\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 8.7 s
Input Error: 8.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(1 \cdot \frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{1 + x}}}\)
  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
    \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.1
  8. Applied simplify to get
    \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{{x}^2 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.1
  9. Using strategy rm
    3.1
  10. Applied *-un-lft-identity to get
    \[\frac{\frac{1}{{x}^2 + x}}{\color{red}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \leadsto \frac{\frac{1}{{x}^2 + x}}{\color{blue}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}}\]
    3.1
  11. Applied *-un-lft-identity to get
    \[\frac{\color{red}{\frac{1}{{x}^2 + x}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)} \leadsto \frac{\color{blue}{1 \cdot \frac{1}{{x}^2 + x}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]
    3.1
  12. Applied times-frac to get
    \[\color{red}{\frac{1 \cdot \frac{1}{{x}^2 + x}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}} \leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
    3.1
  13. Applied simplify to get
    \[\color{red}{\frac{1}{1}} \cdot \frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{1} \cdot \frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.1
  14. Applied simplify to get
    \[1 \cdot \color{red}{\frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \leadsto 1 \cdot \color{blue}{\frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{1 + x}}}}\]
    0.3

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))))))