Average Error: 29.3 → 0.1
Time: 2.4m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14916.372293359223:\\ \;\;\;\;\frac{\frac{-6}{x} - \left(\frac{5}{x \cdot x} + \frac{\frac{16}{x}}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 15126.382106857445:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) - \left(\frac{\left(1 + x\right) \cdot \frac{1 + x}{x - 1}}{x - 1} \cdot \left(1 + x\right)\right) \cdot \frac{1 + x}{\left(-1 + x\right) \cdot \left(-1 + x\right)}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \frac{\left(1 + x\right) \cdot \frac{1 + x}{x - 1}}{x - 1}}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} + \left(\frac{-3}{x} - \frac{1}{x \cdot x}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -14916.372293359223

    1. Initial program 59.3

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

    3. Applied flip--59.3

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

    5. Applied associate-*l/59.3

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

    6. Taylor expanded around inf 0.4

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

    7. 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 -14916.372293359223 < x < 15126.382106857445

    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 associate-*l/0.1

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

    7. Applied flip--0.1

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

    9. Applied flip--0.1

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

    10. Applied associate-/r/0.1

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

    11. Applied associate-*l*0.1

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

    12. Simplified0.1

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

    if 15126.382106857445 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14916.372293359223:\\ \;\;\;\;\frac{\frac{-6}{x} - \left(\frac{5}{x \cdot x} + \frac{\frac{16}{x}}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 15126.382106857445:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) - \left(\frac{\left(1 + x\right) \cdot \frac{1 + x}{x - 1}}{x - 1} \cdot \left(1 + x\right)\right) \cdot \frac{1 + x}{\left(-1 + x\right) \cdot \left(-1 + x\right)}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \frac{\left(1 + x\right) \cdot \frac{1 + x}{x - 1}}{x - 1}}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} + \left(\frac{-3}{x} - \frac{1}{x \cdot x}\right)\\ \end{array}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))