\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 26.3 s
Input Error: 4.0
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{\frac{\frac{2}{x}}{1 + x}}{x - 1}\)
  1. Started with
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    4.0
  2. Using strategy rm
    4.0
  3. Applied frac-sub to get
    \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
    11.4
  4. Applied frac-add to get
    \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
    11.2
  5. Applied simplify to get
    \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
    11.4
  6. Applied taylor to get
    \[\frac{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\left(x + x \cdot x\right) + \left(2 - \left({x}^2 + x\right)\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
    11.2
  7. Taylor expanded around 0 to get
    \[\frac{\left(x + x \cdot x\right) + \color{red}{\left(2 - \left({x}^2 + x\right)\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\left(x + x \cdot x\right) + \color{blue}{\left(2 - \left({x}^2 + x\right)\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
    11.2
  8. Applied simplify to get
    \[\color{red}{\frac{\left(x + x \cdot x\right) + \left(2 - \left({x}^2 + x\right)\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \color{blue}{\frac{\frac{\frac{2}{x}}{1 + x}}{x - 1}}\]
    0.1

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))))