\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 13.3 s
Input Error: 8.5
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{(x * x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} & \text{when } x \le 7.046752f+19 \\ \frac{\frac{\frac{1}{2}}{x \cdot x} - \left(\frac{1}{x} + \frac{\frac{3}{8}}{{x}^3}\right)}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{\frac{1}{2}}{x \cdot x} - \left(\frac{1}{x} + \frac{\frac{3}{8}}{{x}^3}\right)}} & \text{otherwise} \end{cases}\)

    if x < 7.046752f+19

    1. Started with
      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
      7.9
    2. Using strategy rm
      7.9
    3. Applied flip-- to get
      \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
      7.9
    4. Applied simplify to get
      \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      7.9
    5. Using strategy rm
      7.9
    6. Applied frac-sub to get
      \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      6.6
    7. Applied simplify to get
      \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.7
    8. Applied simplify to get
      \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{(x * x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.5

    if 7.046752f+19 < x

    1. Started with
      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
      10.4
    2. Using strategy rm
      10.4
    3. Applied flip-- to get
      \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
      10.4
    4. Applied simplify to get
      \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      10.4
    5. Using strategy rm
      10.4
    6. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{1}{x} - \frac{1}{1 + x}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      10.4
    7. Applied taylor to get
      \[\frac{{\left(\sqrt{\frac{1}{x} - \frac{1}{1 + x}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{{\left(\frac{1}{2} \cdot \frac{1}{{x}^2} - \left(\frac{1}{x} + \frac{3}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.2
    8. Taylor expanded around -inf to get
      \[\frac{{\color{red}{\left(\frac{1}{2} \cdot \frac{1}{{x}^2} - \left(\frac{1}{x} + \frac{3}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^2} - \left(\frac{1}{x} + \frac{3}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
      0.2
    9. Applied simplify to get
      \[\frac{{\left(\frac{1}{2} \cdot \frac{1}{{x}^2} - \left(\frac{1}{x} + \frac{3}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\frac{1}{2}}{x \cdot x} - \left(\frac{1}{x} + \frac{\frac{3}{8}}{{x}^3}\right)}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{\frac{1}{2}}{x \cdot x} - \left(\frac{1}{x} + \frac{\frac{3}{8}}{{x}^3}\right)}}\]
      0.2
    10. Applied final simplification
  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))))))