Average Error: 1.9 → 0.1
Time: 3.5m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 6.087512029638269 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(\left(k + 10\right), k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{99}{k}}{k}\right), \left(\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}\right), \left(\frac{\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}}{k} \cdot -10 + \frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 6.087512029638269 \cdot 10^{+70}:\\
\;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(\left(k + 10\right), k, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r80011153 = a;
        double r80011154 = k;
        double r80011155 = m;
        double r80011156 = pow(r80011154, r80011155);
        double r80011157 = r80011153 * r80011156;
        double r80011158 = 1.0;
        double r80011159 = 10.0;
        double r80011160 = r80011159 * r80011154;
        double r80011161 = r80011158 + r80011160;
        double r80011162 = r80011154 * r80011154;
        double r80011163 = r80011161 + r80011162;
        double r80011164 = r80011157 / r80011163;
        return r80011164;
}
↓
double f(double a, double k, double m) {
        double r80011165 = k;
        double r80011166 = 6.087512029638269e+70;
        bool r80011167 = r80011165 <= r80011166;
        double r80011168 = cbrt(r80011165);
        double r80011169 = m;
        double r80011170 = pow(r80011168, r80011169);
        double r80011171 = a;
        double r80011172 = r80011170 * r80011171;
        double r80011173 = r80011168 * r80011168;
        double r80011174 = pow(r80011173, r80011169);
        double r80011175 = r80011172 * r80011174;
        double r80011176 = 10.0;
        double r80011177 = r80011165 + r80011176;
        double r80011178 = 1.0;
        double r80011179 = fma(r80011177, r80011165, r80011178);
        double r80011180 = r80011175 / r80011179;
        double r80011181 = 99.0;
        double r80011182 = r80011181 / r80011165;
        double r80011183 = r80011182 / r80011165;
        double r80011184 = log(r80011165);
        double r80011185 = r80011169 * r80011184;
        double r80011186 = exp(r80011185);
        double r80011187 = r80011165 / r80011171;
        double r80011188 = r80011186 / r80011187;
        double r80011189 = r80011188 / r80011165;
        double r80011190 = r80011189 / r80011165;
        double r80011191 = -10.0;
        double r80011192 = r80011190 * r80011191;
        double r80011193 = r80011192 + r80011189;
        double r80011194 = fma(r80011183, r80011189, r80011193);
        double r80011195 = r80011167 ? r80011180 : r80011194;
        return r80011195;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 6.087512029638269e+70

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied unpow-prod-down0.0

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

    6. Applied associate-*l*0.0

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

    if 6.087512029638269e+70 < k

    1. Initial program 6.6

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

    2. Simplified6.6

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

    3. Taylor expanded around inf 6.6

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 6.087512029638269 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\mathsf{fma}\left(\left(k + 10\right), k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{99}{k}}{k}\right), \left(\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}\right), \left(\frac{\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}}{k} \cdot -10 + \frac{\frac{e^{m \cdot \log k}}{\frac{k}{a}}}{k}\right)\right)\\ \end{array}\]

Reproduce

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