Average Error: 29.5 → 0.2
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 7.416016917401288 \cdot 10^{-06}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x - 1}\right) \cdot \left(1 + x\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))) < 7.416016917401288e-06

    1. Initial program 58.8

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    3. Applied simplify0.4

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

    if 7.416016917401288e-06 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-sqr-sqrt31.4

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

    6. Applied prod-diff31.4

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

    7. Applied simplify0.1

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

    8. Applied simplify0.1

      \[\leadsto (\left(\frac{-1}{x - 1}\right) \cdot \left(1 + x\right) + \left(\frac{x}{1 + x}\right))_* + \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 7.416016917401288 \cdot 10^{-06}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x - 1}\right) \cdot \left(1 + x\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))