Average Error: 14.4 → 0.1
Time: 10.7s
Precision: 64
Internal Precision: 832
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -179.44688575007123 \lor \neg \left(x \le 172.35174017389497\right):\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} + \frac{-2}{{x}^{6}}\right) + \frac{-2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Error

Bits error versus x

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -179.44688575007123 or 172.35174017389497 < x

    1. Initial program 29.4

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

    3. Applied add-cube-cbrt50.9

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

    4. Applied associate-/r*52.2

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

    5. Taylor expanded around inf 0.8

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

    6. Simplified0.8

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

    8. Applied associate-/r*0.1

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

    if -179.44688575007123 < x < 172.35174017389497

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -179.44688575007123 \lor \neg \left(x \le 172.35174017389497\right):\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} + \frac{-2}{{x}^{6}}\right) + \frac{-2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Runtime

Time bar (total: 10.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes25.50.10.025.599.7%
herbie shell --seed 2018340 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))