Average Error: 29.2 → 0.1
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11149.082968648356 \lor \neg \left(x \le 12057.145931266508\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{5} \cdot \frac{x}{1 + x}}}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -11149.082968648356 or 12057.145931266508 < x

    1. Initial program 59.4

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

    3. Applied div-inv59.5

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

    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 -11149.082968648356 < x < 12057.145931266508

    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 add-cbrt-cube0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Applied cbrt-unprod0.1

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

    8. Simplified0.1

      \[\leadsto \frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\sqrt[3]{\color{blue}{\frac{x}{1 + x} \cdot {\left(\frac{x}{1 + x}\right)}^{5}}} + \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1} + \frac{x}{x + 1} \cdot \frac{x + 1}{x - 1}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11149.082968648356 \lor \neg \left(x \le 12057.145931266508\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \sqrt[3]{{\left(\frac{x}{1 + x}\right)}^{5} \cdot \frac{x}{1 + x}}}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018304 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))