{when } a \cdot x \le -3.2099427716584773 \cdot 10^{-06} \\ \left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right) \cdot \left(x \cdot a\right) & \text{otherwise} \end{cases}\)

    if (* a x) < -3.2099427716584773e-06

    1. Started with
      \[e^{a \cdot x} - 1\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied flip-- to get
      \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1}}\]
      0.1

    if -3.2099427716584773e-06 < (* a x)

    1. Started with
      \[e^{a \cdot x} - 1\]
      47.4
    2. Applied taylor to get
      \[e^{a \cdot x} - 1 \leadsto \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1\]
      43.4
    3. Taylor expanded around 0 to get
      \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1 \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1\]
      43.4
    4. Applied simplify to get
      \[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a}\]
      0.1
    5. Using strategy rm
      0.1
    6. Applied distribute-lft1-in to get
      \[\color{red}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right) \cdot \left(x \cdot a\right)}\]
      0.1
  1. Removed slow pow expressions

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)))