Average Error: 2.2 → 0.1
Time: 28.8s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.3603659287148333 \cdot 10^{+124}:\\ \;\;\;\;\frac{a}{\frac{(k \cdot \left(k + 10\right) + 1)_*}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{e^{\log k \cdot m}}{\frac{k}{a} \cdot k}}{k}\right) \cdot -10 + \left((99 \cdot \left(\frac{e^{\log k \cdot m}}{\frac{{k}^{4}}{a}}\right) + \left(\frac{\frac{a}{k}}{k} \cdot e^{\log k \cdot m}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.3603659287148333e+124

    1. Initial program 0.1

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

    2. Simplified0.1

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

    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

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

    4. Applied div-inv9.1

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

    5. Taylor expanded around inf 9.1

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

    6. Simplified0.6

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

    7. Taylor expanded around inf 9.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

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