Error

Derivation

1. Split input into 2 regimes

2. if k < 6.984339818149475e+132

   1. Initial program 0.1

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

   2. Initial simplification0.0

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

   3. Taylor expanded around inf 19.5

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

   4. Simplified0.0

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

   if 6.984339818149475e+132 < k

   1. Initial program 8.7

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

   2. Initial simplification8.7

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

   4. Applied associate-/l*8.8

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

   5. Taylor expanded around -inf 8.8

      \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{1}{a} + \left(10 \cdot \frac{k}{a} + \frac{{k}^{2}}{a}\right)}}\]

   6. Simplified0.3

      \[\leadsto \frac{{k}^{m}}{\color{blue}{(10 \cdot \left(\frac{k}{a}\right) + \left((\left(\frac{k}{a}\right) \cdot k + \left(\frac{1}{a}\right))_*\right))_*}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 6.984339818149475 \cdot 10^{+132}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{(10 \cdot \left(\frac{k}{a}\right) + \left((\left(\frac{k}{a}\right) \cdot k + \left(\frac{1}{a}\right))_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 28.6s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))