Average Error: 28.4 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -939213.3629633987 \lor \neg \left(x \le 125874.03471477641\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(-3 \cdot x + -1)_*}{\frac{(\left(-1 + x\right) \cdot \left({x}^{3}\right) + \left(-1 + x\right))_*}{\left(1 - x\right) + x \cdot x}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -939213.3629633987 or 125874.03471477641 < x

    1. Initial program 59.6

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

    3. Applied div-inv59.7

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

    4. Applied fma-neg60.5

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

    5. Taylor expanded around inf 0.3

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

    6. 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 -939213.3629633987 < x < 125874.03471477641

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    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 flip3-+0.0

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

    8. Applied associate-*l/0.0

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

    9. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -939213.3629633987 \lor \neg \left(x \le 125874.03471477641\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(-3 \cdot x + -1)_*}{\frac{(\left(-1 + x\right) \cdot \left({x}^{3}\right) + \left(-1 + x\right))_*}{\left(1 - x\right) + x \cdot x}}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))