Average Error: 2.4 → 0.2
Time: 2.4m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.325114164806417 \cdot 10^{+31}:\\ \;\;\;\;\frac{{k}^{\left(\frac{m}{2}\right)} \cdot \left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right)}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{k \cdot k}{\sqrt{e^{m \cdot \log k}}} \cdot \frac{k \cdot k}{\sqrt{e^{m \cdot \log k}}}} + \frac{\frac{a}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}\right) - \frac{\frac{a}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}}{\frac{k}{\sqrt{e^{m \cdot \log k}}}} \cdot \frac{10}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 4.325114164806417 \cdot 10^{+31}:\\
\;\;\;\;\frac{{k}^{\left(\frac{m}{2}\right)} \cdot \left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right)}{\left(k + 10\right) \cdot k + 1}\\

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

\end{array}
double f(double a, double k, double m) {
        double r47816643 = a;
        double r47816644 = k;
        double r47816645 = m;
        double r47816646 = pow(r47816644, r47816645);
        double r47816647 = r47816643 * r47816646;
        double r47816648 = 1.0;
        double r47816649 = 10.0;
        double r47816650 = r47816649 * r47816644;
        double r47816651 = r47816648 + r47816650;
        double r47816652 = r47816644 * r47816644;
        double r47816653 = r47816651 + r47816652;
        double r47816654 = r47816647 / r47816653;
        return r47816654;
}
↓
double f(double a, double k, double m) {
        double r47816655 = k;
        double r47816656 = 4.325114164806417e+31;
        bool r47816657 = r47816655 <= r47816656;
        double r47816658 = m;
        double r47816659 = 2.0;
        double r47816660 = r47816658 / r47816659;
        double r47816661 = pow(r47816655, r47816660);
        double r47816662 = a;
        double r47816663 = r47816662 * r47816661;
        double r47816664 = r47816661 * r47816663;
        double r47816665 = 10.0;
        double r47816666 = r47816655 + r47816665;
        double r47816667 = r47816666 * r47816655;
        double r47816668 = 1.0;
        double r47816669 = r47816667 + r47816668;
        double r47816670 = r47816664 / r47816669;
        double r47816671 = 99.0;
        double r47816672 = r47816671 * r47816662;
        double r47816673 = r47816655 * r47816655;
        double r47816674 = log(r47816655);
        double r47816675 = r47816658 * r47816674;
        double r47816676 = exp(r47816675);
        double r47816677 = sqrt(r47816676);
        double r47816678 = r47816673 / r47816677;
        double r47816679 = r47816678 * r47816678;
        double r47816680 = r47816672 / r47816679;
        double r47816681 = r47816655 / r47816677;
        double r47816682 = r47816662 / r47816681;
        double r47816683 = r47816682 / r47816681;
        double r47816684 = r47816680 + r47816683;
        double r47816685 = r47816665 / r47816655;
        double r47816686 = r47816683 * r47816685;
        double r47816687 = r47816684 - r47816686;
        double r47816688 = r47816657 ? r47816670 : r47816687;
        return r47816688;
}

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 < 4.325114164806417e+31

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied sqr-pow0.1

      \[\leadsto \frac{\color{blue}{\left({k}^{\left(\frac{m}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\right)} \cdot a}{k \cdot \left(k + 10\right) + 1}\]

    5. Applied associate-*l*0.1

      \[\leadsto \frac{\color{blue}{{k}^{\left(\frac{m}{2}\right)} \cdot \left({k}^{\left(\frac{m}{2}\right)} \cdot a\right)}}{k \cdot \left(k + 10\right) + 1}\]

    if 4.325114164806417e+31 < k

    1. Initial program 6.7

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

    2. Simplified6.7

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

    4. Applied sqr-pow6.7

      \[\leadsto \frac{\color{blue}{\left({k}^{\left(\frac{m}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\right)} \cdot a}{k \cdot \left(k + 10\right) + 1}\]

    5. Applied associate-*l*6.7

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

    7. Applied div-inv6.8

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

    9. Applied add-sqr-sqrt6.8

      \[\leadsto \left({k}^{\left(\frac{m}{2}\right)} \cdot \left({k}^{\left(\frac{m}{2}\right)} \cdot a\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{k \cdot \left(k + 10\right) + 1} \cdot \sqrt{k \cdot \left(k + 10\right) + 1}}}\]

    10. Applied associate-/r*6.8

      \[\leadsto \left({k}^{\left(\frac{m}{2}\right)} \cdot \left({k}^{\left(\frac{m}{2}\right)} \cdot a\right)\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{k \cdot \left(k + 10\right) + 1}}}{\sqrt{k \cdot \left(k + 10\right) + 1}}}\]

    11. Taylor expanded around -inf 63.0

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

    12. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.325114164806417 \cdot 10^{+31}:\\ \;\;\;\;\frac{{k}^{\left(\frac{m}{2}\right)} \cdot \left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right)}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{k \cdot k}{\sqrt{e^{m \cdot \log k}}} \cdot \frac{k \cdot k}{\sqrt{e^{m \cdot \log k}}}} + \frac{\frac{a}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}\right) - \frac{\frac{a}{\frac{k}{\sqrt{e^{m \cdot \log k}}}}}{\frac{k}{\sqrt{e^{m \cdot \log k}}}} \cdot \frac{10}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))