Average Error: 2.1 → 0.2
Time: 1.5m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 6.798605532547233 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(a \cdot \sqrt[3]{{k}^{m}}\right)}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < 6.798605532547233e+153

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied associate-*l*0.1

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

    if 6.798605532547233e+153 < k

    1. Initial program 11.0

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

    2. Initial simplification11.0

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

    4. Applied add-cube-cbrt11.0

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

    5. Applied associate-*l*11.0

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

    6. Taylor expanded around inf 11.0

      \[\leadsto \color{blue}{\left(99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]

    7. Simplified0.5

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 6.798605532547233 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(a \cdot \sqrt[3]{{k}^{m}}\right)}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))