Error

Derivation

1. Split input into 2 regimes

2. if k < 1.782154686532667e+143

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified0.1

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

   6. Applied *-un-lft-identity0.1

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

   7. Applied times-frac0.1

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

   8. Simplified0.1

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

   if 1.782154686532667e+143 < k

   1. Initial program 10.1

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

   2. Initial simplification10.1

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

   3. Taylor expanded around -inf 63.0

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

   4. Simplified10.1

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

   6. Applied *-un-lft-identity10.1

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

   7. Applied times-frac10.1

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

   8. Simplified10.1

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

   9. Taylor expanded around -inf 63.0

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

   10. Simplified0.8

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.782154686532667 \cdot 10^{+143}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right))_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 20.7s)Debug logProfile

herbie shell --seed 2018339 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))