Average Error: 29.5 → 0.0
Time: 57.1s
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -54014889895523304.0 \lor \neg \left(x \le 40872809330317.07\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{\left(x \cdot x - 1\right) \cdot \left(x - 1\right)} \cdot \left(x - 1\right)\\ \end{array}\]

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 < -54014889895523304.0 or 40872809330317.07 < x

    1. Initial program 60.4

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

    3. Applied frac-sub62.2

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

    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 -54014889895523304.0 < x < 40872809330317.07

    1. Initial program 1.1

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

    3. Applied frac-sub1.0

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

    4. Taylor expanded around 0 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

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

    8. Applied associate-*l/0.0

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

    9. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -54014889895523304.0 \lor \neg \left(x \le 40872809330317.07\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{\left(x \cdot x - 1\right) \cdot \left(x - 1\right)} \cdot \left(x - 1\right)\\ \end{array}\]

Runtime

Time bar (total: 57.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes20.10.00.020.1100%
herbie shell --seed 2018296 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))