Average Error: 2.1 → 0.1
Time: 4.6m
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 9.517616257504169 \cdot 10^{+74}:\\ \;\;\;\;\frac{a}{\frac{\left(10 \cdot k + 1\right) + k \cdot k}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}}{k}\right))_* + \frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 9.517616257504169 \cdot 10^{+74}:\\
\;\;\;\;\frac{a}{\frac{\left(10 \cdot k + 1\right) + k \cdot k}{{k}^{m}}}\\

\mathbf{else}:\\
\;\;\;\;(\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}}{k}\right))_* + \frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\\

\end{array}
double f(double a, double k, double m) {
        double r91261086 = a;
        double r91261087 = k;
        double r91261088 = m;
        double r91261089 = pow(r91261087, r91261088);
        double r91261090 = r91261086 * r91261089;
        double r91261091 = 1.0;
        double r91261092 = 10.0;
        double r91261093 = r91261092 * r91261087;
        double r91261094 = r91261091 + r91261093;
        double r91261095 = r91261087 * r91261087;
        double r91261096 = r91261094 + r91261095;
        double r91261097 = r91261090 / r91261096;
        return r91261097;
}
↓
double f(double a, double k, double m) {
        double r91261098 = k;
        double r91261099 = 9.517616257504169e+74;
        bool r91261100 = r91261098 <= r91261099;
        double r91261101 = a;
        double r91261102 = 10.0;
        double r91261103 = r91261102 * r91261098;
        double r91261104 = 1.0;
        double r91261105 = r91261103 + r91261104;
        double r91261106 = r91261098 * r91261098;
        double r91261107 = r91261105 + r91261106;
        double r91261108 = m;
        double r91261109 = pow(r91261098, r91261108);
        double r91261110 = r91261107 / r91261109;
        double r91261111 = r91261101 / r91261110;
        double r91261112 = 99.0;
        double r91261113 = r91261112 / r91261106;
        double r91261114 = r91261101 / r91261098;
        double r91261115 = log(r91261098);
        double r91261116 = r91261108 * r91261115;
        double r91261117 = exp(r91261116);
        double r91261118 = r91261114 * r91261117;
        double r91261119 = r91261118 / r91261098;
        double r91261120 = -10.0;
        double r91261121 = r91261119 / r91261098;
        double r91261122 = r91261120 * r91261121;
        double r91261123 = fma(r91261113, r91261119, r91261122);
        double r91261124 = r91261123 + r91261119;
        double r91261125 = r91261100 ? r91261111 : r91261124;
        return r91261125;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 9.517616257504169e+74

    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

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

    if 9.517616257504169e+74 < k

    1. Initial program 7.3

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

    3. Applied associate-/l*7.3

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

    5. Applied clear-num7.5

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

    6. Taylor expanded around -inf 63.0

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

    7. Simplified0.1

      \[\leadsto \color{blue}{(\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}{k}\right))_* + \frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 9.517616257504169 \cdot 10^{+74}:\\ \;\;\;\;\frac{a}{\frac{\left(10 \cdot k + 1\right) + k \cdot k}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}}{k}\right))_* + \frac{\frac{a}{k} \cdot e^{m \cdot \log k}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))