Average Error: 2.0 → 0.1
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.403049411769406 \cdot 10^{+127}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \left(\frac{1}{\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}} \cdot \frac{\frac{k}{a}}{\sqrt[3]{{k}^{m}}}\right) \cdot \left(10 + k\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 5.403049411769406e+127

    1. Initial program 0.1

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

    if 5.403049411769406e+127 < k

    1. Initial program 8.5

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

    3. Applied clear-num8.6

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

    4. Applied simplify8.6

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

    5. Taylor expanded around inf 8.6

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

    6. Applied simplify0.4

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

    8. Applied add-cube-cbrt0.4

      \[\leadsto \frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{\color{blue}{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \sqrt[3]{{k}^{m}}}} \cdot \left(10 + k\right)}\]

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

      \[\leadsto \frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\color{blue}{1 \cdot \frac{k}{a}}}{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \sqrt[3]{{k}^{m}}} \cdot \left(10 + k\right)}\]

    10. Applied times-frac0.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))