Average Error: 2.0 → 0.1
Time: 1.9m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 111796600679780.56:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{\sqrt[3]{\left(\left(k \cdot k + \left(k \cdot 10 + 1\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot e^{\log k \cdot m}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99 + \left(e^{\log k \cdot m} \cdot \frac{\frac{a}{k}}{k} - \left(\frac{\frac{a}{k}}{k} \cdot \frac{10}{k}\right) \cdot e^{\log k \cdot m}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 111796600679780.56:\\
\;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{\sqrt[3]{\left(\left(k \cdot k + \left(k \cdot 10 + 1\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)}}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r55172809 = a;
        double r55172810 = k;
        double r55172811 = m;
        double r55172812 = pow(r55172810, r55172811);
        double r55172813 = r55172809 * r55172812;
        double r55172814 = 1.0;
        double r55172815 = 10.0;
        double r55172816 = r55172815 * r55172810;
        double r55172817 = r55172814 + r55172816;
        double r55172818 = r55172810 * r55172810;
        double r55172819 = r55172817 + r55172818;
        double r55172820 = r55172813 / r55172819;
        return r55172820;
}
↓
double f(double a, double k, double m) {
        double r55172821 = k;
        double r55172822 = 111796600679780.56;
        bool r55172823 = r55172821 <= r55172822;
        double r55172824 = a;
        double r55172825 = m;
        double r55172826 = pow(r55172821, r55172825);
        double r55172827 = r55172824 * r55172826;
        double r55172828 = r55172821 * r55172821;
        double r55172829 = 10.0;
        double r55172830 = r55172821 * r55172829;
        double r55172831 = 1.0;
        double r55172832 = r55172830 + r55172831;
        double r55172833 = r55172828 + r55172832;
        double r55172834 = sqrt(r55172833);
        double r55172835 = r55172827 / r55172834;
        double r55172836 = r55172833 * r55172833;
        double r55172837 = r55172836 * r55172833;
        double r55172838 = cbrt(r55172837);
        double r55172839 = sqrt(r55172838);
        double r55172840 = r55172835 / r55172839;
        double r55172841 = log(r55172821);
        double r55172842 = r55172841 * r55172825;
        double r55172843 = exp(r55172842);
        double r55172844 = r55172824 * r55172843;
        double r55172845 = r55172828 * r55172828;
        double r55172846 = r55172844 / r55172845;
        double r55172847 = 99.0;
        double r55172848 = r55172846 * r55172847;
        double r55172849 = r55172824 / r55172821;
        double r55172850 = r55172849 / r55172821;
        double r55172851 = r55172843 * r55172850;
        double r55172852 = r55172829 / r55172821;
        double r55172853 = r55172850 * r55172852;
        double r55172854 = r55172853 * r55172843;
        double r55172855 = r55172851 - r55172854;
        double r55172856 = r55172848 + r55172855;
        double r55172857 = r55172823 ? r55172840 : r55172856;
        return r55172857;
}

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 < 111796600679780.56

    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-sqr-sqrt0.2

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

    4. Applied associate-/r*0.2

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

    6. Applied add-cbrt-cube0.2

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

    if 111796600679780.56 < k

    1. Initial program 5.5

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

    2. Taylor expanded around -inf 62.9

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

    3. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 111796600679780.56:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{\sqrt[3]{\left(\left(k \cdot k + \left(k \cdot 10 + 1\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)\right) \cdot \left(k \cdot k + \left(k \cdot 10 + 1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot e^{\log k \cdot m}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 99 + \left(e^{\log k \cdot m} \cdot \frac{\frac{a}{k}}{k} - \left(\frac{\frac{a}{k}}{k} \cdot \frac{10}{k}\right) \cdot e^{\log k \cdot m}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))