Error

Derivation

1. Split input into 2 regimes

2. if k < 1.247255618115413e+154

   1. Initial program 0.1

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

   2. Applied simplify0.0

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

   if 1.247255618115413e+154 < k

   1. Initial program 10.2

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

   2. Applied simplify10.2

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

   4. Applied clear-num10.2

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

   5. Taylor expanded around inf 10.2

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

   6. Applied simplify0.6

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

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))