\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 2.5 s
Input Error: 14.1
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\frac{-1}{x + {x}^2}\)
  1. Started with
    \[\frac{1}{x + 1} - \frac{1}{x}\]
    14.1
  2. Using strategy rm
    14.1
  3. Applied frac-sub to get
    \[\color{red}{\frac{1}{x + 1} - \frac{1}{x}} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
    13.6
  4. Applied simplify to get
    \[\frac{\color{red}{1 \cdot x - \left(x + 1\right) \cdot 1}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
    13.6
  5. Applied simplify to get
    \[\frac{x - \left(1 + x\right)}{\color{red}{\left(x + 1\right) \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{x + {x}^2}}\]
    13.6
  6. Applied taylor to get
    \[\frac{x - \left(1 + x\right)}{x + {x}^2} \leadsto \frac{-1}{x + {x}^2}\]
    0.4
  7. Taylor expanded around 0 to get
    \[\frac{\color{red}{-1}}{x + {x}^2} \leadsto \frac{\color{blue}{-1}}{x + {x}^2}\]
    0.4
  8. Removed slow pow expressions

Original test:


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