\[\frac{1 - \cos x}{{x}^2}\]
Test:
NMSE problem 3.4.1
Bits:
128 bits
Bits error versus x
Time: 7.3 s
Input Error: 14.5
Output Error: 0.3
Log: ⚲
Profile: 🕒
\({\left(\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x}\right)}^2\)
  1. Started with
    \[\frac{1 - \cos x}{{x}^2}\]
    14.5
  2. Using strategy rm
    14.5
  3. Applied flip-- to get
    \[\frac{\color{red}{1 - \cos x}}{{x}^2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{{x}^2}\]
    14.5
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{{x}^2} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{{x}^2}\]
    7.2
  5. Using strategy rm
    7.2
  6. Applied add-sqr-sqrt to get
    \[\frac{\frac{{\left(\sin x\right)}^2}{\color{red}{1 + \cos x}}}{{x}^2} \leadsto \frac{\frac{{\left(\sin x\right)}^2}{\color{blue}{{\left(\sqrt{1 + \cos x}\right)}^2}}}{{x}^2}\]
    7.2
  7. Applied square-undiv to get
    \[\frac{\color{red}{\frac{{\left(\sin x\right)}^2}{{\left(\sqrt{1 + \cos x}\right)}^2}}}{{x}^2} \leadsto \frac{\color{blue}{{\left(\frac{\sin x}{\sqrt{1 + \cos x}}\right)}^2}}{{x}^2}\]
    7.2
  8. Applied square-undiv to get
    \[\color{red}{\frac{{\left(\frac{\sin x}{\sqrt{1 + \cos x}}\right)}^2}{{x}^2}} \leadsto \color{blue}{{\left(\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x}\right)}^2}\]
    0.3
  9. Removed slow pow expressions

Original test:


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