Average Error: 29.5 → 0.0
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0027487462279761 \cdot 10^{+33} \lor \neg \left(x \le 616004193554312.5\right):\\ \;\;\;\;\left(-\frac{3}{x}\right) - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) - x}{\left(x - 1\right) \cdot \left(1 + x\right)} - \frac{1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0027487462279761e+33 or 616004193554312.5 < x

    1. Initial program 60.4

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

    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

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

    if -1.0027487462279761e+33 < x < 616004193554312.5

    1. Initial program 2.8

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

    3. Applied frac-sub2.7

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

    5. Applied distribute-lft-in2.7

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

    6. Applied associate--r+2.5

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

    7. Applied simplify0.0

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

    9. Applied div-sub0.0

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

    10. Applied simplify0.0

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

    11. Applied simplify0.0

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

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -1.0027487462279761 \cdot 10^{+33} \lor \neg \left(x \le 616004193554312.5\right):\\ \;\;\;\;\left(-\frac{3}{x}\right) - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) - x}{\left(x - 1\right) \cdot \left(1 + x\right)} - \frac{1}{x - 1}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))