\[\sqrt{x + 1} - \sqrt{x}\]
Test:
NMSE example 3.1
Bits:
128 bits
Bits error versus x
Time: 4.2 s
Input Error: 13.2
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\frac{1}{{\left(\sqrt{\sqrt{x + 1}}\right)}^2 + \sqrt{x}}\)
  1. Started with
    \[\sqrt{x + 1} - \sqrt{x}\]
    13.2
  2. Using strategy rm
    13.2
  3. Applied flip-- to get
    \[\color{red}{\sqrt{x + 1} - \sqrt{x}} \leadsto \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{x + 1} + \sqrt{x}}}\]
    13.2
  4. Applied simplify to get
    \[\frac{\color{red}{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}}{\sqrt{x + 1} + \sqrt{x}} \leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
    0.2
  5. Using strategy rm
    0.2
  6. Applied add-sqr-sqrt to get
    \[\frac{1}{\color{red}{\sqrt{x + 1}} + \sqrt{x}} \leadsto \frac{1}{\color{blue}{{\left(\sqrt{\sqrt{x + 1}}\right)}^2} + \sqrt{x}}\]
    0.3

Original test:


(lambda ((x default))
  #:name "NMSE example 3.1"
  (- (sqrt (+ x 1)) (sqrt x))
  #:target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))