Average Error: 29.2 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x - 1 \le -106744.7941190454 \lor \neg \left(x - 1 \le 25.696485874931508\right):\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} + \left(\frac{-3}{x} - \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(x - 1\right) \cdot \frac{x}{x \cdot x - 1} - \frac{1 + x}{x - 1}}\right)\\ \end{array}\]

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) < -106744.7941190454 or 25.696485874931508 < (- x 1)

    1. Initial program 59.1

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

    3. Applied flip-+60.3

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

    4. Applied associate-/r/60.2

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

    6. Applied add-log-exp60.2

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

    7. Taylor expanded around inf 0.4

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

    8. Simplified0.1

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

    if -106744.7941190454 < (- x 1) < 25.696485874931508

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

    6. Applied add-log-exp0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \le -106744.7941190454 \lor \neg \left(x - 1 \le 25.696485874931508\right):\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} + \left(\frac{-3}{x} - \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(x - 1\right) \cdot \frac{x}{x \cdot x - 1} - \frac{1 + x}{x - 1}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))