Average Error: 2.1 → 0.2
Time: 50.6s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.9827950205668443 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{a}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}} \cdot {k}^{m}}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.9827950205668443e+118

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied times-frac0.1

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

    9. Simplified0.1

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

    if 3.9827950205668443e+118 < k

    1. Initial program 8.8

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

    2. Initial simplification8.8

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

    4. Applied add-sqr-sqrt8.8

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

    5. Applied associate-/r*8.8

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

    6. Taylor expanded around -inf 63.0

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3.9827950205668443 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{a}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}} \cdot {k}^{m}}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 50.6s)Debug logProfile

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