Derivation

Split input into 2 regimes
if k < 5.233993066468966e+98
1. Initial program 0.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Using strategy rm
3. Applied div-inv0.1
  \[\leadsto \color{blue}{\left(a \cdot {k}^{m}\right) \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\]
if 5.233993066468966e+98 < k
1. Initial program 7.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Using strategy rm
3. Applied clear-num7.3
  \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}}\]
4. Taylor expanded around inf 7.3
  \[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a} + \left(\frac{{k}^{2}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a} + \frac{1}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}\right)}}\]
5. Simplified0.6
  \[\leadsto \frac{1}{\color{blue}{\frac{k}{{k}^{m}} \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \frac{{k}^{\left(-m\right)}}{a}}}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.233993066468966 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{k \cdot k + \left(k \cdot 10 + 1\right)} \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \left(\frac{k}{a} + \frac{10}{a}\right) \cdot \frac{k}{{k}^{m}}}\\ \end{array}\]

Runtime

herbie shell --seed 2018234 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

In
Out