\[\frac{x}{{x}^2 + 1}\]
Test:
x / (x^2 + 1)
Bits:
128 bits
Bits error versus x
Time: 5.5 s
Input Error: 6.8
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right) & \text{when } x \le -1783537.0f0 \\ \log_* (1 + (e^{\frac{x}{{x}^2 + 1}} - 1)^*) & \text{when } x \le 19.416372f0 \\ \frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right) & \text{otherwise} \end{cases}\)

    if x < -1783537.0f0 or 19.416372f0 < x

    1. Started with
      \[\frac{x}{{x}^2 + 1}\]
      15.3
    2. Applied taylor to get
      \[\frac{x}{{x}^2 + 1} \leadsto \left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\]
      0.0
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
      0.0
    4. Applied simplify to get
      \[\color{red}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \leadsto \color{blue}{\frac{1}{{x}^{5}} - \left(\frac{1}{{x}^3} - \frac{1}{x}\right)}\]
      0.0

    if -1783537.0f0 < x < 19.416372f0

    1. Started with
      \[\frac{x}{{x}^2 + 1}\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied log1p-expm1-u to get
      \[\color{red}{\frac{x}{{x}^2 + 1}} \leadsto \color{blue}{\log_* (1 + (e^{\frac{x}{{x}^2 + 1}} - 1)^*)}\]
      0.1
  1. Removed slow pow expressions

Original test:


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