Average Error: 2.1 → 0.6
Time: 55.9s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(m \cdot a\right) \cdot \log k + a}{1 + k \cdot \left(k + 10\right)} \le -1.7734617757330396 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \mathbf{elif}\;\frac{\left(m \cdot a\right) \cdot \log k + a}{1 + k \cdot \left(k + 10\right)} \le 5.51750367027517 \cdot 10^{-310}:\\ \;\;\;\;\left(99 \cdot a\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{k}^{m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \end{array}\]

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 (/ (+ a (* (log k) (* m a))) (+ 1 (* k (+ k 10)))) < -1.7734617757330396e-276 or 5.51750367027517e-310 < (/ (+ a (* (log k) (* m a))) (+ 1 (* k (+ k 10))))

    1. Initial program 0.1

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

    2. Initial simplification0.0

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

    4. Applied add-sqr-sqrt0.2

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

    5. Applied associate-/r*0.2

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

    if -1.7734617757330396e-276 < (/ (+ a (* (log k) (* m a))) (+ 1 (* k (+ k 10)))) < 5.51750367027517e-310

    1. Initial program 9.4

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

    2. Initial simplification9.4

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

    3. Taylor expanded around inf 11.2

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

    4. Simplified2.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(m \cdot a\right) \cdot \log k + a}{1 + k \cdot \left(k + 10\right)} \le -1.7734617757330396 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \mathbf{elif}\;\frac{\left(m \cdot a\right) \cdot \log k + a}{1 + k \cdot \left(k + 10\right)} \le 5.51750367027517 \cdot 10^{-310}:\\ \;\;\;\;\left(99 \cdot a\right) \cdot \frac{{k}^{m}}{{k}^{4}} + \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{k}^{m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\\ \end{array}\]

Runtime

Time bar (total: 55.9s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))