\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 19.7 s
Input Error: 14.3
Output Error: 0.1
Log: ⚲
\(\begin{cases} \frac{\frac{1}{x}}{{x}^2} - \frac{\frac{1}{x}}{x} & \text{when } x \le -6572693219723.217 \\ \frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x} & \text{when } x \le 14125114759858.025 \\ \frac{\frac{1}{x}}{{x}^2} - \frac{\frac{1}{x}}{x} & \text{otherwise} \end{cases}\)

    if x < -6572693219723.217 or 14125114759858.025 < x

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      29.5
    2. Applied taylor to get
      \[\frac{1}{x + 1} - \frac{1}{x} \leadsto \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right) - \frac{1}{x}\]
      29.5
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x} \leadsto \color{blue}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x}\]
      29.5
    4. Applied simplify to get
      \[\color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right) - \frac{1}{x}} \leadsto \color{blue}{\frac{\frac{1}{x}}{{x}^2} - \frac{\frac{1}{x}}{x}}\]
      0.1

    if -6572693219723.217 < x < 14125114759858.025

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      0.8
    2. Using strategy rm
      0.8
    3. Applied frac-sub to get
      \[\color{red}{\frac{1}{x + 1} - \frac{1}{x}} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
      0.0
    4. Applied simplify to get
      \[\frac{\color{red}{1 \cdot x - \left(x + 1\right) \cdot 1}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
      0.0
  1. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.1"
  (- (/ 1 (+ x 1)) (/ 1 x)))