Average Error: 2.2 → 0.1
Time: 39.0s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.2565190391246913 \cdot 10^{+153}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{99}{{k}^{4}}\right) \cdot \left(a \cdot {\left(e^{m}\right)}^{\left(\log k\right)}\right) + \left(\left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.2565190391246913e+153

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    if 1.2565190391246913e+153 < k

    1. Initial program 11.4

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

    2. Initial simplification11.4

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

    4. Applied add-sqr-sqrt11.4

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

    5. Applied associate-/r*11.4

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.2565190391246913 \cdot 10^{+153}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(k \cdot \left(k + 10\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{99}{{k}^{4}}\right) \cdot \left(a \cdot {\left(e^{m}\right)}^{\left(\log k\right)}\right) + \left(\left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 39.0s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))