Average Error: 2.0 → 0.2
Time: 57.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.1001003804952939 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-10 \cdot \left(a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)}{k \cdot \left(k \cdot k\right)} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.1001003804952939 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{-10 \cdot \left(a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)}{k \cdot \left(k \cdot k\right)} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)\\

\end{array}
double f(double a, double k, double m) {
        double r75168169 = a;
        double r75168170 = k;
        double r75168171 = m;
        double r75168172 = pow(r75168170, r75168171);
        double r75168173 = r75168169 * r75168172;
        double r75168174 = 1.0;
        double r75168175 = 10.0;
        double r75168176 = r75168175 * r75168170;
        double r75168177 = r75168174 + r75168176;
        double r75168178 = r75168170 * r75168170;
        double r75168179 = r75168177 + r75168178;
        double r75168180 = r75168173 / r75168179;
        return r75168180;
}
↓
double f(double a, double k, double m) {
        double r75168181 = k;
        double r75168182 = 1.1001003804952939e+154;
        bool r75168183 = r75168181 <= r75168182;
        double r75168184 = cbrt(r75168181);
        double r75168185 = m;
        double r75168186 = pow(r75168184, r75168185);
        double r75168187 = a;
        double r75168188 = r75168186 * r75168187;
        double r75168189 = r75168184 * r75168184;
        double r75168190 = pow(r75168189, r75168185);
        double r75168191 = r75168188 * r75168190;
        double r75168192 = 10.0;
        double r75168193 = r75168181 + r75168192;
        double r75168194 = r75168193 * r75168181;
        double r75168195 = 1.0;
        double r75168196 = r75168194 + r75168195;
        double r75168197 = r75168191 / r75168196;
        double r75168198 = -10.0;
        double r75168199 = r75168195 / r75168181;
        double r75168200 = -0.6666666666666666;
        double r75168201 = pow(r75168199, r75168200);
        double r75168202 = pow(r75168201, r75168185);
        double r75168203 = -0.3333333333333333;
        double r75168204 = pow(r75168199, r75168203);
        double r75168205 = pow(r75168204, r75168185);
        double r75168206 = r75168202 * r75168205;
        double r75168207 = r75168187 * r75168206;
        double r75168208 = r75168198 * r75168207;
        double r75168209 = r75168181 * r75168181;
        double r75168210 = r75168181 * r75168209;
        double r75168211 = r75168208 / r75168210;
        double r75168212 = r75168187 / r75168181;
        double r75168213 = r75168181 / r75168212;
        double r75168214 = r75168206 / r75168213;
        double r75168215 = 99.0;
        double r75168216 = r75168215 * r75168187;
        double r75168217 = r75168209 * r75168209;
        double r75168218 = r75168216 / r75168217;
        double r75168219 = r75168218 * r75168206;
        double r75168220 = r75168214 + r75168219;
        double r75168221 = r75168211 + r75168220;
        double r75168222 = r75168183 ? r75168197 : r75168221;
        return r75168222;
}

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 < 1.1001003804952939e+154

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied unpow-prod-down0.1

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

    6. Applied associate-*l*0.1

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

    if 1.1001003804952939e+154 < k

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied add-cube-cbrt10.8

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

    5. Applied unpow-prod-down10.8

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

    6. Applied associate-*l*10.8

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

    7. Taylor expanded around inf 10.8

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

    8. Simplified0.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.1001003804952939 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{k}\right)}^{m} \cdot a\right) \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\left(k + 10\right) \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-10 \cdot \left(a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)}{k \cdot \left(k \cdot k\right)} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))