Average Error: 2.2 → 0.1
Time: 32.9s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.36069794851597 \cdot 10^{+152}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{k}^{m}}}{k} \cdot \frac{\sqrt{{k}^{m}}}{\frac{k}{a}} - \left({k}^{m} \cdot a\right) \cdot \left(\frac{10}{{k}^{3}} - \frac{99}{{k}^{4}}\right)\\ \end{array}\]

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 < 1.36069794851597e+152

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    if 1.36069794851597e+152 < 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. Taylor expanded around inf 11.0

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

    4. Simplified0.6

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k \cdot \frac{k}{a}} - \left(a \cdot {k}^{m}\right) \cdot \left(\frac{10}{{k}^{3}} - \frac{99}{{k}^{4}}\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.6

      \[\leadsto \frac{\color{blue}{\sqrt{{k}^{m}} \cdot \sqrt{{k}^{m}}}}{k \cdot \frac{k}{a}} - \left(a \cdot {k}^{m}\right) \cdot \left(\frac{10}{{k}^{3}} - \frac{99}{{k}^{4}}\right)\]

    7. Applied times-frac0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.36069794851597 \cdot 10^{+152}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{k}^{m}}}{k} \cdot \frac{\sqrt{{k}^{m}}}{\frac{k}{a}} - \left({k}^{m} \cdot a\right) \cdot \left(\frac{10}{{k}^{3}} - \frac{99}{{k}^{4}}\right)\\ \end{array}\]

Runtime

Time bar (total: 32.9s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))