Average Error: 2.0 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.0202661275311305 \cdot 10^{+148}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(k \cdot \left(10 + k\right) + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(1 + \frac{10}{k}\right) \cdot \left(\left(\frac{a}{k} \cdot \sqrt{\frac{{k}^{m}}{k}}\right) \cdot \sqrt{\frac{{k}^{m}}{k}}\right) + \left(\frac{a \cdot 101}{\frac{{k}^{4}}{{k}^{m}}}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.0202661275311305e+148

    1. Initial program 0.1

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

    2. Applied simplify0.1

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

    if 3.0202661275311305e+148 < k

    1. Initial program 9.8

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

    2. Applied simplify9.8

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

    3. Taylor expanded around inf 9.8

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{2}} + \left(10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}} + 101 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}}\right)}\]

    4. Applied simplify0.3

      \[\leadsto \color{blue}{(\left(1 + \frac{10}{k}\right) \cdot \left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right) + \left(\frac{a \cdot 101}{\frac{{k}^{4}}{{k}^{m}}}\right))_*}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.4

      \[\leadsto (\left(1 + \frac{10}{k}\right) \cdot \left(\frac{a}{k} \cdot \color{blue}{\left(\sqrt{\frac{{k}^{m}}{k}} \cdot \sqrt{\frac{{k}^{m}}{k}}\right)}\right) + \left(\frac{a \cdot 101}{\frac{{k}^{4}}{{k}^{m}}}\right))_*\]

    7. Applied associate-*r*0.3

      \[\leadsto (\left(1 + \frac{10}{k}\right) \cdot \color{blue}{\left(\left(\frac{a}{k} \cdot \sqrt{\frac{{k}^{m}}{k}}\right) \cdot \sqrt{\frac{{k}^{m}}{k}}\right)} + \left(\frac{a \cdot 101}{\frac{{k}^{4}}{{k}^{m}}}\right))_*\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))