Error

Derivation

1. Split input into 2 regimes

2. if k < 3.1351933829711156e+56

   1. Initial program 0.1

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

   2. Applied simplify0.0

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

   4. Applied add-sqr-sqrt0.1

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

   5. Applied times-frac0.1

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

   if 3.1351933829711156e+56 < k

   1. Initial program 7.0

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

   2. Applied simplify7.0

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

   4. Applied clear-num7.2

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

   5. Taylor expanded around inf 7.2

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

   6. Applied simplify0.7

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

Runtime

Time bar (total: 24.3s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))