\[\frac{1 - \cos x}{{x}^2}\]
Test:
NMSE problem 3.4.1
Bits:
128 bits
Bits error versus x
Time: 10.1 s
Input Error: 31.5
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{x} \cdot \frac{1 - \cos x}{x} & \text{when } x \le -8.61646249982272 \cdot 10^{-12} \\ \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2 & \text{when } x \le 8.584780819621077 \\ \frac{1}{x} \cdot \frac{1 - \cos x}{x} & \text{otherwise} \end{cases}\)

    if x < -8.61646249982272e-12 or 8.584780819621077 < x

    1. Started with
      \[\frac{1 - \cos x}{{x}^2}\]
      1.7
    2. Using strategy rm
      1.7
    3. Applied square-mult to get
      \[\frac{1 - \cos x}{\color{red}{{x}^2}} \leadsto \frac{1 - \cos x}{\color{blue}{x \cdot x}}\]
      1.7
    4. Applied *-un-lft-identity to get
      \[\frac{\color{red}{1 - \cos x}}{x \cdot x} \leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
      1.7
    5. Applied times-frac to get
      \[\color{red}{\frac{1 \cdot \left(1 - \cos x\right)}{x \cdot x}} \leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
      1.0

    if -8.61646249982272e-12 < x < 8.584780819621077

    1. Started with
      \[\frac{1 - \cos x}{{x}^2}\]
      61.5
    2. Applied taylor to get
      \[\frac{1 - \cos x}{{x}^2} \leadsto \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2\]
      0.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2} \leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2}\]
      0.0
  1. Removed slow pow expressions

Original test:


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