Error

Derivation

1. Split input into 2 regimes

2. if k < 8.789402515263768e+149

   1. Initial program 0.1

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

   2. Simplified0.0

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

   4. Applied add-sqr-sqrt0.1

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

   5. Applied times-frac0.1

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

   if 8.789402515263768e+149 < k

   1. Initial program 10.9

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

   2. Simplified10.9

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

   4. Applied associate-/l*10.9

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

   5. Taylor expanded around inf 10.9

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

   6. Simplified0.4

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.789402515263768 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}} \cdot \frac{{k}^{m}}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{(10 \cdot \left(\frac{k}{a}\right) + \left((\left(\frac{k}{a}\right) \cdot k + \left(\frac{1}{a}\right))_*\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Details

Time bar (total: 27.5s)Debug log