Average Error: 2.1 → 0.1
Time: 16.4s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.388047209286606508977160450051085339464 \cdot 10^{139}:\\ \;\;\;\;\frac{\left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right) \cdot {\left(\sqrt[3]{k}\right)}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}\right) - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 4.388047209286606508977160450051085339464 \cdot 10^{139}:\\
\;\;\;\;\frac{\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}\\

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

\end{array}
double f(double a, double k, double m) {
        double r174338 = a;
        double r174339 = k;
        double r174340 = m;
        double r174341 = pow(r174339, r174340);
        double r174342 = r174338 * r174341;
        double r174343 = 1.0;
        double r174344 = 10.0;
        double r174345 = r174344 * r174339;
        double r174346 = r174343 + r174345;
        double r174347 = r174339 * r174339;
        double r174348 = r174346 + r174347;
        double r174349 = r174342 / r174348;
        return r174349;
}
↓
double f(double a, double k, double m) {
        double r174350 = k;
        double r174351 = 4.3880472092866065e+139;
        bool r174352 = r174350 <= r174351;
        double r174353 = a;
        double r174354 = cbrt(r174350);
        double r174355 = r174354 * r174354;
        double r174356 = m;
        double r174357 = pow(r174355, r174356);
        double r174358 = r174353 * r174357;
        double r174359 = pow(r174354, r174356);
        double r174360 = r174358 * r174359;
        double r174361 = 1.0;
        double r174362 = 10.0;
        double r174363 = r174362 * r174350;
        double r174364 = r174361 + r174363;
        double r174365 = r174350 * r174350;
        double r174366 = r174364 + r174365;
        double r174367 = r174360 / r174366;
        double r174368 = 99.0;
        double r174369 = r174368 * r174353;
        double r174370 = 4.0;
        double r174371 = pow(r174350, r174370);
        double r174372 = 1.0;
        double r174373 = r174372 / r174350;
        double r174374 = -r174356;
        double r174375 = pow(r174373, r174374);
        double r174376 = r174371 / r174375;
        double r174377 = r174369 / r174376;
        double r174378 = r174375 / r174350;
        double r174379 = r174353 / r174350;
        double r174380 = r174378 * r174379;
        double r174381 = r174377 + r174380;
        double r174382 = 3.0;
        double r174383 = pow(r174350, r174382);
        double r174384 = r174353 / r174383;
        double r174385 = r174384 * r174375;
        double r174386 = r174362 * r174385;
        double r174387 = r174381 - r174386;
        double r174388 = r174352 ? r174367 : r174387;
        return r174388;
}

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 < 4.3880472092866065e+139

    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 4.3880472092866065e+139 < k

    1. Initial program 9.6

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

    3. Applied clear-num9.7

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

    4. Simplified9.7

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

    5. Taylor expanded around inf 9.6

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

    6. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.388047209286606508977160450051085339464 \cdot 10^{139}:\\ \;\;\;\;\frac{\left(a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}\right) \cdot {\left(\sqrt[3]{k}\right)}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99 \cdot a}{\frac{{k}^{4}}{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}} + \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k}\right) - 10 \cdot \left(\frac{a}{{k}^{3}} \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))