\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 6.7 s
Input Error: 8.4
Output Error: 0.6
Log: ⚲
Profile: 🕒
\({\left(\frac{\sqrt{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2\)
  1. Started with
    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
    8.4
  2. Using strategy rm
    8.4
  3. Applied add-sqr-sqrt to get
    \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{{\left(\sqrt{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}}\right)}^2}\]
    8.5
  4. Using strategy rm
    8.5
  5. Applied frac-sub to get
    \[{\left(\sqrt{\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}}\right)}^2\]
    8.5
  6. Applied sqrt-div to get
    \[{\color{red}{\left(\sqrt{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}}^2\]
    8.5
  7. Applied simplify to get
    \[{\left(\frac{\color{red}{\sqrt{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2 \leadsto {\left(\frac{\color{blue}{\sqrt{\sqrt{1 + x} - \sqrt{x}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2\]
    8.5
  8. Using strategy rm
    8.5
  9. Applied flip-- to get
    \[{\left(\frac{\sqrt{\color{red}{\sqrt{1 + x} - \sqrt{x}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2 \leadsto {\left(\frac{\sqrt{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2\]
    8.5
  10. Applied simplify to get
    \[{\left(\frac{\sqrt{\frac{\color{red}{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2 \leadsto {\left(\frac{\sqrt{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt{x} \cdot \sqrt{x + 1}}}\right)}^2\]
    0.6

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))))))