\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 21.5 s
Input Error: 8.3
Output Error: 2.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{{x}^2 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} & \text{when } x \le 5.879483f+07 \\ \frac{\frac{\frac{1}{x}}{x} - \frac{1}{{x}^3}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}} & \text{otherwise} \end{cases}\)

    if x < 5.879483f+07

    1. Started with
      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
      2.3
    2. Using strategy rm
      2.3
    3. Applied flip-- to get
      \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
      2.3
    4. Applied simplify to get
      \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      2.3
    5. Using strategy rm
      2.3
    6. Applied frac-sub to get
      \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.8
    7. Applied simplify to get
      \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.5
    8. Applied simplify to get
      \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{{x}^2 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.4

    if 5.879483f+07 < x

    1. Started with
      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
      17.4
    2. Using strategy rm
      17.4
    3. Applied flip-- to get
      \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
      17.4
    4. Applied simplify to get
      \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      17.4
    5. Applied taylor to get
      \[\frac{\frac{1}{x} - \frac{1}{1 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{x} - \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      12.1
    6. Taylor expanded around inf to get
      \[\frac{\frac{1}{x} - \color{red}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{x} - \color{blue}{\left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      12.1
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{1}{x} - \left(\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \frac{1}{{x}^2}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{x} - \frac{\frac{1}{x}}{x \cdot x}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}}}\]
      5.3
    8. Applied simplify to get
      \[\frac{\color{red}{\frac{\frac{1}{x}}{x} - \frac{\frac{1}{x}}{x \cdot x}}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}} \leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{x} - \frac{1}{{x}^3}}}{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}}\]
      5.3
  1. 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))))))