\[e^{x} - 1\]
Test:
NMSE example 3.7
Bits:
128 bits
Bits error versus x
Time: 8.7 s
Input Error: 25.1
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2 + x & \text{when } x \le 0.03150794f0 \\ e^{x} - 1 & \text{otherwise} \end{cases}\)

    if x < 0.03150794f0

    1. Started with
      \[e^{x} - 1\]
      26.3
    2. Applied taylor to get
      \[e^{x} - 1 \leadsto \frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)\]
      0
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)} \leadsto \color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
      0
    4. Applied simplify to get
      \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)} \leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\]
      0
    5. Applied simplify to get
      \[\color{red}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)} + x \leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^2} + x\]
      0

    if 0.03150794f0 < x

    1. Started with
      \[e^{x} - 1\]
      0.6
  1. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "NMSE example 3.7"
  (- (exp x) 1)
  #:target
  (* x (+ (+ 1 (/ x 2)) (/ (sqr x) 6))))