\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 6.2 s
Input Error: 8.4
Output Error: 3.0
Log: ⚲
Profile: 🕒
\(\frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\)
  1. Started with
    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
    8.4
  2. Using strategy rm
    8.4
  3. Applied flip-- to get
    \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
    8.4
  4. Applied simplify to get
    \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    8.4
  5. Using strategy rm
    8.4
  6. Applied frac-sub to get
    \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    7.5
  7. Applied simplify to get
    \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.0
  8. Applied simplify to get
    \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{{x}^2 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.0
  9. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "NMSE example 3.6"
  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))
  #:target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))