Average Error: 2.1 → 0.2
Time: 1.8m
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.4115995146336426 \cdot 10^{-10}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.4115995146336426e-10

    1. Initial program 0.1

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

    if 3.4115995146336426e-10 < k

    1. Initial program 5.2

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

    3. Applied clear-num5.4

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

    4. Applied simplify5.4

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

    5. Taylor expanded around inf 5.4

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

    6. Applied simplify0.5

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))