Average Error: 29.4 → 0.1
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14037.966152180721:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(1 + {\left(\frac{3}{x}\right)}^{3}\right) + \left(-3\right) \cdot \left(\frac{3}{x} \cdot \frac{3}{x} + \left(1 - \frac{3}{x}\right)\right)}{3 \cdot \left(\frac{3}{x} - 1\right) + x}\\ \mathbf{if}\;x \le 12729.18549022099:\\ \;\;\;\;\frac{\frac{\frac{x \cdot x}{1 + x} \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - {\left(1 + x\right)}^{3}}{\left(x + 1\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(1 + {\left(\frac{3}{x}\right)}^{3}\right) + \left(-3\right) \cdot \left(\frac{3}{x} \cdot \frac{3}{x} + \left(1 - \frac{3}{x}\right)\right)}{3 \cdot \left(\frac{3}{x} - 1\right) + x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -14037.966152180721 or 12729.18549022099 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

    5. Applied distribute-neg-frac0.0

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

    6. Applied flip3-+0.0

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

    7. Applied associate-*l/0.0

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

    8. Applied frac-add0.0

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

    9. Applied simplify0.0

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

    10. Applied simplify0.0

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

    11. Applied simplify0.0

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

    if -14037.966152180721 < x < 12729.18549022099

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied frac-times0.1

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

    6. Applied associate-*r/0.1

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

    7. Applied frac-sub0.1

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

    8. Applied simplify0.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{x \cdot x}{1 + x} \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - {\left(1 + x\right)}^{3}}}{\left(x + 1\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.8m)Debug log

herbie shell --seed '#(2961832646 520228599 1275628947 1047906571 1774476463 2890033825)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))