\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 7.3 s
Input Error: 20.0
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\)
  1. Started with
    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
    20.0
  2. Using strategy rm
    20.0
  3. Applied frac-sub to get
    \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
    20.0
  4. Applied simplify to get
    \[\frac{\color{red}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    20.0
  5. Using strategy rm
    20.0
  6. Applied flip-- to get
    \[\frac{\color{red}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    19.8
  7. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    0.4
  8. Using strategy rm
    0.4
  9. Applied add-cube-cbrt to get
    \[\frac{\frac{1}{\color{red}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right)}^3}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    0.6
  10. Applied taylor to get
    \[\frac{\frac{1}{{\left(\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right)}^3}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    0.4
  11. Taylor expanded around 0 to get
    \[\frac{\frac{1}{\color{red}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    0.4
  12. Applied simplify to get
    \[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\]
    0.4
  13. Applied final simplification
  14. 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))))))