Average Error: 2.0 → 0.1
Time: 38.4s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1824118403577.9158:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a \cdot e^{m \cdot \log k}}{k}}{k} + \frac{99 \cdot e^{m \cdot \log k}}{\frac{{k}^{4}}{a}}\right) - \frac{a \cdot e^{m \cdot \log k}}{k} \cdot \frac{10}{k \cdot k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1824118403577.9158:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{a \cdot e^{m \cdot \log k}}{k}}{k} + \frac{99 \cdot e^{m \cdot \log k}}{\frac{{k}^{4}}{a}}\right) - \frac{a \cdot e^{m \cdot \log k}}{k} \cdot \frac{10}{k \cdot k}\\

\end{array}
double f(double a, double k, double m) {
        double r6643660 = a;
        double r6643661 = k;
        double r6643662 = m;
        double r6643663 = pow(r6643661, r6643662);
        double r6643664 = r6643660 * r6643663;
        double r6643665 = 1.0;
        double r6643666 = 10.0;
        double r6643667 = r6643666 * r6643661;
        double r6643668 = r6643665 + r6643667;
        double r6643669 = r6643661 * r6643661;
        double r6643670 = r6643668 + r6643669;
        double r6643671 = r6643664 / r6643670;
        return r6643671;
}
↓
double f(double a, double k, double m) {
        double r6643672 = k;
        double r6643673 = 1824118403577.9158;
        bool r6643674 = r6643672 <= r6643673;
        double r6643675 = a;
        double r6643676 = m;
        double r6643677 = pow(r6643672, r6643676);
        double r6643678 = r6643675 * r6643677;
        double r6643679 = r6643672 * r6643672;
        double r6643680 = 10.0;
        double r6643681 = r6643672 * r6643680;
        double r6643682 = 1.0;
        double r6643683 = r6643681 + r6643682;
        double r6643684 = r6643679 + r6643683;
        double r6643685 = r6643678 / r6643684;
        double r6643686 = log(r6643672);
        double r6643687 = r6643676 * r6643686;
        double r6643688 = exp(r6643687);
        double r6643689 = r6643675 * r6643688;
        double r6643690 = r6643689 / r6643672;
        double r6643691 = r6643690 / r6643672;
        double r6643692 = 99.0;
        double r6643693 = r6643692 * r6643688;
        double r6643694 = 4.0;
        double r6643695 = pow(r6643672, r6643694);
        double r6643696 = r6643695 / r6643675;
        double r6643697 = r6643693 / r6643696;
        double r6643698 = r6643691 + r6643697;
        double r6643699 = r6643680 / r6643679;
        double r6643700 = r6643690 * r6643699;
        double r6643701 = r6643698 - r6643700;
        double r6643702 = r6643674 ? r6643685 : r6643701;
        return r6643702;
}

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 < 1824118403577.9158

    1. Initial program 0.1

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

    if 1824118403577.9158 < k

    1. Initial program 5.5

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

    3. Applied add-sqr-sqrt5.5

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

    4. Applied associate-/r*5.5

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

    5. Taylor expanded around inf 5.5

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1824118403577.9158:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a \cdot e^{m \cdot \log k}}{k}}{k} + \frac{99 \cdot e^{m \cdot \log k}}{\frac{{k}^{4}}{a}}\right) - \frac{a \cdot e^{m \cdot \log k}}{k} \cdot \frac{10}{k \cdot k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))