Average Error: 2.3 → 0.1
Time: 23.1s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.413206367433749081791782278643320672943 \cdot 10^{73}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} + \frac{99}{k \cdot k} \cdot \frac{a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{k \cdot k}\right) - \frac{10}{k} \cdot \frac{a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{k \cdot k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 5.413206367433749081791782278643320672943 \cdot 10^{73}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\

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

\end{array}
double f(double a, double k, double m) {
        double r6933186 = a;
        double r6933187 = k;
        double r6933188 = m;
        double r6933189 = pow(r6933187, r6933188);
        double r6933190 = r6933186 * r6933189;
        double r6933191 = 1.0;
        double r6933192 = 10.0;
        double r6933193 = r6933192 * r6933187;
        double r6933194 = r6933191 + r6933193;
        double r6933195 = r6933187 * r6933187;
        double r6933196 = r6933194 + r6933195;
        double r6933197 = r6933190 / r6933196;
        return r6933197;
}
↓
double f(double a, double k, double m) {
        double r6933198 = k;
        double r6933199 = 5.413206367433749e+73;
        bool r6933200 = r6933198 <= r6933199;
        double r6933201 = cbrt(r6933198);
        double r6933202 = m;
        double r6933203 = pow(r6933201, r6933202);
        double r6933204 = a;
        double r6933205 = r6933201 * r6933201;
        double r6933206 = pow(r6933205, r6933202);
        double r6933207 = r6933204 * r6933206;
        double r6933208 = r6933203 * r6933207;
        double r6933209 = 10.0;
        double r6933210 = r6933209 * r6933198;
        double r6933211 = 1.0;
        double r6933212 = r6933210 + r6933211;
        double r6933213 = r6933198 * r6933198;
        double r6933214 = r6933212 + r6933213;
        double r6933215 = r6933208 / r6933214;
        double r6933216 = r6933204 / r6933198;
        double r6933217 = 1.0;
        double r6933218 = r6933217 / r6933198;
        double r6933219 = -0.6666666666666666;
        double r6933220 = pow(r6933218, r6933219);
        double r6933221 = pow(r6933220, r6933202);
        double r6933222 = -0.3333333333333333;
        double r6933223 = pow(r6933218, r6933222);
        double r6933224 = pow(r6933223, r6933202);
        double r6933225 = r6933221 * r6933224;
        double r6933226 = r6933225 / r6933198;
        double r6933227 = r6933216 * r6933226;
        double r6933228 = 99.0;
        double r6933229 = r6933228 / r6933213;
        double r6933230 = r6933204 * r6933225;
        double r6933231 = r6933230 / r6933213;
        double r6933232 = r6933229 * r6933231;
        double r6933233 = r6933227 + r6933232;
        double r6933234 = r6933209 / r6933198;
        double r6933235 = r6933234 * r6933231;
        double r6933236 = r6933233 - r6933235;
        double r6933237 = r6933200 ? r6933215 : r6933236;
        return r6933237;
}

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 < 5.413206367433749e+73

    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 add-cube-cbrt0.1

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

    4. Applied unpow-prod-down0.1

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

    5. Applied associate-*r*0.1

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

    if 5.413206367433749e+73 < k

    1. Initial program 7.8

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

    3. Applied add-cube-cbrt7.8

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

    4. Applied unpow-prod-down7.8

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

    5. Applied associate-*r*7.8

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

    6. Taylor expanded around inf 7.8

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

    7. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.413206367433749081791782278643320672943 \cdot 10^{73}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m} \cdot \left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right)}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} + \frac{99}{k \cdot k} \cdot \frac{a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{k \cdot k}\right) - \frac{10}{k} \cdot \frac{a \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{k \cdot k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))