Average Error: 29.6 → 0.0
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.2757532121631892 \cdot 10^{+18} \lor \neg \left(x \le 166271.6209549078\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \end{array}\]

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 < -2.2757532121631892e+18 or 166271.6209549078 < x

    1. Initial program 59.9

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

    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 -2.2757532121631892e+18 < x < 166271.6209549078

    1. Initial program 1.0

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

    3. Applied div-inv1.0

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

    5. Applied associate-*r/1.0

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

    6. Applied frac-sub1.0

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

    7. Simplified1.0

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

    8. Taylor expanded around 0 0.0

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

    9. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.2757532121631892 \cdot 10^{+18} \lor \neg \left(x \le 166271.6209549078\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018235 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))