Average Error: 2.1 → 0.1
Time: 2.9m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.343840321626663 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{(\left(k + 10\right) \cdot k + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left((\left(-\frac{a}{k}\right) \cdot \left(\frac{10}{k}\right) + \left(\frac{a}{k}\right))_*\right) \cdot \left(\frac{{k}^{m}}{k}\right) + \left(\frac{{k}^{m}}{\frac{{k}^{4}}{99 \cdot a}}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.343840321626663e+101

    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 *-un-lft-identity0.1

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

    4. Applied times-frac0.1

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

    5. Applied simplify0.1

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

    6. Applied simplify0.0

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

    if 3.343840321626663e+101 < k

    1. Initial program 7.9

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

    2. Taylor expanded around -inf 63.0

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

    3. Applied simplify0.2

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

    4. Taylor expanded around inf 0.2

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

    5. Applied simplify0.3

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))