Average Error: 2.0 → 0.2
Time: 23.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1390911796383614082210567651856566190080:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left({\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right) \cdot \left(a \cdot 99\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \frac{a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k}\right) - \left(\frac{a}{\left(k \cdot k\right) \cdot k} \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right) \cdot 10\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1390911796383614082210567651856566190080:\\
\;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r8882666 = a;
        double r8882667 = k;
        double r8882668 = m;
        double r8882669 = pow(r8882667, r8882668);
        double r8882670 = r8882666 * r8882669;
        double r8882671 = 1.0;
        double r8882672 = 10.0;
        double r8882673 = r8882672 * r8882667;
        double r8882674 = r8882671 + r8882673;
        double r8882675 = r8882667 * r8882667;
        double r8882676 = r8882674 + r8882675;
        double r8882677 = r8882670 / r8882676;
        return r8882677;
}
↓
double f(double a, double k, double m) {
        double r8882678 = k;
        double r8882679 = 1.390911796383614e+39;
        bool r8882680 = r8882678 <= r8882679;
        double r8882681 = a;
        double r8882682 = m;
        double r8882683 = pow(r8882678, r8882682);
        double r8882684 = r8882681 * r8882683;
        double r8882685 = r8882678 * r8882678;
        double r8882686 = 10.0;
        double r8882687 = r8882678 * r8882686;
        double r8882688 = 1.0;
        double r8882689 = r8882687 + r8882688;
        double r8882690 = r8882685 + r8882689;
        double r8882691 = sqrt(r8882690);
        double r8882692 = r8882684 / r8882691;
        double r8882693 = r8882692 / r8882691;
        double r8882694 = 1.0;
        double r8882695 = r8882694 / r8882678;
        double r8882696 = -0.3333333333333333;
        double r8882697 = pow(r8882695, r8882696);
        double r8882698 = pow(r8882697, r8882682);
        double r8882699 = r8882697 * r8882697;
        double r8882700 = pow(r8882699, r8882682);
        double r8882701 = r8882698 * r8882700;
        double r8882702 = 99.0;
        double r8882703 = r8882681 * r8882702;
        double r8882704 = r8882701 * r8882703;
        double r8882705 = r8882685 * r8882685;
        double r8882706 = r8882704 / r8882705;
        double r8882707 = r8882681 / r8882678;
        double r8882708 = r8882701 / r8882678;
        double r8882709 = r8882707 * r8882708;
        double r8882710 = r8882706 + r8882709;
        double r8882711 = r8882685 * r8882678;
        double r8882712 = r8882681 / r8882711;
        double r8882713 = r8882712 * r8882701;
        double r8882714 = r8882713 * r8882686;
        double r8882715 = r8882710 - r8882714;
        double r8882716 = r8882680 ? r8882693 : r8882715;
        return r8882716;
}

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 < 1.390911796383614e+39

    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-sqr-sqrt0.2

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

    4. Applied associate-/r*0.2

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

    if 1.390911796383614e+39 < k

    1. Initial program 5.7

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

    3. Applied add-cube-cbrt5.7

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

    4. Applied unpow-prod-down5.7

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

    5. Applied associate-*r*5.7

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

    6. Taylor expanded around inf 5.7

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1390911796383614082210567651856566190080:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}}{\sqrt{k \cdot k + \left(k \cdot 10 + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left({\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right) \cdot \left(a \cdot 99\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + \frac{a}{k} \cdot \frac{{\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}}{k}\right) - \left(\frac{a}{\left(k \cdot k\right) \cdot k} \cdot \left({\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m} \cdot {\left({\left(\frac{1}{k}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{k}\right)}^{\frac{-1}{3}}\right)}^{m}\right)\right) \cdot 10\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))