\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 4.1 s
Input Error: 6.0
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\frac{\frac{\log \left(\frac{1}{e}\right)}{x + 1}}{x}\)
  1. Started with
    \[\frac{1}{x + 1} - \frac{1}{x}\]
    6.0
  2. Using strategy rm
    6.0
  3. Applied frac-sub to get
    \[\color{red}{\frac{1}{x + 1} - \frac{1}{x}} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
    5.0
  4. Applied simplify to get
    \[\frac{\color{red}{1 \cdot x - \left(x + 1\right) \cdot 1}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
    5.0
  5. Applied simplify to get
    \[\frac{x - \left(1 + x\right)}{\color{red}{\left(x + 1\right) \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{x + {x}^2}}\]
    5.0
  6. Using strategy rm
    5.0
  7. Applied square-mult to get
    \[\frac{x - \left(1 + x\right)}{x + \color{red}{{x}^2}} \leadsto \frac{x - \left(1 + x\right)}{x + \color{blue}{x \cdot x}}\]
    5.0
  8. Applied distribute-rgt1-in to get
    \[\frac{x - \left(1 + x\right)}{\color{red}{x + x \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{\left(x + 1\right) \cdot x}}\]
    5.0
  9. Applied associate-/r* to get
    \[\color{red}{\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}} \leadsto \color{blue}{\frac{\frac{x - \left(1 + x\right)}{x + 1}}{x}}\]
    5.0
  10. Using strategy rm
    5.0
  11. Applied add-log-exp to get
    \[\frac{\frac{\color{red}{x - \left(1 + x\right)}}{x + 1}}{x} \leadsto \frac{\frac{\color{blue}{\log \left(e^{x - \left(1 + x\right)}\right)}}{x + 1}}{x}\]
    5.0
  12. Applied simplify to get
    \[\frac{\frac{\log \color{red}{\left(e^{x - \left(1 + x\right)}\right)}}{x + 1}}{x} \leadsto \frac{\frac{\log \color{blue}{\left(\frac{1}{e}\right)}}{x + 1}}{x}\]
    0.0
  13. Removed slow pow expressions

Original test:


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