Average Error: 15.1 → 0.0
Time: 13.2s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -32821892570143.49:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 342287.57643896976:\\ \;\;\;\;\frac{1}{(x \cdot x + 1)_*} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -32821892570143.49 or 342287.57643896976 < x

    1. Initial program 31.4

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

    2. Simplified31.4

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt31.4

      \[\leadsto \frac{x}{\color{blue}{\sqrt{(x \cdot x + 1)_*} \cdot \sqrt{(x \cdot x + 1)_*}}}\]

    5. Applied associate-/r*31.3

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}}\]

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -32821892570143.49 < x < 342287.57643896976

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{x}{\color{blue}{\sqrt{(x \cdot x + 1)_*} \cdot \sqrt{(x \cdot x + 1)_*}}}\]

    5. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}}\]
    6. Using strategy rm

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

      \[\leadsto \frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{\color{blue}{1 \cdot (x \cdot x + 1)_*}}}\]

    8. Applied sqrt-prod0.0

      \[\leadsto \frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\color{blue}{\sqrt{1} \cdot \sqrt{(x \cdot x + 1)_*}}}\]

    9. Applied div-inv0.0

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{\sqrt{(x \cdot x + 1)_*}}}}{\sqrt{1} \cdot \sqrt{(x \cdot x + 1)_*}}\]

    10. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{x}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}}\]

    11. Simplified0.0

      \[\leadsto \color{blue}{x} \cdot \frac{\frac{1}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}\]

    12. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -32821892570143.49:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 342287.57643896976:\\ \;\;\;\;\frac{1}{(x \cdot x + 1)_*} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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