\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 21.2 s
Input Error: 4.1
Output Error: 4.1
Log: ⚲
Profile: 🕒
\(\left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot 1\)
  1. Started with
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    4.1
  2. Using strategy rm
    4.1
  3. Applied flip-+ to get
    \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
    11.1
  4. Using strategy rm
    11.1
  5. Applied *-un-lft-identity to get
    \[\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \leadsto \frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\color{blue}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}}\]
    11.1
  6. Applied difference-of-squares to get
    \[\frac{\color{red}{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)} \leadsto \frac{\color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right) \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}\]
    11.1
  7. Applied times-frac to get
    \[\color{red}{\frac{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right) \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}} \leadsto \color{blue}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}}{1} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
    4.1
  8. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}}{1}} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \color{blue}{\left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right)} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
    4.1
  9. Applied simplify to get
    \[\left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot \color{red}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \leadsto \left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot \color{blue}{1}\]
    4.1

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))))