Average Error: 2.2 → 0.3
Time: 33.7s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 121296497.93342482:\\ \;\;\;\;\frac{1}{\frac{k \cdot k + \left(1 + 10 \cdot k\right)}{{k}^{m} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 99\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < 121296497.93342482

    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 clear-num0.2

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

    if 121296497.93342482 < k

    1. Initial program 5.9

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

    2. Taylor expanded around inf 5.9

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 121296497.93342482:\\ \;\;\;\;\frac{1}{\frac{k \cdot k + \left(1 + 10 \cdot k\right)}{{k}^{m} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 99\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{a}{k} \cdot \frac{10}{k}\right)\\ \end{array}\]

Runtime

Time bar (total: 33.7s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))