Average Error: 2.2 → 0.1
Time: 30.4s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 7.641740909562312 \cdot 10^{+152}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{a}{k}}{k}\right) \cdot \left({\left(e^{m}\right)}^{\left(\log k\right)}\right) + \left(\left(\frac{99}{{k}^{4}} - \frac{10}{{k}^{3}}\right) \cdot \left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot a\right)\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 7.641740909562312e+152

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    if 7.641740909562312e+152 < k

    1. Initial program 11.1

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

    2. Initial simplification11.1

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 7.641740909562312 \cdot 10^{+152}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{a}{k}}{k}\right) \cdot \left({\left(e^{m}\right)}^{\left(\log k\right)}\right) + \left(\left(\frac{99}{{k}^{4}} - \frac{10}{{k}^{3}}\right) \cdot \left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot a\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 30.4s)Debug logProfile

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