Error

Derivation

1. Split input into 2 regimes

2. if k < 5.89303279033011e+85

   1. Initial program 0.1

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

   2. Simplified0.1

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

   4. Applied add-cube-cbrt0.1

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

   5. Applied associate-*l*0.1

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

   if 5.89303279033011e+85 < k

   1. Initial program 7.3

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

   2. Simplified7.3

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

   4. Applied associate-/l*7.4

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

   5. Taylor expanded around 0 7.4

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

   6. Simplified0.6

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.89303279033011 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(a \cdot \sqrt[3]{{k}^{m}}\right)}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{(10 \cdot \left(\frac{k}{a}\right) + \left((\left(\frac{k}{a}\right) \cdot k + \left(\frac{1}{a}\right))_*\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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: 21.1s)Debug log