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Derivation

1. Split input into 2 regimes

2. if k < 1.4326697182135415e+31

   1. Initial program 0.1

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

   2. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{1 + k \cdot \left(10 + k\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \sqrt[3]{{k}^{m}}\right)} \cdot a}{1 + k \cdot \left(10 + k\right)}\]

   5. Applied associate-*l*0.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot a\right)}}{1 + k \cdot \left(10 + k\right)}\]

   if 1.4326697182135415e+31 < k

   1. Initial program 5.7

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

   2. Applied simplify5.7

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

   3. Taylor expanded around inf 5.7

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{2}} + \left(10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}} + 101 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}}\right)}\]

   4. Applied simplify0.3

      \[\leadsto \color{blue}{\frac{101 \cdot a}{\frac{{k}^{4}}{{k}^{m}}} + \frac{\frac{a}{k}}{k} \cdot \left({k}^{m} + \frac{{k}^{m}}{\frac{k}{10}}\right)}\]
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))