Average Error: 1.8 → 0.1
Time: 21.5s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 9.792334190194593 \cdot 10^{+89}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;(100 \cdot \left(\frac{{k}^{m}}{{k}^{4}} \cdot a\right) + \left(\frac{{k}^{m}}{k} \cdot \frac{a}{k}\right))_* - (\left(\frac{{k}^{m}}{\frac{k \cdot k}{\frac{a}{k}}}\right) \cdot 10 + \left(\frac{{\left(e^{1 + -2}\right)}^{\left(\log k \cdot \left(-m\right)\right)}}{\frac{{k}^{4}}{a}}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 9.792334190194593e+89

    1. Initial program 0.1

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

    if 9.792334190194593e+89 < k

    1. Initial program 6.7

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

    2. Applied simplify6.7

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

    4. Applied clear-num6.8

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

    5. Taylor expanded around inf 6.8

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

    6. Applied simplify0.6

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

    7. Taylor expanded around inf 27.9

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - \left(\frac{e^{\log \left(\frac{1}{k}\right) \cdot m} \cdot \left(e^{-2 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a\right)}{{k}^{4}} + 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}\right)}\]

    8. Applied simplify0.1

      \[\leadsto \color{blue}{(100 \cdot \left(\frac{{k}^{m}}{{k}^{4}} \cdot a\right) + \left(\frac{{k}^{m}}{k} \cdot \frac{a}{k}\right))_* - (\left(\frac{{k}^{m}}{\frac{k \cdot k}{\frac{a}{k}}}\right) \cdot 10 + \left(\frac{{\left(e^{1 + -2}\right)}^{\left(\log k \cdot \left(-m\right)\right)}}{\frac{{k}^{4}}{a}}\right))_*}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 21.5s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))