Average Error: 1.9 → 0.1
Time: 1.7m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 18566395.147397373:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k} - \left(\frac{a}{k} \cdot \frac{10}{k}\right) \cdot e^{m \cdot \log k}}{k} + 99 \cdot \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 18566395.147397373:\\
\;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k} - \left(\frac{a}{k} \cdot \frac{10}{k}\right) \cdot e^{m \cdot \log k}}{k} + 99 \cdot \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\

\end{array}
double f(double a, double k, double m) {
        double r73489330 = a;
        double r73489331 = k;
        double r73489332 = m;
        double r73489333 = pow(r73489331, r73489332);
        double r73489334 = r73489330 * r73489333;
        double r73489335 = 1.0;
        double r73489336 = 10.0;
        double r73489337 = r73489336 * r73489331;
        double r73489338 = r73489335 + r73489337;
        double r73489339 = r73489331 * r73489331;
        double r73489340 = r73489338 + r73489339;
        double r73489341 = r73489334 / r73489340;
        return r73489341;
}
↓
double f(double a, double k, double m) {
        double r73489342 = k;
        double r73489343 = 18566395.147397373;
        bool r73489344 = r73489342 <= r73489343;
        double r73489345 = cbrt(r73489342);
        double r73489346 = m;
        double r73489347 = pow(r73489345, r73489346);
        double r73489348 = a;
        double r73489349 = r73489347 * r73489348;
        double r73489350 = r73489345 * r73489345;
        double r73489351 = pow(r73489350, r73489346);
        double r73489352 = r73489349 * r73489351;
        double r73489353 = 10.0;
        double r73489354 = r73489342 + r73489353;
        double r73489355 = r73489354 * r73489342;
        double r73489356 = 1.0;
        double r73489357 = r73489355 + r73489356;
        double r73489358 = r73489352 / r73489357;
        double r73489359 = r73489348 / r73489342;
        double r73489360 = log(r73489342);
        double r73489361 = r73489346 * r73489360;
        double r73489362 = exp(r73489361);
        double r73489363 = r73489359 * r73489362;
        double r73489364 = r73489353 / r73489342;
        double r73489365 = r73489359 * r73489364;
        double r73489366 = r73489365 * r73489362;
        double r73489367 = r73489363 - r73489366;
        double r73489368 = r73489367 / r73489342;
        double r73489369 = 99.0;
        double r73489370 = r73489348 * r73489362;
        double r73489371 = r73489342 * r73489342;
        double r73489372 = r73489371 * r73489371;
        double r73489373 = r73489370 / r73489372;
        double r73489374 = r73489369 * r73489373;
        double r73489375 = r73489368 + r73489374;
        double r73489376 = r73489344 ? r73489358 : r73489375;
        return r73489376;
}

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 < 18566395.147397373

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied unpow-prod-down0.0

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

    6. Applied associate-*l*0.0

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

    if 18566395.147397373 < k

    1. Initial program 5.2

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

    2. Simplified5.2

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

    3. Taylor expanded around -inf 62.9

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

    4. Simplified0.1

      \[\leadsto \color{blue}{99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \left(0 + \log k\right)} - \left(\frac{10}{k} \cdot \frac{\frac{a}{k}}{k}\right) \cdot e^{m \cdot \left(0 + \log k\right)}\right)}\]
    5. Using strategy rm

    6. Applied associate-*r/0.1

      \[\leadsto 99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \left(0 + \log k\right)} - \color{blue}{\frac{\frac{10}{k} \cdot \frac{a}{k}}{k}} \cdot e^{m \cdot \left(0 + \log k\right)}\right)\]

    7. Applied associate-*l/0.1

      \[\leadsto 99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \left(0 + \log k\right)} - \color{blue}{\frac{\left(\frac{10}{k} \cdot \frac{a}{k}\right) \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}\right)\]

    8. Applied associate-*l/0.1

      \[\leadsto 99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \left(\color{blue}{\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}} - \frac{\left(\frac{10}{k} \cdot \frac{a}{k}\right) \cdot e^{m \cdot \left(0 + \log k\right)}}{k}\right)\]

    9. Applied sub-div0.1

      \[\leadsto 99 \cdot \frac{e^{m \cdot \left(0 + \log k\right)} \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \color{blue}{\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)} - \left(\frac{10}{k} \cdot \frac{a}{k}\right) \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 18566395.147397373:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k} - \left(\frac{a}{k} \cdot \frac{10}{k}\right) \cdot e^{m \cdot \log k}}{k} + 99 \cdot \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))