Average Error: 31.1 → 30.7
Time: 21.2s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)}}{\sqrt[3]{\sqrt[3]{x + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)} - \left(\left(-1 - x\right) - x \cdot x\right)\]

Error

Bits error versus x

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Results

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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

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

  5. Applied add-cube-cbrt30.7

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

  6. Applied associate-/r*30.7

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}}} - \left(\left(-1 - x\right) - x \cdot x\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt30.7

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

  9. Applied associate-*l*30.7

    \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1}\right)}}}{\sqrt[3]{x + 1}} - \left(\left(-1 - x\right) - x \cdot x\right)\]
  10. Using strategy rm

  11. Applied add-cube-cbrt30.7

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

  12. Final simplification30.7

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

Runtime

Time bar (total: 21.2s)Debug logProfile

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