Average Error: 29.5 → 0.1
Time: 5.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13330.589790459633 \lor \neg \left(x \le 18853.62828990301\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}}\right) + \log \left(\sqrt{e^{(\left((x \cdot x + \left(1 + x\right))_*\right) \cdot \left(\frac{-1 - x}{(x \cdot \left(x \cdot x\right) + -1)_*}\right) + \left(\frac{x}{1 + x}\right))_*}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13330.589790459633 or 18853.62828990301 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.0

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

    if -13330.589790459633 < x < 18853.62828990301

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

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

    8. Applied flip3--0.1

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

    9. Applied associate-/r/0.1

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

    10. Applied add-sqr-sqrt30.3

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

    11. Applied prod-diff30.3

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

    12. Simplified0.1

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

    13. Simplified0.1

      \[\leadsto \log \left(\sqrt{e^{(\left((x \cdot x + \left(1 + x\right))_*\right) \cdot \left(\frac{-1 - x}{(x \cdot \left(x \cdot x\right) + -1)_*}\right) + \left(\frac{x}{1 + x}\right))_* + \color{blue}{0}}}\right) + \log \left(\sqrt{e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13330.589790459633 \lor \neg \left(x \le 18853.62828990301\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}}\right) + \log \left(\sqrt{e^{(\left((x \cdot x + \left(1 + x\right))_*\right) \cdot \left(\frac{-1 - x}{(x \cdot \left(x \cdot x\right) + -1)_*}\right) + \left(\frac{x}{1 + x}\right))_*}}\right)\\ \end{array}\]

Runtime

Time bar (total: 5.6m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.50.10.029.599.9%
herbie shell --seed 2018297 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))