Average Error: 2.1 → 0.2
Time: 16.0s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 14158963.13343324698507785797119140625:\\ \;\;\;\;\frac{\frac{a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\frac{\sqrt{1 + k \cdot \left(k + 10\right)}}{{\left(\sqrt[3]{k}\right)}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)} \cdot 99}{\frac{{k}^{4}}{a}} + \left(\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k} - \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)} \cdot \left(a \cdot 10\right)}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 14158963.13343324698507785797119140625:\\
\;\;\;\;\frac{\frac{a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\frac{\sqrt{1 + k \cdot \left(k + 10\right)}}{{\left(\sqrt[3]{k}\right)}^{m}}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r208307 = a;
        double r208308 = k;
        double r208309 = m;
        double r208310 = pow(r208308, r208309);
        double r208311 = r208307 * r208310;
        double r208312 = 1.0;
        double r208313 = 10.0;
        double r208314 = r208313 * r208308;
        double r208315 = r208312 + r208314;
        double r208316 = r208308 * r208308;
        double r208317 = r208315 + r208316;
        double r208318 = r208311 / r208317;
        return r208318;
}
↓
double f(double a, double k, double m) {
        double r208319 = k;
        double r208320 = 14158963.133433247;
        bool r208321 = r208319 <= r208320;
        double r208322 = a;
        double r208323 = cbrt(r208319);
        double r208324 = r208323 * r208323;
        double r208325 = m;
        double r208326 = pow(r208324, r208325);
        double r208327 = r208322 * r208326;
        double r208328 = 1.0;
        double r208329 = 10.0;
        double r208330 = r208319 + r208329;
        double r208331 = r208319 * r208330;
        double r208332 = r208328 + r208331;
        double r208333 = sqrt(r208332);
        double r208334 = r208327 / r208333;
        double r208335 = pow(r208323, r208325);
        double r208336 = r208333 / r208335;
        double r208337 = r208334 / r208336;
        double r208338 = 1.0;
        double r208339 = r208338 / r208319;
        double r208340 = -r208325;
        double r208341 = pow(r208339, r208340);
        double r208342 = 99.0;
        double r208343 = r208341 * r208342;
        double r208344 = 4.0;
        double r208345 = pow(r208319, r208344);
        double r208346 = r208345 / r208322;
        double r208347 = r208343 / r208346;
        double r208348 = r208341 / r208319;
        double r208349 = r208322 / r208319;
        double r208350 = r208348 * r208349;
        double r208351 = r208322 * r208329;
        double r208352 = r208341 * r208351;
        double r208353 = 3.0;
        double r208354 = pow(r208319, r208353);
        double r208355 = r208352 / r208354;
        double r208356 = r208350 - r208355;
        double r208357 = r208347 + r208356;
        double r208358 = r208321 ? r208337 : r208357;
        return r208358;
}

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 < 14158963.133433247

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied unpow-prod-down0.0

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied associate-/r*0.1

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

    9. Simplified0.1

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

    if 14158963.133433247 < k

    1. Initial program 5.5

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

    2. Simplified5.5

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

    4. Applied clear-num5.7

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

    5. Simplified5.7

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

    6. Taylor expanded around inf 5.5

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 14158963.13343324698507785797119140625:\\ \;\;\;\;\frac{\frac{a \cdot {\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\frac{\sqrt{1 + k \cdot \left(k + 10\right)}}{{\left(\sqrt[3]{k}\right)}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)} \cdot 99}{\frac{{k}^{4}}{a}} + \left(\frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k} \cdot \frac{a}{k} - \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)} \cdot \left(a \cdot 10\right)}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))