Average Error: 2.2 → 0.1
Time: 8.5s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.8838862069065277 \cdot 10^{106}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot a}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.8838862069065277 \cdot 10^{106}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot a}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}}\\

\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 r365534 = a;
        double r365535 = k;
        double r365536 = m;
        double r365537 = pow(r365535, r365536);
        double r365538 = r365534 * r365537;
        double r365539 = 1.0;
        double r365540 = 10.0;
        double r365541 = r365540 * r365535;
        double r365542 = r365539 + r365541;
        double r365543 = r365535 * r365535;
        double r365544 = r365542 + r365543;
        double r365545 = r365538 / r365544;
        return r365545;
}
↓
double f(double a, double k, double m) {
        double r365546 = k;
        double r365547 = 1.8838862069065277e+106;
        bool r365548 = r365546 <= r365547;
        double r365549 = cbrt(r365546);
        double r365550 = r365549 * r365549;
        double r365551 = m;
        double r365552 = pow(r365550, r365551);
        double r365553 = a;
        double r365554 = r365552 * r365553;
        double r365555 = 10.0;
        double r365556 = r365555 + r365546;
        double r365557 = r365546 * r365556;
        double r365558 = 1.0;
        double r365559 = r365557 + r365558;
        double r365560 = pow(r365549, r365551);
        double r365561 = r365559 / r365560;
        double r365562 = r365554 / r365561;
        double r365563 = 99.0;
        double r365564 = 1.0;
        double r365565 = r365564 / r365546;
        double r365566 = -0.6666666666666666;
        double r365567 = pow(r365565, r365566);
        double r365568 = pow(r365567, r365551);
        double r365569 = r365568 * r365553;
        double r365570 = -0.3333333333333333;
        double r365571 = pow(r365565, r365570);
        double r365572 = pow(r365571, r365551);
        double r365573 = r365569 * r365572;
        double r365574 = r365563 * r365573;
        double r365575 = 4.0;
        double r365576 = pow(r365546, r365575);
        double r365577 = r365574 / r365576;
        double r365578 = r365569 / r365546;
        double r365579 = r365572 / r365546;
        double r365580 = r365578 * r365579;
        double r365581 = r365555 * r365573;
        double r365582 = 3.0;
        double r365583 = pow(r365546, r365582);
        double r365584 = r365581 / r365583;
        double r365585 = r365580 - r365584;
        double r365586 = r365577 + r365585;
        double r365587 = r365548 ? r365562 : r365586;
        return r365587;
}

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 < 1.8838862069065277e+106

    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-down0.1

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

    6. Applied associate-/l*0.1

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

    8. Applied associate-*l/0.0

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

    if 1.8838862069065277e+106 < k

    1. Initial program 8.6

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

    2. Simplified8.6

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

    4. Applied add-cube-cbrt8.6

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

    5. Applied unpow-prod-down8.6

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

    6. Applied associate-/l*8.6

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

    7. Taylor expanded around inf 8.6

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.8838862069065277 \cdot 10^{106}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{m} \cdot a}{\frac{k \cdot \left(10 + k\right) + 1}{{\left(\sqrt[3]{k}\right)}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{99 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{4}} + \left(\frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k} - \frac{10 \cdot \left(\left({\left({\left(\frac{1}{k}\right)}^{\frac{-2}{3}}\right)}^{m} \cdot a\right) \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)}{{k}^{3}}\right)\\ \end{array}\]

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))))