\[\frac{1 - \cos x}{\sin x}\]
Test:
NMSE example 3.4
Bits:
128 bits
Bits error versus x
Time: 20.0 s
Input Error: 34.5
Output Error: 0.3
Log: ⚲
Profile: 🕒
\((e^{\log_* (1 + \frac{\sin x}{\cos x + 1})} - 1)^*\)
  1. Started with
    \[\frac{1 - \cos x}{\sin x}\]
    34.5
  2. Using strategy rm
    34.5
  3. Applied flip-- to get
    \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]
    34.7
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{\sin x} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
    17.0
  5. Using strategy rm
    17.0
  6. Applied expm1-log1p-u to get
    \[\color{red}{\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}} \leadsto \color{blue}{(e^{\log_* (1 + \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x})} - 1)^*}\]
    17.0
  7. Applied simplify to get
    \[(e^{\color{red}{\log_* (1 + \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x})}} - 1)^* \leadsto (e^{\color{blue}{\log_* (1 + \frac{\sin x}{\cos x + 1})}} - 1)^*\]
    0.3

Original test:


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