Average Error: 28.9 → 0.1
Time: 46.1s
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15984.259334689732 \lor \neg \left(x \le 9084.462165022695\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - x}{-1 + x} \cdot \left({x}^{3} + 1\right) + x \cdot \left(\left(1 - x\right) + x \cdot x\right)}{\left(\left(1 - x\right) + x \cdot x\right) \cdot \left(1 + x\right)}\\ \end{array}\]

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 < -15984.259334689732 or 9084.462165022695 < x

    1. Initial program 59.2

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

    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 -15984.259334689732 < x < 9084.462165022695

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    5. Applied flip3-+0.1

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

    6. Applied associate-*l/0.1

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

    7. Applied frac-sub0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15984.259334689732 \lor \neg \left(x \le 9084.462165022695\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - x}{-1 + x} \cdot \left({x}^{3} + 1\right) + x \cdot \left(\left(1 - x\right) + x \cdot x\right)}{\left(\left(1 - x\right) + x \cdot x\right) \cdot \left(1 + x\right)}\\ \end{array}\]

Runtime

Time bar (total: 46.1s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))