Average Error: 2.2 → 0.1
Time: 8.2s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.27894232057722871 \cdot 10^{154}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.27894232057722871 \cdot 10^{154}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}} \cdot a\\

\mathbf{else}:\\
\;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\

\end{array}
double f(double a, double k, double m) {
        double r363181 = a;
        double r363182 = k;
        double r363183 = m;
        double r363184 = pow(r363182, r363183);
        double r363185 = r363181 * r363184;
        double r363186 = 1.0;
        double r363187 = 10.0;
        double r363188 = r363187 * r363182;
        double r363189 = r363186 + r363188;
        double r363190 = r363182 * r363182;
        double r363191 = r363189 + r363190;
        double r363192 = r363185 / r363191;
        return r363192;
}
↓
double f(double a, double k, double m) {
        double r363193 = k;
        double r363194 = 1.2789423205772287e+154;
        bool r363195 = r363193 <= r363194;
        double r363196 = cbrt(r363193);
        double r363197 = r363196 * r363196;
        double r363198 = m;
        double r363199 = pow(r363197, r363198);
        double r363200 = 10.0;
        double r363201 = r363200 + r363193;
        double r363202 = r363193 * r363201;
        double r363203 = 1.0;
        double r363204 = r363202 + r363203;
        double r363205 = pow(r363196, r363198);
        double r363206 = r363204 / r363205;
        double r363207 = r363199 / r363206;
        double r363208 = a;
        double r363209 = r363207 * r363208;
        double r363210 = 99.0;
        double r363211 = 1.0;
        double r363212 = r363211 / r363193;
        double r363213 = -0.6666666666666666;
        double r363214 = pow(r363212, r363213);
        double r363215 = pow(r363214, r363198);
        double r363216 = r363215 * r363208;
        double r363217 = -0.3333333333333333;
        double r363218 = pow(r363212, r363217);
        double r363219 = pow(r363218, r363198);
        double r363220 = r363216 * r363219;
        double r363221 = r363210 * r363220;
        double r363222 = 4.0;
        double r363223 = pow(r363193, r363222);
        double r363224 = r363221 / r363223;
        double r363225 = r363216 / r363193;
        double r363226 = r363219 / r363193;
        double r363227 = r363225 * r363226;
        double r363228 = r363200 * r363220;
        double r363229 = 3.0;
        double r363230 = pow(r363193, r363229);
        double r363231 = r363228 / r363230;
        double r363232 = r363227 - r363231;
        double r363233 = r363224 + r363232;
        double r363234 = r363195 ? r363209 : r363233;
        return r363234;
}

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 < 1.2789423205772287e+154

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied unpow-prod-down0.1

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

    6. Applied associate-/l*0.1

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

    if 1.2789423205772287e+154 < k

    1. Initial program 11.4

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

    2. Simplified11.4

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

    4. Applied add-cube-cbrt11.4

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

    5. Applied unpow-prod-down11.4

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

    6. Applied associate-/l*11.4

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

    7. Taylor expanded around inf 11.4

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

    8. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.27894232057722871 \cdot 10^{154}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))