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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 5.659800354484786 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(a \cdot \sqrt{{k}^{m}}\right) \cdot \sqrt{{k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if k < 5.659800354484786e+48
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 add-sqr-sqrt0.1
  \[\leadsto \frac{a \cdot \color{blue}{\left(\sqrt{{k}^{m}} \cdot \sqrt{{k}^{m}}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
4. Applied associate-*r*0.1
  \[\leadsto \frac{\color{blue}{\left(a \cdot \sqrt{{k}^{m}}\right) \cdot \sqrt{{k}^{m}}}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
if 5.659800354484786e+48 < k
1. Initial program 5.9
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Taylor expanded around inf 5.9
  \[\leadsto \color{blue}{\left(99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]
3. Applied simplify0.3
  \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2019053 
(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 < 5.659800354484786e+48`

`if 5.659800354484786e+48 < k`

Runtime

In
Out