\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 12.4 s
Input Error: 19.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\frac{\frac{1}{x}}{\left(\sqrt{1 + x} + \sqrt{x}\right) + \sqrt{\frac{1}{x}}}\)
  1. Started with
    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
    19.4
  2. Using strategy rm
    19.4
  3. Applied flip-- to get
    \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
    19.4
  4. Applied simplify to get
    \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    19.4
  5. Using strategy rm
    19.4
  6. Applied frac-sub to get
    \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
    18.8
  7. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \leadsto \color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(1 + x\right)\right)}}\]
    18.7
  8. Using strategy rm
    18.7
  9. Applied add-cube-cbrt to get
    \[\color{red}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(1 + x\right)\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(1 + x\right)\right)}}\right)}^3}\]
    19.2
  10. Applied simplify to get
    \[{\color{red}{\left(\sqrt[3]{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(1 + x\right)\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{x}}{\frac{1 + x}{\sqrt{1 + x}} + \frac{1 + x}{\sqrt{x}}}}\right)}}^3\]
    1.1
  11. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\frac{1}{x}}{\frac{1 + x}{\sqrt{1 + x}} + \frac{1 + x}{\sqrt{x}}}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{1}{x \cdot \left(\sqrt{x} + \left(\sqrt{1 + x} + \sqrt{\frac{1}{x}}\right)\right)}}\right)}^3\]
    1.6
  12. Taylor expanded around 0 to get
    \[{\color{red}{\left(\sqrt[3]{\frac{1}{x \cdot \left(\sqrt{x} + \left(\sqrt{1 + x} + \sqrt{\frac{1}{x}}\right)\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{x \cdot \left(\sqrt{x} + \left(\sqrt{1 + x} + \sqrt{\frac{1}{x}}\right)\right)}}\right)}}^3\]
    1.6
  13. Applied simplify to get
    \[{\left(\sqrt[3]{\frac{1}{x \cdot \left(\sqrt{x} + \left(\sqrt{1 + x} + \sqrt{\frac{1}{x}}\right)\right)}}\right)}^3 \leadsto \frac{\frac{1}{x}}{\left(\sqrt{1 + x} + \sqrt{x}\right) + \sqrt{\frac{1}{x}}}\]
    0.3
  14. Applied final simplification
  15. 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))))))