Average Error: 2.0 → 0.1
Time: 17.4s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.442302470468942821043172495313077911614 \cdot 10^{138}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k} + \left(\left(\frac{a}{{k}^{4}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right) \cdot 99 - \frac{10 \cdot a}{\frac{{k}^{3}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.442302470468942821043172495313077911614 \cdot 10^{138}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\

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

\end{array}
double f(double a, double k, double m) {
        double r146780 = a;
        double r146781 = k;
        double r146782 = m;
        double r146783 = pow(r146781, r146782);
        double r146784 = r146780 * r146783;
        double r146785 = 1.0;
        double r146786 = 10.0;
        double r146787 = r146786 * r146781;
        double r146788 = r146785 + r146787;
        double r146789 = r146781 * r146781;
        double r146790 = r146788 + r146789;
        double r146791 = r146784 / r146790;
        return r146791;
}
↓
double f(double a, double k, double m) {
        double r146792 = k;
        double r146793 = 1.4423024704689428e+138;
        bool r146794 = r146792 <= r146793;
        double r146795 = a;
        double r146796 = m;
        double r146797 = pow(r146792, r146796);
        double r146798 = r146795 * r146797;
        double r146799 = 1.0;
        double r146800 = 10.0;
        double r146801 = r146800 * r146792;
        double r146802 = r146799 + r146801;
        double r146803 = r146792 * r146792;
        double r146804 = r146802 + r146803;
        double r146805 = r146798 / r146804;
        double r146806 = 1.0;
        double r146807 = r146806 / r146792;
        double r146808 = -r146796;
        double r146809 = pow(r146807, r146808);
        double r146810 = r146809 / r146792;
        double r146811 = r146795 / r146792;
        double r146812 = r146810 * r146811;
        double r146813 = 4.0;
        double r146814 = pow(r146792, r146813);
        double r146815 = r146795 / r146814;
        double r146816 = r146815 * r146809;
        double r146817 = 99.0;
        double r146818 = r146816 * r146817;
        double r146819 = r146800 * r146795;
        double r146820 = 3.0;
        double r146821 = pow(r146792, r146820);
        double r146822 = r146821 / r146809;
        double r146823 = r146819 / r146822;
        double r146824 = r146818 - r146823;
        double r146825 = r146812 + r146824;
        double r146826 = r146794 ? r146805 : r146825;
        return r146826;
}

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 < 1.4423024704689428e+138

    1. Initial program 0.1

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

    if 1.4423024704689428e+138 < k

    1. Initial program 8.9

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

    3. Applied add-sqr-sqrt8.9

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

    4. Applied associate-/r*8.9

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

    5. Simplified8.9

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

    7. Applied associate-+l+8.9

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

    8. Simplified8.9

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

    9. Taylor expanded around inf 8.9

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

    10. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.442302470468942821043172495313077911614 \cdot 10^{138}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k} + \left(\left(\frac{a}{{k}^{4}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right) \cdot 99 - \frac{10 \cdot a}{\frac{{k}^{3}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))