Average Error: 31.1 → 30.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - (\left(-1 - x\right) \cdot x + -1)_*\]

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Taylor expanded around 0 30.7

    \[\leadsto \frac{1}{x + 1} - \color{blue}{\left(-\left(x + \left({x}^{2} + 1\right)\right)\right)}\]

  3. Simplified30.7

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

  5. Applied add-cube-cbrt30.7

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - (\left(-1 - x\right) \cdot x + -1)_*\]

  6. Applied associate-/r*30.7

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}}} - (\left(-1 - x\right) \cdot x + -1)_*\]

  7. Final simplification30.7

    \[\leadsto \frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - (\left(-1 - x\right) \cdot x + -1)_*\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))