Average Error: 2.1 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.603101957022493 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{k \cdot k + \left(k \cdot 10 + 1\right)} \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99 + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k} - e^{m \cdot \log k} \cdot \left(\frac{10}{k} \cdot \frac{\frac{a}{k}}{k}\right)\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.603101957022493e+58

    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 1.603101957022493e+58 < k

    1. Initial program 6.5

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

    3. Applied div-inv6.5

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

    4. Taylor expanded around -inf 63.0

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

    5. Simplified0.2

      \[\leadsto \color{blue}{99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \left(0 + \log k\right)} - \left(\frac{10}{k} \cdot \frac{\frac{a}{k}}{k}\right) \cdot e^{m \cdot \left(0 + \log k\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.603101957022493 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{k \cdot k + \left(k \cdot 10 + 1\right)} \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99 + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k} - e^{m \cdot \log k} \cdot \left(\frac{10}{k} \cdot \frac{\frac{a}{k}}{k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))