Average Error: 2.1 → 0.1
Time: 5.2s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4.5162262000245942 \cdot 10^{147}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 4.5162262000245942 \cdot 10^{147}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\

\end{array}
double f(double a, double k, double m) {
        double r329334 = a;
        double r329335 = k;
        double r329336 = m;
        double r329337 = pow(r329335, r329336);
        double r329338 = r329334 * r329337;
        double r329339 = 1.0;
        double r329340 = 10.0;
        double r329341 = r329340 * r329335;
        double r329342 = r329339 + r329341;
        double r329343 = r329335 * r329335;
        double r329344 = r329342 + r329343;
        double r329345 = r329338 / r329344;
        return r329345;
}
↓
double f(double a, double k, double m) {
        double r329346 = k;
        double r329347 = 4.516226200024594e+147;
        bool r329348 = r329346 <= r329347;
        double r329349 = a;
        double r329350 = m;
        double r329351 = pow(r329346, r329350);
        double r329352 = r329349 * r329351;
        double r329353 = 1.0;
        double r329354 = 10.0;
        double r329355 = r329354 * r329346;
        double r329356 = r329353 + r329355;
        double r329357 = r329346 * r329346;
        double r329358 = r329356 + r329357;
        double r329359 = r329352 / r329358;
        double r329360 = -1.0;
        double r329361 = 1.0;
        double r329362 = r329361 / r329346;
        double r329363 = log(r329362);
        double r329364 = r329350 * r329363;
        double r329365 = r329360 * r329364;
        double r329366 = exp(r329365);
        double r329367 = r329366 / r329346;
        double r329368 = r329349 / r329346;
        double r329369 = 99.0;
        double r329370 = r329349 * r329366;
        double r329371 = 4.0;
        double r329372 = pow(r329346, r329371);
        double r329373 = r329370 / r329372;
        double r329374 = r329369 * r329373;
        double r329375 = 3.0;
        double r329376 = pow(r329346, r329375);
        double r329377 = r329370 / r329376;
        double r329378 = r329354 * r329377;
        double r329379 = r329374 - r329378;
        double r329380 = fma(r329367, r329368, r329379);
        double r329381 = r329348 ? r329359 : r329380;
        return r329381;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 4.516226200024594e+147

    1. Initial program 0.1

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

    if 4.516226200024594e+147 < k

    1. Initial program 9.9

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

    2. Taylor expanded around inf 9.9

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

    3. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4.5162262000245942 \cdot 10^{147}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))