Average Error: 2.1 → 0.1
Time: 42.1s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 163036002.53249753:\\ \;\;\;\;\frac{a}{\sqrt[3]{\left(k + 10\right) \cdot k + 1} \cdot \sqrt[3]{\left(k + 10\right) \cdot k + 1}} \cdot \frac{{k}^{m}}{\sqrt[3]{1 + \left(10 + k\right) \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k} \cdot \left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) + \frac{{k}^{m} \cdot 99}{\frac{{k}^{4}}{a}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 163036002.53249753

    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 add-cube-cbrt0.1

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}}\]

    4. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}}\]

    5. Applied simplify0.1

      \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(k + 10\right) \cdot k + 1} \cdot \sqrt[3]{\left(k + 10\right) \cdot k + 1}}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}\]

    6. Applied simplify0.0

      \[\leadsto \frac{a}{\sqrt[3]{\left(k + 10\right) \cdot k + 1} \cdot \sqrt[3]{\left(k + 10\right) \cdot k + 1}} \cdot \color{blue}{\frac{{k}^{m}}{\sqrt[3]{1 + \left(10 + k\right) \cdot k}}}\]

    if 163036002.53249753 < k

    1. Initial program 5.5

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

    2. Taylor expanded around inf 5.5

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

    3. Applied simplify0.1

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

Runtime

Time bar (total: 42.1s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))