Average Error: 2.0 → 0.1
Time: 58.0s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.233577604930774 \cdot 10^{+136}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot a\right) \cdot \left(\frac{99}{{k}^{4}} - \frac{\frac{10}{k}}{k \cdot k}\right) + \left(\frac{a}{k} \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 5.233577604930774e+136

    1. Initial program 0.1

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

    2. Applied simplify0.1

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

    if 5.233577604930774e+136 < k

    1. Initial program 9.2

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

    2. Applied simplify9.2

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

    4. Applied clear-num9.2

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

    5. Taylor expanded around inf 9.2

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

    6. Applied simplify0.4

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

    7. Taylor expanded around -inf 63.0

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

    8. Applied simplify0.2

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

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))