Average Error: 2.0 → 0.1
Time: 18.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 31303448207475440:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{1}{{\left(\sqrt[3]{\sqrt[3]{k}} \cdot \sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot \left(\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 31303448207475440:\\
\;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{1}{{\left(\sqrt[3]{\sqrt[3]{k}} \cdot \sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot \left(\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot a\right)\\

\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 r132292 = a;
        double r132293 = k;
        double r132294 = m;
        double r132295 = pow(r132293, r132294);
        double r132296 = r132292 * r132295;
        double r132297 = 1.0;
        double r132298 = 10.0;
        double r132299 = r132298 * r132293;
        double r132300 = r132297 + r132299;
        double r132301 = r132293 * r132293;
        double r132302 = r132300 + r132301;
        double r132303 = r132296 / r132302;
        return r132303;
}
↓
double f(double a, double k, double m) {
        double r132304 = k;
        double r132305 = 3.130344820747544e+16;
        bool r132306 = r132304 <= r132305;
        double r132307 = cbrt(r132304);
        double r132308 = m;
        double r132309 = pow(r132307, r132308);
        double r132310 = 1.0;
        double r132311 = cbrt(r132307);
        double r132312 = r132311 * r132311;
        double r132313 = pow(r132312, r132308);
        double r132314 = r132310 / r132313;
        double r132315 = r132309 / r132314;
        double r132316 = 10.0;
        double r132317 = r132316 + r132304;
        double r132318 = r132304 * r132317;
        double r132319 = 1.0;
        double r132320 = r132318 + r132319;
        double r132321 = pow(r132311, r132308);
        double r132322 = r132320 / r132321;
        double r132323 = r132309 / r132322;
        double r132324 = a;
        double r132325 = r132323 * r132324;
        double r132326 = r132315 * r132325;
        double r132327 = 99.0;
        double r132328 = r132310 / r132304;
        double r132329 = -0.6666666666666666;
        double r132330 = pow(r132328, r132329);
        double r132331 = pow(r132330, r132308);
        double r132332 = r132331 * r132324;
        double r132333 = -0.3333333333333333;
        double r132334 = pow(r132328, r132333);
        double r132335 = pow(r132334, r132308);
        double r132336 = r132332 * r132335;
        double r132337 = r132327 * r132336;
        double r132338 = 4.0;
        double r132339 = pow(r132304, r132338);
        double r132340 = r132337 / r132339;
        double r132341 = r132332 / r132304;
        double r132342 = r132335 / r132304;
        double r132343 = r132341 * r132342;
        double r132344 = r132316 * r132336;
        double r132345 = 3.0;
        double r132346 = pow(r132304, r132345);
        double r132347 = r132344 / r132346;
        double r132348 = r132343 - r132347;
        double r132349 = r132340 + r132348;
        double r132350 = r132306 ? r132326 : r132349;
        return r132350;
}

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 < 3.130344820747544e+16

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied unpow-prod-down0.0

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

    6. Applied associate-/l*0.0

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

    8. Applied add-cube-cbrt0.0

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

    9. Applied unpow-prod-down0.0

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

    10. Applied *-un-lft-identity0.0

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

    11. Applied times-frac0.0

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

    12. Applied unpow-prod-down0.1

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

    13. Applied times-frac0.1

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

    14. Applied associate-*l*0.0

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

    if 3.130344820747544e+16 < k

    1. Initial program 5.4

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

    2. Simplified5.4

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

    4. Applied add-cube-cbrt5.4

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

    5. Applied unpow-prod-down5.4

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

    6. Applied associate-/l*5.4

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

    7. Taylor expanded around inf 5.4

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

    8. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 31303448207475440:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{1}{{\left(\sqrt[3]{\sqrt[3]{k}} \cdot \sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot \left(\frac{{\left(\sqrt[3]{k}\right)}^{m}}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{\sqrt[3]{k}}\right)}^{m}}} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))