\[\frac{1 - \cos x}{\sin x}\]
Test:
NMSE example 3.4
Bits:
128 bits
Bits error versus x
Time: 46.9 s
Input Error: 30.1
Output Error: 1.0
Log: ⚲
Profile: 🕒
\(\frac{\sin x}{1} \cdot \log \left(e^{\frac{1}{1 + \cos x}}\right)\)
  1. Started with
    \[\frac{1 - \cos x}{\sin x}\]
    30.1
  2. Using strategy rm
    30.1
  3. Applied flip-- to get
    \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]
    30.4
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{\sin x} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
    14.8
  5. Using strategy rm
    14.8
  6. Applied *-un-lft-identity to get
    \[\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{red}{\sin x}} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{blue}{1 \cdot \sin x}}\]
    14.8
  7. Applied *-un-lft-identity to get
    \[\frac{\frac{{\left(\sin x\right)}^2}{\color{red}{1 + \cos x}}}{1 \cdot \sin x} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{\color{blue}{1 \cdot \left(1 + \cos x\right)}}}{1 \cdot \sin x}\]
    14.8
  8. Applied square-mult to get
    \[\frac{\frac{\color{red}{{\left(\sin x\right)}^2}}{1 \cdot \left(1 + \cos x\right)}}{1 \cdot \sin x} \leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 \cdot \left(1 + \cos x\right)}}{1 \cdot \sin x}\]
    14.8
  9. Applied times-frac to get
    \[\frac{\color{red}{\frac{\sin x \cdot \sin x}{1 \cdot \left(1 + \cos x\right)}}}{1 \cdot \sin x} \leadsto \frac{\color{blue}{\frac{\sin x}{1} \cdot \frac{\sin x}{1 + \cos x}}}{1 \cdot \sin x}\]
    14.8
  10. Applied times-frac to get
    \[\color{red}{\frac{\frac{\sin x}{1} \cdot \frac{\sin x}{1 + \cos x}}{1 \cdot \sin x}} \leadsto \color{blue}{\frac{\frac{\sin x}{1}}{1} \cdot \frac{\frac{\sin x}{1 + \cos x}}{\sin x}}\]
    0.5
  11. Applied simplify to get
    \[\color{red}{\frac{\frac{\sin x}{1}}{1}} \cdot \frac{\frac{\sin x}{1 + \cos x}}{\sin x} \leadsto \color{blue}{\frac{\sin x}{1}} \cdot \frac{\frac{\sin x}{1 + \cos x}}{\sin x}\]
    0.5
  12. Applied simplify to get
    \[\frac{\sin x}{1} \cdot \color{red}{\frac{\frac{\sin x}{1 + \cos x}}{\sin x}} \leadsto \frac{\sin x}{1} \cdot \color{blue}{\frac{1}{1 + \cos x}}\]
    0.5
  13. Using strategy rm
    0.5
  14. Applied add-log-exp to get
    \[\frac{\sin x}{1} \cdot \color{red}{\frac{1}{1 + \cos x}} \leadsto \frac{\sin x}{1} \cdot \color{blue}{\log \left(e^{\frac{1}{1 + \cos x}}\right)}\]
    1.0

Original test:


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