\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 11.1 s
Input Error: 15.5
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\left(-2\right) \cdot (\left((x * x + 1)_*\right) * x + \left(\frac{1}{x}\right))_*\)
  1. Started with
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    15.5
  2. Applied taylor to get
    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leadsto -\left(2 \cdot {x}^{3} + \left(2 \cdot \frac{1}{x} + 2 \cdot x\right)\right)\]
    0.0
  3. Taylor expanded around 0 to get
    \[\color{red}{-\left(2 \cdot {x}^{3} + \left(2 \cdot \frac{1}{x} + 2 \cdot x\right)\right)} \leadsto \color{blue}{-\left(2 \cdot {x}^{3} + \left(2 \cdot \frac{1}{x} + 2 \cdot x\right)\right)}\]
    0.0
  4. Applied simplify to get
    \[\color{red}{-\left(2 \cdot {x}^{3} + \left(2 \cdot \frac{1}{x} + 2 \cdot x\right)\right)} \leadsto \color{blue}{\left(-2\right) \cdot (\left((x * x + 1)_*\right) * x + \left(\frac{1}{x}\right))_*}\]
    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))))