Average Error: 2.0 → 0.1
Time: 1.4m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.3601387289399805 \cdot 10^{+100}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k} - e^{m \cdot \log k} \cdot \left(\frac{\frac{a}{k}}{k} \cdot \frac{10}{k}\right)\right) + \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.3601387289399805 \cdot 10^{+100}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\

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

\end{array}
double f(double a, double k, double m) {
        double r54258886 = a;
        double r54258887 = k;
        double r54258888 = m;
        double r54258889 = pow(r54258887, r54258888);
        double r54258890 = r54258886 * r54258889;
        double r54258891 = 1.0;
        double r54258892 = 10.0;
        double r54258893 = r54258892 * r54258887;
        double r54258894 = r54258891 + r54258893;
        double r54258895 = r54258887 * r54258887;
        double r54258896 = r54258894 + r54258895;
        double r54258897 = r54258890 / r54258896;
        return r54258897;
}
↓
double f(double a, double k, double m) {
        double r54258898 = k;
        double r54258899 = 1.3601387289399805e+100;
        bool r54258900 = r54258898 <= r54258899;
        double r54258901 = cbrt(r54258898);
        double r54258902 = m;
        double r54258903 = pow(r54258901, r54258902);
        double r54258904 = a;
        double r54258905 = r54258901 * r54258901;
        double r54258906 = pow(r54258905, r54258902);
        double r54258907 = r54258904 * r54258906;
        double r54258908 = r54258903 * r54258907;
        double r54258909 = 10.0;
        double r54258910 = r54258909 * r54258898;
        double r54258911 = 1.0;
        double r54258912 = r54258910 + r54258911;
        double r54258913 = r54258898 * r54258898;
        double r54258914 = r54258912 + r54258913;
        double r54258915 = r54258908 / r54258914;
        double r54258916 = r54258904 / r54258898;
        double r54258917 = r54258916 / r54258898;
        double r54258918 = log(r54258898);
        double r54258919 = r54258902 * r54258918;
        double r54258920 = exp(r54258919);
        double r54258921 = r54258917 * r54258920;
        double r54258922 = r54258909 / r54258898;
        double r54258923 = r54258917 * r54258922;
        double r54258924 = r54258920 * r54258923;
        double r54258925 = r54258921 - r54258924;
        double r54258926 = r54258904 * r54258920;
        double r54258927 = r54258913 * r54258913;
        double r54258928 = r54258926 / r54258927;
        double r54258929 = 99.0;
        double r54258930 = r54258928 * r54258929;
        double r54258931 = r54258925 + r54258930;
        double r54258932 = r54258900 ? r54258915 : r54258931;
        return r54258932;
}

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 < 1.3601387289399805e+100

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied unpow-prod-down0.1

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

    5. Applied associate-*r*0.1

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

    if 1.3601387289399805e+100 < k

    1. Initial program 7.7

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

    3. Applied flip3-+63.0

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

    4. Applied associate-/r/63.0

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

    5. Taylor expanded around -inf 63.0

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.3601387289399805 \cdot 10^{+100}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot e^{m \cdot \log k} - e^{m \cdot \log k} \cdot \left(\frac{\frac{a}{k}}{k} \cdot \frac{10}{k}\right)\right) + \frac{a \cdot e^{m \cdot \log k}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))