Average Error: 1.7 → 0.1
Time: 17.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 6.005951583124144 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}} \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k}}{k} \cdot -10 + \left(\frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}} \cdot 99 + \frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 6.005951583124144 \cdot 10^{+111}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}} \cdot \left({k}^{m} \cdot a\right)\\

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

\end{array}
double f(double a, double k, double m) {
        double r4736902 = a;
        double r4736903 = k;
        double r4736904 = m;
        double r4736905 = pow(r4736903, r4736904);
        double r4736906 = r4736902 * r4736905;
        double r4736907 = 1.0;
        double r4736908 = 10.0;
        double r4736909 = r4736908 * r4736903;
        double r4736910 = r4736907 + r4736909;
        double r4736911 = r4736903 * r4736903;
        double r4736912 = r4736910 + r4736911;
        double r4736913 = r4736906 / r4736912;
        return r4736913;
}
↓
double f(double a, double k, double m) {
        double r4736914 = k;
        double r4736915 = 6.005951583124144e+111;
        bool r4736916 = r4736914 <= r4736915;
        double r4736917 = 1.0;
        double r4736918 = 10.0;
        double r4736919 = r4736914 + r4736918;
        double r4736920 = r4736919 * r4736914;
        double r4736921 = r4736917 + r4736920;
        double r4736922 = sqrt(r4736921);
        double r4736923 = r4736917 / r4736922;
        double r4736924 = r4736923 / r4736922;
        double r4736925 = m;
        double r4736926 = pow(r4736914, r4736925);
        double r4736927 = a;
        double r4736928 = r4736926 * r4736927;
        double r4736929 = r4736924 * r4736928;
        double r4736930 = r4736927 / r4736914;
        double r4736931 = log(r4736914);
        double r4736932 = r4736931 * r4736925;
        double r4736933 = exp(r4736932);
        double r4736934 = r4736933 / r4736914;
        double r4736935 = r4736930 * r4736934;
        double r4736936 = r4736935 / r4736914;
        double r4736937 = -10.0;
        double r4736938 = r4736936 * r4736937;
        double r4736939 = r4736914 * r4736914;
        double r4736940 = r4736939 * r4736939;
        double r4736941 = r4736940 / r4736927;
        double r4736942 = r4736933 / r4736941;
        double r4736943 = 99.0;
        double r4736944 = r4736942 * r4736943;
        double r4736945 = r4736944 + r4736935;
        double r4736946 = r4736938 + r4736945;
        double r4736947 = r4736916 ? r4736929 : r4736946;
        return r4736947;
}

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 < 6.005951583124144e+111

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied div-inv0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied associate-/r*0.1

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

    if 6.005951583124144e+111 < k

    1. Initial program 6.9

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

    2. Simplified6.9

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

    3. Taylor expanded around inf 6.9

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 6.005951583124144 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}} \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k}}{k} \cdot -10 + \left(\frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}} \cdot 99 + \frac{a}{k} \cdot \frac{e^{\log k \cdot m}}{k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))