Error

Derivation

1. Split input into 2 regimes

2. if k < 8.236251385880573e+87

   1. Initial program 0.1

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

   2. Simplified0.0

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

   if 8.236251385880573e+87 < k

   1. Initial program 7.8

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

   2. Simplified7.8

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

   3. Taylor expanded around inf 7.8

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

   4. Simplified0.6

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.236251385880573 \cdot 10^{+87}:\\ \;\;\;\;\frac{a}{\frac{(k \cdot \left(k + 10\right) + 1)_*}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{e^{\log k \cdot m}}{\frac{k}{a} \cdot k}}{k}\right) \cdot -10 + \left((99 \cdot \left(\frac{e^{\log k \cdot m}}{\frac{{k}^{4}}{a}}\right) + \left(\frac{e^{\log k \cdot m}}{\frac{k}{a} \cdot k}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))