Average Error: 1.9 → 0.2
Time: 52.1s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.348174326170382 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{k \cdot k + \left(k \cdot 10 + 1\right)} \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{a \cdot e^{m \cdot \log k}}\right) \cdot k + \left((\left(\frac{k}{a \cdot e^{m \cdot \log k}}\right) \cdot 10 + \left(\frac{1}{a \cdot e^{m \cdot \log k}}\right))_*\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 4.348174326170382e+98

    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 div-inv0.1

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

    if 4.348174326170382e+98 < k

    1. Initial program 7.1

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

    3. Applied clear-num7.3

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

    4. Taylor expanded around inf 7.3

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

    5. Simplified0.7

      \[\leadsto \frac{1}{\color{blue}{(\left(\frac{k}{e^{\log k \cdot m} \cdot a}\right) \cdot k + \left((\left(\frac{k}{e^{\log k \cdot m} \cdot a}\right) \cdot 10 + \left(\frac{1}{e^{\log k \cdot m} \cdot a}\right))_*\right))_*}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.348174326170382 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{k \cdot k + \left(k \cdot 10 + 1\right)} \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{a \cdot e^{m \cdot \log k}}\right) \cdot k + \left((\left(\frac{k}{a \cdot e^{m \cdot \log k}}\right) \cdot 10 + \left(\frac{1}{a \cdot e^{m \cdot \log k}}\right))_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 52.1s)Debug logProfile

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