Error

Derivation

1. Split input into 2 regimes

2. if k < 1.3283693179329341e+154

   1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

   2. Simplified0.0

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}}\]
   3. Using strategy rm

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

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{1 \cdot (k \cdot \left(k + 10\right) + 1)_*}}\]

   5. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{{k}^{m}}{1} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}}\]

   6. Simplified0.0

      \[\leadsto \color{blue}{{k}^{m}} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\]

   if 1.3283693179329341e+154 < k

   1. Initial program 10.3

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

   2. Simplified10.3

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity10.3

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{1 \cdot (k \cdot \left(k + 10\right) + 1)_*}}\]

   5. Applied times-frac10.2

      \[\leadsto \color{blue}{\frac{{k}^{m}}{1} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}}\]

   6. Simplified10.2

      \[\leadsto \color{blue}{{k}^{m}} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt10.2

      \[\leadsto \color{blue}{\left(\sqrt{{k}^{m}} \cdot \sqrt{{k}^{m}}\right)} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\]

   9. Applied associate-*l*10.2

      \[\leadsto \color{blue}{\sqrt{{k}^{m}} \cdot \left(\sqrt{{k}^{m}} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\right)}\]

   10. Taylor expanded around -inf 63.0

       \[\leadsto \color{blue}{\left(99 \cdot \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{4}} + \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{2}}\right) - 10 \cdot \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{3}}}\]

   11. Simplified0.6

       \[\leadsto \color{blue}{(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right))_*\right))_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.3283693179329341 \cdot 10^{+154}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Details

Time bar (total: 23.4s)Debug log