Average Error: 2.1 → 0.2
Time: 16.6s
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 r190578 = a;
        double r190579 = k;
        double r190580 = m;
        double r190581 = pow(r190579, r190580);
        double r190582 = r190578 * r190581;
        double r190583 = 1.0;
        double r190584 = 10.0;
        double r190585 = r190584 * r190579;
        double r190586 = r190583 + r190585;
        double r190587 = r190579 * r190579;
        double r190588 = r190586 + r190587;
        double r190589 = r190582 / r190588;
        return r190589;
}
↓
double f(double a, double k, double m) {
        double r190590 = k;
        double r190591 = 14158963.133433247;
        bool r190592 = r190590 <= r190591;
        double r190593 = a;
        double r190594 = cbrt(r190590);
        double r190595 = r190594 * r190594;
        double r190596 = m;
        double r190597 = pow(r190595, r190596);
        double r190598 = r190593 * r190597;
        double r190599 = 1.0;
        double r190600 = 10.0;
        double r190601 = r190590 + r190600;
        double r190602 = r190590 * r190601;
        double r190603 = r190599 + r190602;
        double r190604 = sqrt(r190603);
        double r190605 = r190598 / r190604;
        double r190606 = pow(r190594, r190596);
        double r190607 = r190604 / r190606;
        double r190608 = r190605 / r190607;
        double r190609 = 1.0;
        double r190610 = r190609 / r190590;
        double r190611 = -r190596;
        double r190612 = pow(r190610, r190611);
        double r190613 = 99.0;
        double r190614 = r190612 * r190613;
        double r190615 = 4.0;
        double r190616 = pow(r190590, r190615);
        double r190617 = r190616 / r190593;
        double r190618 = r190614 / r190617;
        double r190619 = r190612 / r190590;
        double r190620 = r190593 / r190590;
        double r190621 = r190619 * r190620;
        double r190622 = r190593 * r190600;
        double r190623 = r190612 * r190622;
        double r190624 = 3.0;
        double r190625 = pow(r190590, r190624);
        double r190626 = r190623 / r190625;
        double r190627 = r190621 - r190626;
        double r190628 = r190618 + r190627;
        double r190629 = r190592 ? r190608 : r190628;
        return r190629;
}

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))))