Average Error: 1.9 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3656945447.528863:\\ \;\;\;\;\frac{a}{\sqrt{1 + \left(k + 10\right) \cdot k}} \cdot \frac{{k}^{m}}{\sqrt{1 + \left(k + 10\right) \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot \frac{-10}{k}\right) \cdot e^{m \cdot \log k} + \left(\frac{\frac{a}{k}}{k} + \frac{a \cdot 99}{{k}^{4}}\right) \cdot e^{m \cdot \log k}\\ \end{array}\]

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 < 3656945447.528863

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

    if 3656945447.528863 < k

    1. Initial program 5.0

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

    3. Applied clear-num5.2

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

    4. Taylor expanded around -inf 62.9

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

    5. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3656945447.528863:\\ \;\;\;\;\frac{a}{\sqrt{1 + \left(k + 10\right) \cdot k}} \cdot \frac{{k}^{m}}{\sqrt{1 + \left(k + 10\right) \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot \frac{-10}{k}\right) \cdot e^{m \cdot \log k} + \left(\frac{\frac{a}{k}}{k} + \frac{a \cdot 99}{{k}^{4}}\right) \cdot e^{m \cdot \log k}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))