Average Error: 2.1 → 0.1
Time: 58.3s
Precision: 64
Internal Precision: 576
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.8412110490843214 \cdot 10^{+107}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\frac{\left(a - a \cdot \frac{10}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 5.8412110490843214e+107

    1. Initial program 0.1

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

    2. Initial simplification0.0

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

    if 5.8412110490843214e+107 < k

    1. Initial program 7.9

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

    2. Initial simplification7.9

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

    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. Simplified0.2

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

    6. Applied associate-*r/0.2

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

    7. Applied sub-div0.2

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

    8. Applied associate-*r/0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.8412110490843214 \cdot 10^{+107}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\frac{\left(a - a \cdot \frac{10}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}}{k}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 58.3s)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))