Error

Derivation

1. Split input into 2 regimes

2. if (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))) < 2.1316266130167845e-219

   1. Initial program 59.6

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

   2. Taylor expanded around inf 0.5

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

   3. Applied simplify0.2

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

   if 2.1316266130167845e-219 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

   1. Initial program 0.6

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

   3. Applied flip--0.6

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
   4. Using strategy rm

   5. Applied frac-times0.6

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

   6. Applied associate-*r/0.6

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

   7. Applied frac-sub0.6

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))