Average Error: 15.1 → 0.0
Time: 15.8s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -32821892570143.49:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 342287.57643896976:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original15.1
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. 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 -32821892570143.49 < x < 342287.57643896976

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
    5. Using strategy rm

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

      \[\leadsto \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\color{blue}{1 \cdot \sqrt{x \cdot x + 1}}}\]

    7. Applied div-inv0.0

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{\sqrt{x \cdot x + 1}}}}{1 \cdot \sqrt{x \cdot x + 1}}\]

    8. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\frac{1}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]

    9. Simplified0.0

      \[\leadsto \color{blue}{x} \cdot \frac{\frac{1}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\]

    10. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

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

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

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