\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 18.9 s
Input Error: 8.4
Output Error: 3.0
Log: ⚲
Profile: 🕒
\(\frac{\frac{1}{(x * x + x)_*}}{\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{1 + x}}}\)
  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
    \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    3.0
  8. Applied simplify to get
    \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{(x * x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    2.9
  9. Using strategy rm
    2.9
  10. Applied add-cube-cbrt to get
    \[\frac{\frac{1}{(x * x + x)_*}}{\color{red}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \leadsto \frac{\frac{1}{(x * x + x)_*}}{\color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\right)}^3}}\]
    3.3
  11. Applied add-cube-cbrt to get
    \[\frac{\color{red}{\frac{1}{(x * x + x)_*}}}{{\left(\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{1}{(x * x + x)_*}}\right)}^3}}{{\left(\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\right)}^3}\]
    3.5
  12. Applied cube-undiv to get
    \[\color{red}{\frac{{\left(\sqrt[3]{\frac{1}{(x * x + x)_*}}\right)}^3}{{\left(\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\right)}^3}\]
    3.5
  13. Applied taylor to get
    \[{\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\sqrt{\frac{1}{1 + x}} + \sqrt{\frac{1}{x}}}}\right)}^3\]
    3.5
  14. Taylor expanded around 0 to get
    \[\color{red}{{\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\sqrt{\frac{1}{1 + x}} + \sqrt{\frac{1}{x}}}}\right)}^3} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\sqrt{\frac{1}{1 + x}} + \sqrt{\frac{1}{x}}}}\right)}^3}\]
    3.5
  15. Applied simplify to get
    \[{\left(\frac{\sqrt[3]{\frac{1}{(x * x + x)_*}}}{\sqrt[3]{\sqrt{\frac{1}{1 + x}} + \sqrt{\frac{1}{x}}}}\right)}^3 \leadsto \frac{\frac{1}{(x * x + x)_*}}{\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{1 + x}}}\]
    3.0
  16. Applied final simplification
  17. 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))))))