Average Error: 14.4 → 0.2
Time: 12.4s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -203.6589479322794:\\ \;\;\;\;\left(\frac{-2}{x \cdot x} + \frac{-2}{{x}^{6}}\right) + \frac{-2}{{x}^{4}}\\ \mathbf{elif}\;x \le 226.7865855542354:\\ \;\;\;\;\frac{1}{1 + x} - \sqrt[3]{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} + \frac{\frac{-2}{x}}{x}\right) + \frac{-2}{{x}^{4}}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -203.6589479322794

    1. Initial program 28.4

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

    2. Taylor expanded around -inf 0.6

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\frac{-2}{{x}^{4}} + \left(\frac{-2}{{x}^{6}} + \frac{-2}{x \cdot x}\right)}\]

    if -203.6589479322794 < x < 226.7865855542354

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    if 226.7865855542354 < x

    1. Initial program 28.6

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

    2. Taylor expanded around -inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    3. Simplified0.8

      \[\leadsto \color{blue}{\frac{-2}{{x}^{4}} + \left(\frac{-2}{{x}^{6}} + \frac{-2}{x \cdot x}\right)}\]

    4. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{6}}\right)\right)}\]

    5. Simplified0.1

      \[\leadsto \color{blue}{\frac{-2}{{x}^{4}} + \left(\frac{-2}{{x}^{6}} + \frac{\frac{-2}{x}}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -203.6589479322794:\\ \;\;\;\;\left(\frac{-2}{x \cdot x} + \frac{-2}{{x}^{6}}\right) + \frac{-2}{{x}^{4}}\\ \mathbf{elif}\;x \le 226.7865855542354:\\ \;\;\;\;\frac{1}{1 + x} - \sqrt[3]{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} + \frac{\frac{-2}{x}}{x}\right) + \frac{-2}{{x}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))