Average Error: 28.6 → 0.1
Time: 5.4m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11423.365833095679 \lor \neg \left(x \le 12602.174557478349\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right) \cdot x - \left(-1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(1 + x\right)}\\ \end{array}\]

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 < -11423.365833095679 or 12602.174557478349 < x

    1. Initial program 59.2

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

    3. Applied div-inv59.3

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

    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 -11423.365833095679 < x < 12602.174557478349

    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 flip-+0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}} \cdot \frac{1}{x - 1}\]

    6. Applied frac-times0.1

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

    7. Applied frac-sub0.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11423.365833095679 \lor \neg \left(x \le 12602.174557478349\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-1 + x\right) \cdot \left(-1 + x\right)\right) \cdot x - \left(-1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot \left(1 + x\right)}\\ \end{array}\]

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))