Average Error: 2.1 → 0.2
Time: 14.3s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.172773149861676 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{a \cdot {\left(\frac{1}{k}\right)}^{\left(\frac{m}{2} \cdot -2\right)}}{\sqrt{k \cdot \left(10 + k\right) + 1}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k} + 99 \cdot \frac{{k}^{m}}{\frac{{k}^{4}}{a}}\right) - \frac{\left(10 \cdot {k}^{m}\right) \cdot a}{{k}^{3}}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 2.172773149861676 \cdot 10^{123}:\\
\;\;\;\;\frac{\frac{a \cdot {\left(\frac{1}{k}\right)}^{\left(\frac{m}{2} \cdot -2\right)}}{\sqrt{k \cdot \left(10 + k\right) + 1}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r151177 = a;
        double r151178 = k;
        double r151179 = m;
        double r151180 = pow(r151178, r151179);
        double r151181 = r151177 * r151180;
        double r151182 = 1.0;
        double r151183 = 10.0;
        double r151184 = r151183 * r151178;
        double r151185 = r151182 + r151184;
        double r151186 = r151178 * r151178;
        double r151187 = r151185 + r151186;
        double r151188 = r151181 / r151187;
        return r151188;
}
↓
double f(double a, double k, double m) {
        double r151189 = k;
        double r151190 = 2.172773149861676e+123;
        bool r151191 = r151189 <= r151190;
        double r151192 = a;
        double r151193 = 1.0;
        double r151194 = r151193 / r151189;
        double r151195 = m;
        double r151196 = 2.0;
        double r151197 = r151195 / r151196;
        double r151198 = -2.0;
        double r151199 = r151197 * r151198;
        double r151200 = pow(r151194, r151199);
        double r151201 = r151192 * r151200;
        double r151202 = 10.0;
        double r151203 = r151202 + r151189;
        double r151204 = r151189 * r151203;
        double r151205 = 1.0;
        double r151206 = r151204 + r151205;
        double r151207 = sqrt(r151206);
        double r151208 = r151201 / r151207;
        double r151209 = r151202 * r151189;
        double r151210 = r151205 + r151209;
        double r151211 = r151189 * r151189;
        double r151212 = r151210 + r151211;
        double r151213 = sqrt(r151212);
        double r151214 = r151208 / r151213;
        double r151215 = r151192 / r151189;
        double r151216 = pow(r151189, r151195);
        double r151217 = r151216 / r151189;
        double r151218 = r151215 * r151217;
        double r151219 = 99.0;
        double r151220 = 4.0;
        double r151221 = pow(r151189, r151220);
        double r151222 = r151221 / r151192;
        double r151223 = r151216 / r151222;
        double r151224 = r151219 * r151223;
        double r151225 = r151218 + r151224;
        double r151226 = r151202 * r151216;
        double r151227 = r151226 * r151192;
        double r151228 = 3.0;
        double r151229 = pow(r151189, r151228);
        double r151230 = r151227 / r151229;
        double r151231 = r151225 - r151230;
        double r151232 = r151191 ? r151214 : r151231;
        return r151232;
}

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 < 2.172773149861676e+123

    1. Initial program 0.1

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

    2. Taylor expanded around inf 20.6

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

    3. Simplified0.1

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

    5. Applied sqr-pow0.1

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

    6. Applied associate-*r*0.1

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

    8. Applied add-sqr-sqrt0.2

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

    9. Applied associate-/r*0.2

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

    10. Simplified0.2

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

    if 2.172773149861676e+123 < k

    1. Initial program 9.2

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.2

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

    4. Taylor expanded around inf 9.3

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

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.172773149861676 \cdot 10^{123}:\\ \;\;\;\;\frac{\frac{a \cdot {\left(\frac{1}{k}\right)}^{\left(\frac{m}{2} \cdot -2\right)}}{\sqrt{k \cdot \left(10 + k\right) + 1}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k} + 99 \cdot \frac{{k}^{m}}{\frac{{k}^{4}}{a}}\right) - \frac{\left(10 \cdot {k}^{m}\right) \cdot a}{{k}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))