\[\frac{x}{{x}^2 + 1}\]
Test:
x / (x^2 + 1)
Bits:
128 bits
Bits error versus x
Time: 8.3 s
Input Error: 15.2
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right) & \text{when } x \le -4.3423375737525415 \cdot 10^{+46} \\ \frac{x}{{\left({x}^2\right)}^2 - {1}^2} \cdot \left({x}^2 - 1\right) & \text{when } x \le 5.107891217442523 \cdot 10^{+18} \\ \frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right) & \text{otherwise} \end{cases}\)

    if x < -4.3423375737525415e+46 or 5.107891217442523e+18 < x

    1. Started with
      \[\frac{x}{{x}^2 + 1}\]
      33.7
    2. Applied taylor to get
      \[\frac{x}{{x}^2 + 1} \leadsto \left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\]
      0
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
      0
    4. Applied simplify to get
      \[\color{red}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \leadsto \color{blue}{\frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right)}\]
      0

    if -4.3423375737525415e+46 < x < 5.107891217442523e+18

    1. Started with
      \[\frac{x}{{x}^2 + 1}\]
      0.0
    2. Using strategy rm
      0.0
    3. Applied flip-+ to get
      \[\frac{x}{\color{red}{{x}^2 + 1}} \leadsto \frac{x}{\color{blue}{\frac{{\left({x}^2\right)}^2 - {1}^2}{{x}^2 - 1}}}\]
      0.0
    4. Applied associate-/r/ to get
      \[\color{red}{\frac{x}{\frac{{\left({x}^2\right)}^2 - {1}^2}{{x}^2 - 1}}} \leadsto \color{blue}{\frac{x}{{\left({x}^2\right)}^2 - {1}^2} \cdot \left({x}^2 - 1\right)}\]
      0.0
  1. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "x / (x^2 + 1)"
  (/ x (+ (sqr x) 1))
  #:target
  (/ 1 (+ x (/ 1 x))))