Average Error: 2.1 → 0.1
Time: 10.2s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.09022971803593299 \cdot 10^{147}:\\ \;\;\;\;\left(\left(\frac{a}{k \cdot \left(10 + k\right) + 1} \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{a}{e^{m \cdot \left(-\log k\right)}}}{k}}{k} - \frac{10}{{k}^{3}} \cdot \frac{a}{e^{m \cdot \left(-\log k\right)}}\right) + \frac{a}{{k}^{4} \cdot e^{m \cdot \left(-\log k\right)}} \cdot 99\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 5.09022971803593299 \cdot 10^{147}:\\
\;\;\;\;\left(\left(\frac{a}{k \cdot \left(10 + k\right) + 1} \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\\

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

\end{array}
double f(double a, double k, double m) {
        double r269942 = a;
        double r269943 = k;
        double r269944 = m;
        double r269945 = pow(r269943, r269944);
        double r269946 = r269942 * r269945;
        double r269947 = 1.0;
        double r269948 = 10.0;
        double r269949 = r269948 * r269943;
        double r269950 = r269947 + r269949;
        double r269951 = r269943 * r269943;
        double r269952 = r269950 + r269951;
        double r269953 = r269946 / r269952;
        return r269953;
}
↓
double f(double a, double k, double m) {
        double r269954 = k;
        double r269955 = 5.090229718035933e+147;
        bool r269956 = r269954 <= r269955;
        double r269957 = a;
        double r269958 = 10.0;
        double r269959 = r269958 + r269954;
        double r269960 = r269954 * r269959;
        double r269961 = 1.0;
        double r269962 = r269960 + r269961;
        double r269963 = r269957 / r269962;
        double r269964 = m;
        double r269965 = 2.0;
        double r269966 = r269964 / r269965;
        double r269967 = pow(r269954, r269966);
        double r269968 = r269963 * r269967;
        double r269969 = sqrt(r269967);
        double r269970 = r269968 * r269969;
        double r269971 = r269970 * r269969;
        double r269972 = log(r269954);
        double r269973 = -r269972;
        double r269974 = r269964 * r269973;
        double r269975 = exp(r269974);
        double r269976 = r269957 / r269975;
        double r269977 = r269976 / r269954;
        double r269978 = r269977 / r269954;
        double r269979 = 3.0;
        double r269980 = pow(r269954, r269979);
        double r269981 = r269958 / r269980;
        double r269982 = r269981 * r269976;
        double r269983 = r269978 - r269982;
        double r269984 = 4.0;
        double r269985 = pow(r269954, r269984);
        double r269986 = r269985 * r269975;
        double r269987 = r269957 / r269986;
        double r269988 = 99.0;
        double r269989 = r269987 * r269988;
        double r269990 = r269983 + r269989;
        double r269991 = r269956 ? r269971 : r269990;
        return r269991;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < 5.090229718035933e+147

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied sqr-pow0.1

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

    5. Applied associate-*r*0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied associate-*r*0.1

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

    if 5.090229718035933e+147 < k

    1. Initial program 10.2

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

    2. Simplified10.2

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

    3. Taylor expanded around inf 10.2

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.09022971803593299 \cdot 10^{147}:\\ \;\;\;\;\left(\left(\frac{a}{k \cdot \left(10 + k\right) + 1} \cdot {k}^{\left(\frac{m}{2}\right)}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\right) \cdot \sqrt{{k}^{\left(\frac{m}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{a}{e^{m \cdot \left(-\log k\right)}}}{k}}{k} - \frac{10}{{k}^{3}} \cdot \frac{a}{e^{m \cdot \left(-\log k\right)}}\right) + \frac{a}{{k}^{4} \cdot e^{m \cdot \left(-\log k\right)}} \cdot 99\\ \end{array}\]

Reproduce

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