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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 2.209127604221486 \cdot 10^{+130}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 99\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if k < 2.209127604221486e+130
1. Initial program 0.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
if 2.209127604221486e+130 < k
1. Initial program 9.9
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Taylor expanded around inf 9.9
  \[\leadsto \color{blue}{\left(99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]
3. Simplified0.3
  \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.209127604221486 \cdot 10^{+130}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 99\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018216 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Error

Try it out

Derivation

`if k < 2.209127604221486e+130`

`if 2.209127604221486e+130 < k`

Runtime

In
Out