Error

Derivation

1. Split input into 2 regimes

2. if k < 6.810503467121912e+50

   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 6.810503467121912e+50 < 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 63.0

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

   3. Applied simplify0.2

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))