Average Error: 2.2 → 0.2
Time: 35.4s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.5740549747991725 \cdot 10^{+42}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 99}{\frac{{k}^{4}}{e^{\log k \cdot m}}} + \left(\frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k} - \frac{e^{\log k \cdot m}}{k \cdot \frac{k}{a}} \cdot \frac{10}{k}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 4.5740549747991725e+42

    1. Initial program 0.1

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

    2. Simplified0.0

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

    if 4.5740549747991725e+42 < k

    1. Initial program 6.6

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

    2. Simplified6.6

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

    4. Applied add-sqr-sqrt24.0

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

    5. Taylor expanded around inf 6.6

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

    6. Simplified0.9

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

    8. Applied *-un-lft-identity0.9

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

    9. Applied times-frac0.5

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

    10. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.5740549747991725 \cdot 10^{+42}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(k + 10\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 99}{\frac{{k}^{4}}{e^{\log k \cdot m}}} + \left(\frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k} - \frac{e^{\log k \cdot m}}{k \cdot \frac{k}{a}} \cdot \frac{10}{k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))