Average Error: 29.4 → 0.1
Time: 37.1s
Precision: 64
Internal Precision: 576
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12713.063048510125 \lor \neg \left(x \le 12649.733194213431\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right)}} \cdot \sqrt[3]{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right) \cdot \sqrt[3]{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\\ \end{array}\]

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 < -12713.063048510125 or 12649.733194213431 < 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

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

    if -12713.063048510125 < x < 12649.733194213431

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    5. Applied add-cbrt-cube0.1

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

    7. Applied add-log-exp0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12713.063048510125 \lor \neg \left(x \le 12649.733194213431\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right)}} \cdot \sqrt[3]{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right) \cdot \sqrt[3]{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\\ \end{array}\]

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))