Average Error: 2.2 → 0.1
Time: 6.8m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.3603659287148333 \cdot 10^{+124}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot a}{\frac{{k}^{4}}{e^{\log k \cdot m}}} + \left(e^{\log k \cdot m} - \frac{e^{\log k \cdot m}}{k} \cdot 10\right) \cdot \frac{\frac{a}{k}}{k}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.3603659287148333e+124

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 1.3603659287148333e+124 < k

    1. Initial program 9.1

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

    2. Simplified9.1

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified0.9

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

    5. Taylor expanded around -inf 63.0

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

    6. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.3603659287148333 \cdot 10^{+124}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot a}{\frac{{k}^{4}}{e^{\log k \cdot m}}} + \left(e^{\log k \cdot m} - \frac{e^{\log k \cdot m}}{k} \cdot 10\right) \cdot \frac{\frac{a}{k}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))