\[\frac{1 - \cos x}{{x}^2}\]
Test:
NMSE problem 3.4.1
Bits:
128 bits
Bits error versus x
Time: 11.9 s
Input Error: 31.2
Output Error: 15.7
Log: ⚲
Profile: 🕒
\(\frac{1}{x} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}\)
  1. Started with
    \[\frac{1 - \cos x}{{x}^2}\]
    31.2
  2. Using strategy rm
    31.2
  3. Applied flip-- to get
    \[\frac{\color{red}{1 - \cos x}}{{x}^2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{{x}^2}\]
    31.3
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{{x}^2} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{{x}^2}\]
    15.8
  5. Using strategy rm
    15.8
  6. Applied square-mult to get
    \[\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{red}{{x}^2}} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\color{blue}{x \cdot x}}\]
    15.8
  7. Applied *-un-lft-identity to get
    \[\frac{\frac{{\left(\sin x\right)}^2}{\color{red}{1 + \cos x}}}{x \cdot x} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{\color{blue}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x}\]
    15.8
  8. Applied *-un-lft-identity to get
    \[\frac{\frac{{\color{red}{\left(\sin x\right)}}^2}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x} \leadsto \frac{\frac{{\color{blue}{\left(1 \cdot \sin x\right)}}^2}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x}\]
    15.8
  9. Applied square-prod to get
    \[\frac{\frac{\color{red}{{\left(1 \cdot \sin x\right)}^2}}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x} \leadsto \frac{\frac{\color{blue}{{1}^2 \cdot {\left(\sin x\right)}^2}}{1 \cdot \left(1 + \cos x\right)}}{x \cdot x}\]
    15.8
  10. Applied times-frac to get
    \[\frac{\color{red}{\frac{{1}^2 \cdot {\left(\sin x\right)}^2}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x} \leadsto \frac{\color{blue}{\frac{{1}^2}{1} \cdot \frac{{\left(\sin x\right)}^2}{1 + \cos x}}}{x \cdot x}\]
    15.8
  11. Applied times-frac to get
    \[\color{red}{\frac{\frac{{1}^2}{1} \cdot \frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x \cdot x}} \leadsto \color{blue}{\frac{\frac{{1}^2}{1}}{x} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}}\]
    15.7
  12. Applied simplify to get
    \[\color{red}{\frac{\frac{{1}^2}{1}}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x} \leadsto \color{blue}{\frac{1}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}\]
    15.7

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.4.1"
  (/ (- 1 (cos x)) (sqr x)))