Error

Derivation

1. Split input into 2 regimes

2. if k < 19905263.164965536

   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 div-inv0.0

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

   if 19905263.164965536 < k

   1. Initial program 5.5

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

   2. Simplified5.5

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

   3. Taylor expanded around -inf 62.9

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

   4. Simplified0.6

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

   5. Taylor expanded around -inf 62.9

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

   6. Simplified0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 19905263.164965536:\\ \;\;\;\;\frac{1}{(k \cdot \left(k + 10\right) + 1)_*} \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{\frac{a}{k} \cdot {k}^{m}}{k}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +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: 22.3s)Debug log