Average Error: 1.9 → 0.1
Time: 1.5m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 18566395.147397373:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{m \cdot \log k} - \left(10 \cdot \frac{\frac{a}{k}}{k}\right) \cdot e^{m \cdot \log k}}{k} + 99 \cdot \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array}\]
\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(10 \cdot \frac{\frac{a}{k}}{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 r83408513 = a;
        double r83408514 = k;
        double r83408515 = m;
        double r83408516 = pow(r83408514, r83408515);
        double r83408517 = r83408513 * r83408516;
        double r83408518 = 1.0;
        double r83408519 = 10.0;
        double r83408520 = r83408519 * r83408514;
        double r83408521 = r83408518 + r83408520;
        double r83408522 = r83408514 * r83408514;
        double r83408523 = r83408521 + r83408522;
        double r83408524 = r83408517 / r83408523;
        return r83408524;
}
↓
double f(double a, double k, double m) {
        double r83408525 = k;
        double r83408526 = 18566395.147397373;
        bool r83408527 = r83408525 <= r83408526;
        double r83408528 = cbrt(r83408525);
        double r83408529 = m;
        double r83408530 = pow(r83408528, r83408529);
        double r83408531 = a;
        double r83408532 = r83408530 * r83408531;
        double r83408533 = r83408528 * r83408528;
        double r83408534 = pow(r83408533, r83408529);
        double r83408535 = r83408532 * r83408534;
        double r83408536 = 10.0;
        double r83408537 = r83408525 + r83408536;
        double r83408538 = r83408537 * r83408525;
        double r83408539 = 1.0;
        double r83408540 = r83408538 + r83408539;
        double r83408541 = r83408535 / r83408540;
        double r83408542 = r83408531 / r83408525;
        double r83408543 = log(r83408525);
        double r83408544 = r83408529 * r83408543;
        double r83408545 = exp(r83408544);
        double r83408546 = r83408542 * r83408545;
        double r83408547 = r83408542 / r83408525;
        double r83408548 = r83408536 * r83408547;
        double r83408549 = r83408548 * r83408545;
        double r83408550 = r83408546 - r83408549;
        double r83408551 = r83408550 / r83408525;
        double r83408552 = 99.0;
        double r83408553 = r83408531 * r83408545;
        double r83408554 = r83408525 * r83408525;
        double r83408555 = r83408554 * r83408554;
        double r83408556 = r83408553 / r83408555;
        double r83408557 = r83408552 * r83408556;
        double r83408558 = r83408551 + r83408557;
        double r83408559 = r83408527 ? r83408541 : r83408558;
        return r83408559;
}

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-*l/0.1

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

    7. Applied associate-*l/0.1

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

    8. Applied associate-*l/0.1

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

    9. Applied sub-div0.1

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

  4. Final simplification0.1

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

Reproduce

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