\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 10.4 s
Input Error: 4.3
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^3}\right) + \frac{2}{{x}^{5}} & \text{when } x \le -45.253582f0 \\ \frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) + \left(x + 1\right) \cdot \left(\left(-2\right) \cdot \left(x - 1\right) + x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)} & \text{when } x \le 293.25146f0 \\ \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^3}\right) + \frac{2}{{x}^{5}} & \text{otherwise} \end{cases}\)

    if x < -45.253582f0 or 293.25146f0 < x

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      8.5
    2. Applied taylor to get
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leadsto 2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)\]
      0.7
    3. Taylor expanded around inf to get
      \[\color{red}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]
      0.7
    4. Applied simplify to get
      \[\color{red}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^3}\right) + \frac{2}{{x}^{5}}}\]
      0.7

    if -45.253582f0 < x < 293.25146f0

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      0.3
    2. Using strategy rm
      0.3
    3. Applied sub-neg to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\left(\frac{1}{x + 1} + \left(-\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\]
      0.3
    4. Applied associate-+l+ to get
      \[\color{red}{\left(\frac{1}{x + 1} + \left(-\frac{2}{x}\right)\right) + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{1}{x + 1} + \left(\left(-\frac{2}{x}\right) + \frac{1}{x - 1}\right)}\]
      0.3
    5. Using strategy rm
      0.3
    6. Applied distribute-neg-frac to get
      \[\frac{1}{x + 1} + \left(\color{red}{\left(-\frac{2}{x}\right)} + \frac{1}{x - 1}\right) \leadsto \frac{1}{x + 1} + \left(\color{blue}{\frac{-2}{x}} + \frac{1}{x - 1}\right)\]
      0.3
    7. Applied frac-add to get
      \[\frac{1}{x + 1} + \color{red}{\left(\frac{-2}{x} + \frac{1}{x - 1}\right)} \leadsto \frac{1}{x + 1} + \color{blue}{\frac{\left(-2\right) \cdot \left(x - 1\right) + x \cdot 1}{x \cdot \left(x - 1\right)}}\]
      0.3
    8. Applied frac-add to get
      \[\color{red}{\frac{1}{x + 1} + \frac{\left(-2\right) \cdot \left(x - 1\right) + x \cdot 1}{x \cdot \left(x - 1\right)}} \leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) + \left(x + 1\right) \cdot \left(\left(-2\right) \cdot \left(x - 1\right) + x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}}\]
      0.1
  1. Removed slow pow expressions

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))))