\[e^{x} - 1\]
Test:
NMSE example 3.7
Bits:
128 bits
Bits error versus x
Time: 4.7 s
Input Error: 59.3
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)\)
  1. Started with
    \[e^{x} - 1\]
    59.3
  2. Using strategy rm
    59.3
  3. Applied flip-- to get
    \[\color{red}{e^{x} - 1} \leadsto \color{blue}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}\]
    59.3
  4. Applied taylor to get
    \[\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1} \leadsto \frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)\]
    0.1
  5. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)} \leadsto \color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
    0.1
  6. Applied simplify to get
    \[\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right) \leadsto \left(\frac{1}{6} \cdot {x}^3 + x\right) + \left(x \cdot x\right) \cdot \frac{1}{2}\]
    0.1
  7. Applied final simplification
  8. Applied simplify to get
    \[\color{red}{\left(\frac{1}{6} \cdot {x}^3 + x\right) + \left(x \cdot x\right) \cdot \frac{1}{2}} \leadsto \color{blue}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\]
    0.1
  9. 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))))