\[\sqrt{x - 1} \cdot \sqrt{x}\]
Test:
sqrt times
Bits:
128 bits
Bits error versus x
Time: 9.0 s
Input Error: 9.7
Output Error: 1.5
Log: ⚲
Profile: 🕒
\(\frac{\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}}{\frac{\sqrt{1 + x}}{\sqrt{x}}}\)
  1. Started with
    \[\sqrt{x - 1} \cdot \sqrt{x}\]
    9.7
  2. Using strategy rm
    9.7
  3. Applied flip-- to get
    \[\sqrt{\color{red}{x - 1}} \cdot \sqrt{x} \leadsto \sqrt{\color{blue}{\frac{{x}^2 - {1}^2}{x + 1}}} \cdot \sqrt{x}\]
    19.4
  4. Applied sqrt-div to get
    \[\color{red}{\sqrt{\frac{{x}^2 - {1}^2}{x + 1}}} \cdot \sqrt{x} \leadsto \color{blue}{\frac{\sqrt{{x}^2 - {1}^2}}{\sqrt{x + 1}}} \cdot \sqrt{x}\]
    19.3
  5. Applied associate-*l/ to get
    \[\color{red}{\frac{\sqrt{{x}^2 - {1}^2}}{\sqrt{x + 1}} \cdot \sqrt{x}} \leadsto \color{blue}{\frac{\sqrt{{x}^2 - {1}^2} \cdot \sqrt{x}}{\sqrt{x + 1}}}\]
    19.3
  6. Applied simplify to get
    \[\frac{\color{red}{\sqrt{{x}^2 - {1}^2} \cdot \sqrt{x}}}{\sqrt{x + 1}} \leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{{x}^2 - 1}}}{\sqrt{x + 1}}\]
    19.3
  7. Applied taylor to get
    \[\frac{\sqrt{x} \cdot \sqrt{{x}^2 - 1}}{\sqrt{x + 1}} \leadsto \frac{\sqrt{x} \cdot \left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}{\sqrt{x + 1}}\]
    1.0
  8. Taylor expanded around inf to get
    \[\frac{\sqrt{x} \cdot \color{red}{\left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}}{\sqrt{x + 1}} \leadsto \frac{\sqrt{x} \cdot \color{blue}{\left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}}{\sqrt{x + 1}}\]
    1.0
  9. Applied simplify to get
    \[\frac{\sqrt{x} \cdot \left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}{\sqrt{x + 1}} \leadsto \frac{\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}}{\frac{\sqrt{1 + x}}{\sqrt{x}}}\]
    1.5
  10. Applied final simplification
  11. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))