Average Error: 29.1 → 0.1
Time: 50.2s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -18144.82389771105 \lor \neg \left(x \le 17078.541050548312\right):\\ \;\;\;\;\frac{\frac{-6}{x} - \left(\frac{5}{x \cdot x} + \frac{\frac{16}{x}}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(x \cdot x\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \end{array}\]

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 < -18144.82389771105 or 17078.541050548312 < x

    1. Initial program 59.4

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

    3. Applied flip--59.4

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

    4. Taylor expanded around inf 0.3

      \[\leadsto \frac{\color{blue}{-\left(16 \cdot \frac{1}{{x}^{3}} + \left(5 \cdot \frac{1}{{x}^{2}} + 6 \cdot \frac{1}{x}\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    5. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{-6}{x} - \left(\frac{\frac{16}{x}}{x \cdot x} + \frac{5}{x \cdot x}\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    if -18144.82389771105 < x < 17078.541050548312

    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 frac-times0.1

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

    7. Applied frac-sub0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -18144.82389771105 \lor \neg \left(x \le 17078.541050548312\right):\\ \;\;\;\;\frac{\frac{-6}{x} - \left(\frac{5}{x \cdot x} + \frac{\frac{16}{x}}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(x \cdot x\right) - \left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \end{array}\]

Runtime

Time bar (total: 50.2s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))