Average Error: 2.8 → 0.1
Time: 11.7s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.33933545914045452 \cdot 10^{153}:\\ \;\;\;\;\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 1.33933545914045452 \cdot 10^{153}:\\
\;\;\;\;\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\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 r445267 = a;
        double r445268 = k;
        double r445269 = m;
        double r445270 = pow(r445268, r445269);
        double r445271 = r445267 * r445270;
        double r445272 = 1.0;
        double r445273 = 10.0;
        double r445274 = r445273 * r445268;
        double r445275 = r445272 + r445274;
        double r445276 = r445268 * r445268;
        double r445277 = r445275 + r445276;
        double r445278 = r445271 / r445277;
        return r445278;
}
↓
double f(double a, double k, double m) {
        double r445279 = k;
        double r445280 = 1.3393354591404545e+153;
        bool r445281 = r445279 <= r445280;
        double r445282 = a;
        double r445283 = m;
        double r445284 = 2.0;
        double r445285 = r445283 / r445284;
        double r445286 = pow(r445279, r445285);
        double r445287 = r445282 * r445286;
        double r445288 = r445287 * r445286;
        double r445289 = 1.0;
        double r445290 = 10.0;
        double r445291 = r445290 * r445279;
        double r445292 = r445289 + r445291;
        double r445293 = r445279 * r445279;
        double r445294 = r445292 + r445293;
        double r445295 = r445288 / r445294;
        double r445296 = -1.0;
        double r445297 = 1.0;
        double r445298 = r445297 / r445279;
        double r445299 = log(r445298);
        double r445300 = r445283 * r445299;
        double r445301 = r445296 * r445300;
        double r445302 = exp(r445301);
        double r445303 = r445302 / r445279;
        double r445304 = r445282 / r445279;
        double r445305 = 99.0;
        double r445306 = r445282 * r445302;
        double r445307 = 4.0;
        double r445308 = pow(r445279, r445307);
        double r445309 = r445306 / r445308;
        double r445310 = r445305 * r445309;
        double r445311 = 3.0;
        double r445312 = pow(r445279, r445311);
        double r445313 = r445306 / r445312;
        double r445314 = r445290 * r445313;
        double r445315 = r445310 - r445314;
        double r445316 = fma(r445303, r445304, r445315);
        double r445317 = r445281 ? r445295 : r445316;
        return r445317;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 1.3393354591404545e+153

    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 sqr-pow0.1

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

    4. Applied associate-*r*0.1

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

    if 1.3393354591404545e+153 < k

    1. Initial program 10.5

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

    2. Taylor expanded around inf 10.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.33933545914045452 \cdot 10^{153}:\\ \;\;\;\;\frac{\left(a \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot {k}^{\left(\frac{m}{2}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))