Average Error: 29.2 → 0.0
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x - 1 \le -4.031170491600452 \cdot 10^{+16} \lor \neg \left(x - 1 \le 890817713.4035921\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{(x \cdot \left(x - \left(2 + x\right)\right) + \left(-1 - x\right))_*}{\left(x - 1\right) \cdot \left(x \cdot x - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- x 1) < -4.031170491600452e+16 or 890817713.4035921 < (- x 1)

    1. Initial program 60.2

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

    3. Applied add-log-exp60.2

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

    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

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

    if -4.031170491600452e+16 < (- x 1) < 890817713.4035921

    1. Initial program 0.8

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

    3. Applied add-log-exp0.8

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

    5. Applied div-inv0.8

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

    6. Applied exp-prod2.0

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

    7. Applied log-pow2.0

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

    9. Applied associate-*l/2.0

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

    10. Applied frac-sub2.1

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

    11. Simplified0.1

      \[\leadsto \frac{\color{blue}{(x \cdot \left(x - \left(x + 2\right)\right) + \left(-1 - x\right))_*}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
    12. Using strategy rm

    13. Applied flip-+0.1

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

    14. Applied associate-*l/0.1

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

    15. Applied associate-/r/0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \le -4.031170491600452 \cdot 10^{+16} \lor \neg \left(x - 1 \le 890817713.4035921\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{(x \cdot \left(x - \left(2 + x\right)\right) + \left(-1 - x\right))_*}{\left(x - 1\right) \cdot \left(x \cdot x - 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))