\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 6.1 s
Input Error: 11.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{\frac{2}{x}}{x \cdot x}\)
  1. Started with
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    11.2
  2. Applied taylor to get
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leadsto 2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)\]
    0.6
  3. Taylor expanded around inf to get
    \[\color{red}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]
    0.6
  4. Applied simplify to get
    \[\color{red}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \leadsto \color{blue}{\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^3}}\]
    0.7
  5. Using strategy rm
    0.7
  6. Applied cube-mult to get
    \[\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{2}{\color{red}{{x}^3}} \leadsto \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}}\]
    0.7
  7. Applied associate-/r* to get
    \[\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \color{red}{\frac{2}{x \cdot \left(x \cdot x\right)}} \leadsto \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \color{blue}{\frac{\frac{2}{x}}{x \cdot x}}\]
    0.2

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.3"
  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
  #:target
  (/ 2 (* x (- (sqr x) 1))))