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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 3.16223621866072129 \cdot 10^{65}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \frac{k}{\frac{a}{k}}}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;k \le 3.16223621866072129 \cdot 10^{65}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\

\mathbf{else}:\\
\;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \frac{k}{\frac{a}{k}}}\\

\end{array}

double f(double a, double k, double m) {
        double r221309 = a;
        double r221310 = k;
        double r221311 = m;
        double r221312 = pow(r221310, r221311);
        double r221313 = r221309 * r221312;
        double r221314 = 1.0;
        double r221315 = 10.0;
        double r221316 = r221315 * r221310;
        double r221317 = r221314 + r221316;
        double r221318 = r221310 * r221310;
        double r221319 = r221317 + r221318;
        double r221320 = r221313 / r221319;
        return r221320;
}

↓

double f(double a, double k, double m) {
        double r221321 = k;
        double r221322 = 3.1622362186607213e+65;
        bool r221323 = r221321 <= r221322;
        double r221324 = a;
        double r221325 = m;
        double r221326 = pow(r221321, r221325);
        double r221327 = r221324 * r221326;
        double r221328 = 1.0;
        double r221329 = 10.0;
        double r221330 = r221329 * r221321;
        double r221331 = r221328 + r221330;
        double r221332 = r221321 * r221321;
        double r221333 = r221331 + r221332;
        double r221334 = r221327 / r221333;
        double r221335 = 1.0;
        double r221336 = r221335 / r221324;
        double r221337 = r221321 / r221324;
        double r221338 = r221329 * r221337;
        double r221339 = fma(r221328, r221336, r221338);
        double r221340 = r221324 / r221321;
        double r221341 = r221321 / r221340;
        double r221342 = r221339 + r221341;
        double r221343 = r221326 / r221342;
        double r221344 = r221323 ? r221334 : r221343;
        return r221344;
}

Derivation

Split input into 2 regimes
if k < 3.1622362186607213e+65
1. Initial program 0.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
if 3.1622362186607213e+65 < k
1. Initial program 7.0
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Simplified7.2
  \[\leadsto \color{blue}{\frac{{k}^{m}}{\frac{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}{a}}}\]
3. Taylor expanded around 0 7.2
  \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{{k}^{2}}{a} + \left(1 \cdot \frac{1}{a} + 10 \cdot \frac{k}{a}\right)}}\]
4. Simplified7.2
  \[\leadsto \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \frac{{k}^{2}}{a}}}\]
5. Using strategy rm
6. Applied unpow27.2
  \[\leadsto \frac{{k}^{m}}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \frac{\color{blue}{k \cdot k}}{a}}\]
7. Applied associate-/l*0.7
  \[\leadsto \frac{{k}^{m}}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \color{blue}{\frac{k}{\frac{a}{k}}}}\]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3.16223621866072129 \cdot 10^{65}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(1, \frac{1}{a}, 10 \cdot \frac{k}{a}\right) + \frac{k}{\frac{a}{k}}}\\ \end{array}\]

Error

Derivation

`if k < 3.1622362186607213e+65`

`if 3.1622362186607213e+65 < k`

Reproduce