Average Error: 14.7 → 0.0
Time: 19.4s
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -241882970083935.34 \lor \neg \left(x \le 123440394.66957925\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -241882970083935.34 or 123440394.66957925 < x

    1. Initial program 30.6

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

    2. Initial simplification30.6

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

    3. Taylor expanded around inf 0.0

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

    if -241882970083935.34 < x < 123440394.66957925

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

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

    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 *-un-lft-identity0.0

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

    10. Applied times-frac0.0

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

    11. Simplified0.0

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

    12. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -241882970083935.34 \lor \neg \left(x \le 123440394.66957925\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Runtime

Time bar (total: 19.4s)Debug logProfile

herbie shell --seed 2018250 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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