Error

Derivation

1. Split input into 2 regimes

2. if x < -7533157453258729.0 or 116650.87853212819 < x

   1. Initial program 59.9

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

   2. Taylor expanded around inf 0.3

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

   3. 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 -7533157453258729.0 < x < 116650.87853212819

   1. Initial program 0.6

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

   3. Applied add-cube-cbrt0.7

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

   4. Applied *-un-lft-identity0.7

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

   5. Applied times-frac0.7

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{x + 1}{\sqrt[3]{x - 1}}}\]
   6. Using strategy rm

   7. Applied frac-times0.7

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

   8. Applied frac-sub0.7

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

   9. Simplified0.1

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

   10. Simplified0.0

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

   12. Applied *-un-lft-identity0.0

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

   13. Applied *-un-lft-identity0.0

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

   14. Applied times-frac0.0

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

   15. Simplified0.0

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

   16. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7533157453258729.0 \lor \neg \left(x \le 116650.87853212819\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 \cdot x + -1}{(x \cdot x + -1)_*}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018285 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))