Average Error: 2.0 → 0.1
Time: 32.9s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 8.476232649528033 \cdot 10^{+29}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(\left(k + 10\right) \cdot k + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k} + (\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\right) + \left(\frac{\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}}{k} \cdot -10\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 8.476232649528033e+29

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    if 8.476232649528033e+29 < k

    1. Initial program 5.7

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

    2. Simplified5.7

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.476232649528033 \cdot 10^{+29}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{(\left(k + 10\right) \cdot k + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k} + (\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\right) + \left(\frac{\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}}{k} \cdot -10\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))