Average Error: 2.1 → 0.1
Time: 26.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.050984953394394 \cdot 10^{+107}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k \cdot k}, -10, \mathsf{fma}\left(99, \frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k}\right)\right)\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 3.050984953394394 \cdot 10^{+107}:\\
\;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{{k}^{m}}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r15887782 = a;
        double r15887783 = k;
        double r15887784 = m;
        double r15887785 = pow(r15887783, r15887784);
        double r15887786 = r15887782 * r15887785;
        double r15887787 = 1.0;
        double r15887788 = 10.0;
        double r15887789 = r15887788 * r15887783;
        double r15887790 = r15887787 + r15887789;
        double r15887791 = r15887783 * r15887783;
        double r15887792 = r15887790 + r15887791;
        double r15887793 = r15887786 / r15887792;
        return r15887793;
}
↓
double f(double a, double k, double m) {
        double r15887794 = k;
        double r15887795 = 3.050984953394394e+107;
        bool r15887796 = r15887794 <= r15887795;
        double r15887797 = a;
        double r15887798 = 10.0;
        double r15887799 = r15887794 + r15887798;
        double r15887800 = 1.0;
        double r15887801 = fma(r15887799, r15887794, r15887800);
        double r15887802 = m;
        double r15887803 = pow(r15887794, r15887802);
        double r15887804 = r15887801 / r15887803;
        double r15887805 = r15887797 / r15887804;
        double r15887806 = log(r15887794);
        double r15887807 = r15887806 * r15887802;
        double r15887808 = exp(r15887807);
        double r15887809 = r15887808 / r15887794;
        double r15887810 = r15887794 * r15887794;
        double r15887811 = r15887797 / r15887810;
        double r15887812 = r15887809 * r15887811;
        double r15887813 = -10.0;
        double r15887814 = 99.0;
        double r15887815 = r15887810 * r15887810;
        double r15887816 = r15887815 / r15887797;
        double r15887817 = r15887808 / r15887816;
        double r15887818 = r15887797 / r15887794;
        double r15887819 = r15887809 * r15887818;
        double r15887820 = fma(r15887814, r15887817, r15887819);
        double r15887821 = fma(r15887812, r15887813, r15887820);
        double r15887822 = r15887796 ? r15887805 : r15887821;
        return r15887822;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.050984953394394e+107

    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 associate-/l*0.1

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

    4. Simplified0.0

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

    if 3.050984953394394e+107 < k

    1. Initial program 8.2

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
    2. Using strategy rm

    3. Applied associate-/l*8.2

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

    4. Simplified8.2

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

    5. Taylor expanded around inf 8.2

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3.050984953394394 \cdot 10^{+107}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k \cdot k}, -10, \mathsf{fma}\left(99, \frac{e^{\log k \cdot m}}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{a}}, \frac{e^{\log k \cdot m}}{k} \cdot \frac{a}{k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))