Average Error: 2.2 → 0.1
Time: 42.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1.0 + 10.0 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 108481932297843.05:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10.0, 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} + \frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} \cdot \left(\frac{99.0}{k \cdot k} - \frac{10.0}{k}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1.0 + 10.0 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 108481932297843.05:\\
\;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10.0, 1.0\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} + \frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} \cdot \left(\frac{99.0}{k \cdot k} - \frac{10.0}{k}\right)\\

\end{array}
double f(double a, double k, double m) {
        double r8362382 = a;
        double r8362383 = k;
        double r8362384 = m;
        double r8362385 = pow(r8362383, r8362384);
        double r8362386 = r8362382 * r8362385;
        double r8362387 = 1.0;
        double r8362388 = 10.0;
        double r8362389 = r8362388 * r8362383;
        double r8362390 = r8362387 + r8362389;
        double r8362391 = r8362383 * r8362383;
        double r8362392 = r8362390 + r8362391;
        double r8362393 = r8362386 / r8362392;
        return r8362393;
}
↓
double f(double a, double k, double m) {
        double r8362394 = k;
        double r8362395 = 108481932297843.05;
        bool r8362396 = r8362394 <= r8362395;
        double r8362397 = m;
        double r8362398 = pow(r8362394, r8362397);
        double r8362399 = a;
        double r8362400 = r8362398 * r8362399;
        double r8362401 = 10.0;
        double r8362402 = r8362394 + r8362401;
        double r8362403 = 1.0;
        double r8362404 = fma(r8362394, r8362402, r8362403);
        double r8362405 = r8362400 / r8362404;
        double r8362406 = log(r8362394);
        double r8362407 = r8362397 * r8362406;
        double r8362408 = exp(r8362407);
        double r8362409 = r8362399 / r8362394;
        double r8362410 = r8362408 * r8362409;
        double r8362411 = r8362410 / r8362394;
        double r8362412 = 99.0;
        double r8362413 = r8362394 * r8362394;
        double r8362414 = r8362412 / r8362413;
        double r8362415 = r8362401 / r8362394;
        double r8362416 = r8362414 - r8362415;
        double r8362417 = r8362411 * r8362416;
        double r8362418 = r8362411 + r8362417;
        double r8362419 = r8362396 ? r8362405 : r8362418;
        return r8362419;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 108481932297843.05

    1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1.0 + 10.0 \cdot k\right) + k \cdot k}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10.0, 1.0\right)}}\]

    if 108481932297843.05 < k

    1. Initial program 5.7

      \[\frac{a \cdot {k}^{m}}{\left(1.0 + 10.0 \cdot k\right) + k \cdot k}\]

    2. Simplified5.7

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10.0, 1.0\right)}}\]

    3. Taylor expanded around inf 5.7

      \[\leadsto \color{blue}{\left(99.0 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}} + \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}}\right) - 10.0 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]

    4. Simplified0.1

      \[\leadsto \color{blue}{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k} + \frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k} \cdot \left(\frac{99.0}{k \cdot k} - \frac{10.0}{k}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 108481932297843.05:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10.0, 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} + \frac{e^{m \cdot \log k} \cdot \frac{a}{k}}{k} \cdot \left(\frac{99.0}{k \cdot k} - \frac{10.0}{k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))