Average Error: 28.7 → 0.1
Time: 26.2s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -17671.62355682104:\\ \;\;\;\;\frac{\left(\left(-3\right) + \left(\frac{3}{x} - 1\right) \cdot \left(\frac{3}{x} \cdot \left(-3\right)\right)\right) + \left({\left(\frac{3}{x}\right)}^{3} + 1\right) \cdot \frac{-1}{x}}{\left(\frac{3}{x} \cdot x\right) \cdot \left(\frac{3}{x} - 1\right) + x}\\ \mathbf{if}\;x \le 11845.347394511926:\\ \;\;\;\;\frac{x}{{x}^{3} + 1} \cdot \left(\left(1 - x\right) + x \cdot x\right) - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \frac{-3}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -17671.62355682104

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

    5. Applied distribute-neg-frac0.0

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

    6. Applied flip3-+0.0

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

    7. Applied associate-*l/0.0

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

    8. Applied frac-add0.0

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

    9. Applied simplify0.0

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

    10. Applied simplify0.0

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

    if -17671.62355682104 < x < 11845.347394511926

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied simplify0.1

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

    if 11845.347394511926 < x

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -17671.62355682104:\\ \;\;\;\;\frac{\left(\left(-3\right) + \left(\frac{3}{x} - 1\right) \cdot \left(\frac{3}{x} \cdot \left(-3\right)\right)\right) + \left({\left(\frac{3}{x}\right)}^{3} + 1\right) \cdot \frac{-1}{x}}{\left(\frac{3}{x} \cdot x\right) \cdot \left(\frac{3}{x} - 1\right) + x}\\ \mathbf{if}\;x \le 11845.347394511926:\\ \;\;\;\;\frac{x}{{x}^{3} + 1} \cdot \left(\left(1 - x\right) + x \cdot x\right) - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \frac{-3}{x}\\ \end{array}}\]

Runtime

Time bar (total: 26.2s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))