Average Error: 1.9 → 0.1
Time: 17.4s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.309780576654702709208487445056226971465 \cdot 10^{132}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{1}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 4.309780576654702709208487445056226971465 \cdot 10^{132}:\\
\;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{1}{1 + k \cdot \left(10 + k\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}\\

\end{array}
double f(double a, double k, double m) {
        double r270770 = a;
        double r270771 = k;
        double r270772 = m;
        double r270773 = pow(r270771, r270772);
        double r270774 = r270770 * r270773;
        double r270775 = 1.0;
        double r270776 = 10.0;
        double r270777 = r270776 * r270771;
        double r270778 = r270775 + r270777;
        double r270779 = r270771 * r270771;
        double r270780 = r270778 + r270779;
        double r270781 = r270774 / r270780;
        return r270781;
}
↓
double f(double a, double k, double m) {
        double r270782 = k;
        double r270783 = 4.309780576654703e+132;
        bool r270784 = r270782 <= r270783;
        double r270785 = a;
        double r270786 = m;
        double r270787 = pow(r270782, r270786);
        double r270788 = r270785 * r270787;
        double r270789 = 1.0;
        double r270790 = 1.0;
        double r270791 = 10.0;
        double r270792 = r270791 + r270782;
        double r270793 = r270782 * r270792;
        double r270794 = r270790 + r270793;
        double r270795 = r270789 / r270794;
        double r270796 = r270788 * r270795;
        double r270797 = 99.0;
        double r270798 = r270797 * r270785;
        double r270799 = 4.0;
        double r270800 = pow(r270782, r270799);
        double r270801 = r270789 / r270782;
        double r270802 = -r270786;
        double r270803 = pow(r270801, r270802);
        double r270804 = r270800 / r270803;
        double r270805 = r270798 / r270804;
        double r270806 = 3.0;
        double r270807 = pow(r270782, r270806);
        double r270808 = r270785 / r270807;
        double r270809 = r270808 * r270803;
        double r270810 = r270791 * r270809;
        double r270811 = r270805 - r270810;
        double r270812 = r270803 / r270782;
        double r270813 = r270785 / r270782;
        double r270814 = r270812 * r270813;
        double r270815 = r270811 + r270814;
        double r270816 = r270784 ? r270796 : r270815;
        return r270816;
}

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 < 4.309780576654703e+132

    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 div-inv0.1

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

    4. Simplified0.1

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

    if 4.309780576654703e+132 < k

    1. Initial program 8.3

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

    3. Applied div-inv8.3

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

    4. Simplified8.3

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

    6. Applied add-sqr-sqrt8.3

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

    7. Applied associate-/r*8.3

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

    8. Simplified8.3

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

    9. Taylor expanded around inf 8.3

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

    10. Simplified0.3

      \[\leadsto \color{blue}{\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.309780576654702709208487445056226971465 \cdot 10^{132}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{1}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))