Average Error: 29.2 → 0.0
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.285358430037734 \cdot 10^{+17} \lor \neg \left(x \le 112879.67300606739\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(-3 \cdot x + -1)_*}{\left(x \cdot x - 1\right) \cdot \left(x \cdot x - 1\right)} \cdot \left(\left(x - 1\right) \cdot \left(1 + x\right)\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.285358430037734e+17 or 112879.67300606739 < x

    1. Initial program 60.0

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

    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

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

    if -2.285358430037734e+17 < x < 112879.67300606739

    1. Initial program 0.8

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

    3. Applied frac-sub0.8

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

    4. Taylor expanded around -inf 0.0

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

    5. Simplified0.0

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

    7. Applied flip--0.0

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

    8. Applied flip-+0.0

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

    9. Applied frac-times0.0

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

    10. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.285358430037734 \cdot 10^{+17} \lor \neg \left(x \le 112879.67300606739\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(-3 \cdot x + -1)_*}{\left(x \cdot x - 1\right) \cdot \left(x \cdot x - 1\right)} \cdot \left(\left(x - 1\right) \cdot \left(1 + x\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes23.40.00.023.399.9%
herbie shell --seed 2018339 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))