Average Error: 2.1 → 0.2
Time: 33.8s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.7315380803037458 \cdot 10^{+79}:\\ \;\;\;\;\frac{a}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}} \cdot \frac{{k}^{m}}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}}\\ \mathbf{if}\;k \le +\infty:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.7315380803037458e+79

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied times-frac0.1

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

    5. Applied simplify0.1

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

    6. Applied simplify0.1

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

    if 2.7315380803037458e+79 < k

    1. Initial program 7.2

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

    3. Applied clear-num7.3

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

    4. Applied simplify7.3

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

    5. Taylor expanded around inf 7.3

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

    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: 33.8s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))