Average Error: 2.1 → 0.1
Time: 27.8s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 73317554942727741440:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{k}, \frac{e^{m \cdot \log k}}{k}, \left(\frac{a}{k} \cdot \frac{e^{m \cdot \log k}}{k}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 73317554942727741440:\\
\;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r10166713 = a;
        double r10166714 = k;
        double r10166715 = m;
        double r10166716 = pow(r10166714, r10166715);
        double r10166717 = r10166713 * r10166716;
        double r10166718 = 1.0;
        double r10166719 = 10.0;
        double r10166720 = r10166719 * r10166714;
        double r10166721 = r10166718 + r10166720;
        double r10166722 = r10166714 * r10166714;
        double r10166723 = r10166721 + r10166722;
        double r10166724 = r10166717 / r10166723;
        return r10166724;
}
↓
double f(double a, double k, double m) {
        double r10166725 = k;
        double r10166726 = 7.331755494272774e+19;
        bool r10166727 = r10166725 <= r10166726;
        double r10166728 = m;
        double r10166729 = pow(r10166725, r10166728);
        double r10166730 = a;
        double r10166731 = r10166729 * r10166730;
        double r10166732 = 10.0;
        double r10166733 = r10166725 + r10166732;
        double r10166734 = 1.0;
        double r10166735 = fma(r10166725, r10166733, r10166734);
        double r10166736 = r10166731 / r10166735;
        double r10166737 = r10166730 / r10166725;
        double r10166738 = log(r10166725);
        double r10166739 = r10166728 * r10166738;
        double r10166740 = exp(r10166739);
        double r10166741 = r10166740 / r10166725;
        double r10166742 = r10166737 * r10166741;
        double r10166743 = 99.0;
        double r10166744 = r10166725 * r10166725;
        double r10166745 = r10166743 / r10166744;
        double r10166746 = r10166732 / r10166725;
        double r10166747 = r10166745 - r10166746;
        double r10166748 = r10166742 * r10166747;
        double r10166749 = fma(r10166737, r10166741, r10166748);
        double r10166750 = r10166727 ? r10166736 : r10166749;
        return r10166750;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 7.331755494272774e+19

    1. Initial program 0.1

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

    2. Simplified0.0

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

    if 7.331755494272774e+19 < k

    1. Initial program 5.7

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

    2. Simplified5.7

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

    3. Taylor expanded around inf 5.7

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 73317554942727741440:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{k}, \frac{e^{m \cdot \log k}}{k}, \left(\frac{a}{k} \cdot \frac{e^{m \cdot \log k}}{k}\right) \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))