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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 1.6334911283145693 \cdot 10^{+87}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{k}{a} + \frac{10}{a}\right) + \left(\frac{\frac{1}{a}}{{k}^{m}}\right))_*}\\ \end{array}\]

Derivation

Split input into 2 regimes
if k < 1.6334911283145693e+87
1. Initial program 0.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Applied simplify0.0
  \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}}\]
if 1.6334911283145693e+87 < k
1. Initial program 7.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Applied simplify7.1
  \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}}\]
3. Using strategy rm
4. Applied clear-num7.3
  \[\leadsto \color{blue}{\frac{1}{\frac{(\left(10 + k\right) \cdot k + 1)_*}{{k}^{m} \cdot a}}}\]
5. Taylor expanded around inf 7.3
  \[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + \left(\frac{1}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a} + \frac{{k}^{2}}{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}\right)}}\]
6. Applied simplify0.4
  \[\leadsto \color{blue}{\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{k}{a} + \frac{10}{a}\right) + \left(\frac{\frac{1}{a}}{{k}^{m}}\right))_*}}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Error

Derivation

`if k < 1.6334911283145693e+87`

`if 1.6334911283145693e+87 < k`

Runtime