Average Error: 15.0 → 0.0
Time: 11.4s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.166459689224054 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 436.9147618452836:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original15.0
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.166459689224054e+31 or 436.9147618452836 < x

    1. Initial program 31.8

      \[\frac{x}{x \cdot x + 1}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}}\]

    if -2.166459689224054e+31 < x < 436.9147618452836

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.0

      \[\leadsto \frac{x}{\color{blue}{1 \cdot \left(x \cdot x + 1\right)}}\]

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{1}}{x \cdot x + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.166459689224054 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 436.9147618452836:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))