\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 7.1 s
Input Error: 14.5
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) & \text{when } x \le -2.8723509911375156 \\ \frac{1}{x + 1} - \frac{1}{x} & \text{when } x \le 18533971.955403153 \\ \frac{\frac{1}{x}}{x \cdot x} - \frac{\frac{1}{x}}{x} & \text{otherwise} \end{cases}\)

    if x < -2.8723509911375156

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      29.2
    2. Applied taylor to get
      \[\frac{1}{x + 1} - \frac{1}{x} \leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right)\]
      0.7
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right)} \leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right)}\]
      0.7

    if -2.8723509911375156 < x < 18533971.955403153

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      0.1

    if 18533971.955403153 < x

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      29.2
    2. Applied taylor to get
      \[\frac{1}{x + 1} - \frac{1}{x} \leadsto \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right) - \frac{1}{x}\]
      29.2
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x} \leadsto \color{blue}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x}\]
      29.2
    4. Applied simplify to get
      \[\color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right) - \frac{1}{x}} \leadsto \color{blue}{\frac{\frac{1}{x}}{x \cdot x} - \frac{\frac{1}{x}}{x}}\]
      0.1
  1. Removed slow pow expressions

Original test:


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