\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 5.0 s
Input Error: 5.8
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{x}}{x \cdot x} - \frac{\frac{1}{x}}{x} & \text{when } x \le -12565212.0f0 \\ \frac{\frac{x - \left(1 + x\right)}{x + 1}}{x} & \text{when } x \le 9068344.0f0 \\ \frac{\frac{1}{x}}{x \cdot x} - \frac{\frac{1}{x}}{x} & \text{otherwise} \end{cases}\)

    if x < -12565212.0f0 or 9068344.0f0 < x

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      11.9
    2. Applied taylor to get
      \[\frac{1}{x + 1} - \frac{1}{x} \leadsto \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right) - \frac{1}{x}\]
      6.2
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x} \leadsto \color{blue}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)} - \frac{1}{x}\]
      6.2
    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

    if -12565212.0f0 < x < 9068344.0f0

    1. Started with
      \[\frac{1}{x + 1} - \frac{1}{x}\]
      1.5
    2. Using strategy rm
      1.5
    3. Applied frac-sub to get
      \[\color{red}{\frac{1}{x + 1} - \frac{1}{x}} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
      0.1
    4. Applied simplify to get
      \[\frac{\color{red}{1 \cdot x - \left(x + 1\right) \cdot 1}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
      0.1
    5. Applied simplify to get
      \[\frac{x - \left(1 + x\right)}{\color{red}{\left(x + 1\right) \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{x + {x}^2}}\]
      0.1
    6. Using strategy rm
      0.1
    7. Applied square-mult to get
      \[\frac{x - \left(1 + x\right)}{x + \color{red}{{x}^2}} \leadsto \frac{x - \left(1 + x\right)}{x + \color{blue}{x \cdot x}}\]
      0.1
    8. Applied distribute-rgt1-in to get
      \[\frac{x - \left(1 + x\right)}{\color{red}{x + x \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{\left(x + 1\right) \cdot x}}\]
      0.1
    9. Applied associate-/r* to get
      \[\color{red}{\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}} \leadsto \color{blue}{\frac{\frac{x - \left(1 + x\right)}{x + 1}}{x}}\]
      0.1
  1. Removed slow pow expressions

Original test:


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