Average Error: 29.2 → 0.0
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -20771570235.688507 \lor \neg \left(x \le 125538.89122461753\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -9 \cdot \left(x \cdot x\right)}{\left(x \cdot 3 + -1\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -20771570235.688507 or 125538.89122461753 < x

    1. Initial program 59.7

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

    3. Applied frac-sub61.5

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

    4. Taylor expanded around -inf 0.3

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

    5. Simplified0.0

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

    if -20771570235.688507 < x < 125538.89122461753

    1. Initial program 0.3

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

    3. Applied frac-sub0.3

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

    7. Applied flip--0.0

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

    8. Applied associate-/l/0.0

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

    9. Simplified0.0

      \[\leadsto \frac{\color{blue}{1 + -9 \cdot \left(x \cdot x\right)}}{\left(\left(x + 1\right) \cdot \left(x - 1\right)\right) \cdot \left(-1 + 3 \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -20771570235.688507 \lor \neg \left(x \le 125538.89122461753\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -9 \cdot \left(x \cdot x\right)}{\left(x \cdot 3 + -1\right) \cdot \left(\left(1 + x\right) \cdot \left(x - 1\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))