Average Error: 29.2 → 0.1
Time: 3.2m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12818.04570684953 \lor \neg \left(x \le 13658.098196859137\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}}}{{\left(\frac{x}{1 + x}\right)}^{3} \cdot {\left(\frac{x}{1 + x}\right)}^{3} + \left({\left(\frac{x}{1 + x}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3} + {\left(\frac{1 + x}{x - 1}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

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 < -12818.04570684953 or 13658.098196859137 < x

    1. Initial program 59.4

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

    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

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

    if -12818.04570684953 < x < 13658.098196859137

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    5. Applied flip3--0.1

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

    7. Applied pow-pow0.1

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

    9. Applied add-cube-cbrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12818.04570684953 \lor \neg \left(x \le 13658.098196859137\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{9} - {\left({\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}^{3}}}{{\left(\frac{x}{1 + x}\right)}^{3} \cdot {\left(\frac{x}{1 + x}\right)}^{3} + \left({\left(\frac{x}{1 + x}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3} + {\left(\frac{1 + x}{x - 1}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3}\right)}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018258 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))