Average Error: 2.2 → 0.2
Time: 7.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 9.967300159477422353423086957172453598505 \cdot 10^{119}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{10}{k}\right) \cdot \left(\frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{k} \cdot \frac{a}{k}\right) + \frac{99 \cdot a}{k} \cdot \frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{{k}^{3}}\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{k} \cdot \frac{a}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 9.967300159477422353423086957172453598505 \cdot 10^{119}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r205634 = a;
        double r205635 = k;
        double r205636 = m;
        double r205637 = pow(r205635, r205636);
        double r205638 = r205634 * r205637;
        double r205639 = 1.0;
        double r205640 = 10.0;
        double r205641 = r205640 * r205635;
        double r205642 = r205639 + r205641;
        double r205643 = r205635 * r205635;
        double r205644 = r205642 + r205643;
        double r205645 = r205638 / r205644;
        return r205645;
}
↓
double f(double a, double k, double m) {
        double r205646 = k;
        double r205647 = 9.967300159477422e+119;
        bool r205648 = r205646 <= r205647;
        double r205649 = a;
        double r205650 = m;
        double r205651 = pow(r205646, r205650);
        double r205652 = r205649 * r205651;
        double r205653 = 1.0;
        double r205654 = 10.0;
        double r205655 = r205654 + r205646;
        double r205656 = r205646 * r205655;
        double r205657 = r205653 + r205656;
        double r205658 = r205652 / r205657;
        double r205659 = r205654 / r205646;
        double r205660 = -r205659;
        double r205661 = 1.0;
        double r205662 = r205661 / r205646;
        double r205663 = -1.0;
        double r205664 = r205663 * r205650;
        double r205665 = pow(r205662, r205664);
        double r205666 = r205665 / r205646;
        double r205667 = r205649 / r205646;
        double r205668 = r205666 * r205667;
        double r205669 = r205660 * r205668;
        double r205670 = 99.0;
        double r205671 = r205670 * r205649;
        double r205672 = r205671 / r205646;
        double r205673 = 3.0;
        double r205674 = pow(r205646, r205673);
        double r205675 = r205665 / r205674;
        double r205676 = r205672 * r205675;
        double r205677 = r205669 + r205676;
        double r205678 = r205677 + r205668;
        double r205679 = r205648 ? r205658 : r205678;
        return r205679;
}

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 < 9.967300159477422e+119

    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 associate-+l+0.1

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

    4. Simplified0.1

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

    if 9.967300159477422e+119 < k

    1. Initial program 8.7

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

    3. Applied associate-+l+8.7

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

    4. Simplified8.7

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

    5. Taylor expanded around inf 8.7

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

    6. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 9.967300159477422353423086957172453598505 \cdot 10^{119}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{10}{k}\right) \cdot \left(\frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{k} \cdot \frac{a}{k}\right) + \frac{99 \cdot a}{k} \cdot \frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{{k}^{3}}\right) + \frac{{\left(\frac{1}{k}\right)}^{\left(-1 \cdot m\right)}}{k} \cdot \frac{a}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))