\[\frac{1 - \cos x}{\sin x}\]
Test:
NMSE example 3.4
Bits:
128 bits
Bits error versus x
Time: 21.0 s
Input Error: 30.2
Output Error: 1.0
Log: ⚲
Profile: 🕒
\(\log_* (1 + (e^{\frac{\sin x}{\cos x + 1}} - 1)^*)\)
  1. Started with
    \[\frac{1 - \cos x}{\sin x}\]
    30.2
  2. Using strategy rm
    30.2
  3. Applied flip-- to get
    \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]
    30.5
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{\sin x} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
    14.5
  5. Using strategy rm
    14.5
  6. Applied log1p-expm1-u to get
    \[\color{red}{\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}} \leadsto \color{blue}{\log_* (1 + (e^{\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}} - 1)^*)}\]
    15.1
  7. Applied simplify to get
    \[\log_* (1 + \color{red}{(e^{\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}} - 1)^*}) \leadsto \log_* (1 + \color{blue}{(e^{\frac{\sin x}{\cos x + 1}} - 1)^*})\]
    1.0

Original test:


(lambda ((x default))
  #:name "NMSE example 3.4"
  (/ (- 1 (cos x)) (sin x))
  #:target
  (tan (/ x 2)))