Average Error: 0.0 → 0.0
Time: 14.4s
Precision: 64
Internal Precision: 320
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[(\left(\frac{1}{x \cdot x - 1}\right) \cdot \left(x + 1\right) + \left(\log \left(e^{\frac{x}{x + 1}}\right)\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Initial simplification0.0

    \[\leadsto \frac{1}{x - 1} + \frac{x}{x + 1}\]
  3. Using strategy rm

  4. Applied flip--0.0

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

  5. Applied associate-/r/0.0

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

  6. Applied fma-def0.0

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

  7. Simplified0.0

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

  9. Applied add-log-exp0.0

    \[\leadsto (\left(\frac{1}{x \cdot x - 1}\right) \cdot \left(x + 1\right) + \color{blue}{\left(\log \left(e^{\frac{x}{x + 1}}\right)\right)})_*\]

  10. Final simplification0.0

    \[\leadsto (\left(\frac{1}{x \cdot x - 1}\right) \cdot \left(x + 1\right) + \left(\log \left(e^{\frac{x}{x + 1}}\right)\right))_*\]

Runtime

Time bar (total: 14.4s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))