Average Error: 2.0 → 0.1
Time: 25.9s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.6905430635633034 \cdot 10^{+145}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99}{k \cdot k} - \frac{10}{k}\right) \cdot \frac{\frac{e^{m \cdot \log k}}{k} \cdot a}{k} + \frac{\frac{e^{m \cdot \log k}}{k} \cdot a}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 2.6905430635633034 \cdot 10^{+145}:\\
\;\;\;\;\frac{a \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\\

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

\end{array}
double f(double a, double k, double m) {
        double r6913061 = a;
        double r6913062 = k;
        double r6913063 = m;
        double r6913064 = pow(r6913062, r6913063);
        double r6913065 = r6913061 * r6913064;
        double r6913066 = 1.0;
        double r6913067 = 10.0;
        double r6913068 = r6913067 * r6913062;
        double r6913069 = r6913066 + r6913068;
        double r6913070 = r6913062 * r6913062;
        double r6913071 = r6913069 + r6913070;
        double r6913072 = r6913065 / r6913071;
        return r6913072;
}
↓
double f(double a, double k, double m) {
        double r6913073 = k;
        double r6913074 = 2.6905430635633034e+145;
        bool r6913075 = r6913073 <= r6913074;
        double r6913076 = a;
        double r6913077 = m;
        double r6913078 = 2.0;
        double r6913079 = r6913077 / r6913078;
        double r6913080 = pow(r6913073, r6913079);
        double r6913081 = r6913076 * r6913080;
        double r6913082 = 1.0;
        double r6913083 = 10.0;
        double r6913084 = r6913073 + r6913083;
        double r6913085 = r6913073 * r6913084;
        double r6913086 = r6913082 + r6913085;
        double r6913087 = r6913081 / r6913086;
        double r6913088 = r6913087 * r6913080;
        double r6913089 = 99.0;
        double r6913090 = r6913073 * r6913073;
        double r6913091 = r6913089 / r6913090;
        double r6913092 = r6913083 / r6913073;
        double r6913093 = r6913091 - r6913092;
        double r6913094 = log(r6913073);
        double r6913095 = r6913077 * r6913094;
        double r6913096 = exp(r6913095);
        double r6913097 = r6913096 / r6913073;
        double r6913098 = r6913097 * r6913076;
        double r6913099 = r6913098 / r6913073;
        double r6913100 = r6913093 * r6913099;
        double r6913101 = r6913100 + r6913099;
        double r6913102 = r6913075 ? r6913088 : r6913101;
        return r6913102;
}

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 < 2.6905430635633034e+145

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied sqr-pow0.0

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

    5. Applied associate-*l*0.1

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

    6. Taylor expanded around 0 0.1

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

    7. Simplified0.1

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

    9. Applied add-sqr-sqrt0.1

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

    10. Applied associate-/r*0.1

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

    12. Applied *-un-lft-identity0.1

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

    13. Applied sqrt-prod0.1

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

    14. Applied *-un-lft-identity0.1

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

    15. Applied sqrt-prod0.1

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

    16. Applied times-frac0.1

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

    17. Applied times-frac0.1

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

    18. Simplified0.1

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

    19. Simplified0.1

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

    if 2.6905430635633034e+145 < k

    1. Initial program 9.7

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

    2. Simplified9.7

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

    4. Applied sqr-pow9.7

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

    5. Applied associate-*l*9.7

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

    6. Taylor expanded around 0 9.7

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

    7. Simplified9.7

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

    8. Taylor expanded around -inf 63.0

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

    9. Simplified0.1

      \[\leadsto \color{blue}{\frac{a \cdot \frac{e^{\left(\left(0 + \log k\right) \cdot m\right) \cdot 1}}{k}}{k} \cdot \left(\frac{99}{k \cdot k} - \frac{10}{k}\right) + \frac{a \cdot \frac{e^{\left(\left(0 + \log k\right) \cdot m\right) \cdot 1}}{k}}{k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.6905430635633034 \cdot 10^{+145}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(\frac{m}{2}\right)}}{1 + k \cdot \left(k + 10\right)} \cdot {k}^{\left(\frac{m}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{99}{k \cdot k} - \frac{10}{k}\right) \cdot \frac{\frac{e^{m \cdot \log k}}{k} \cdot a}{k} + \frac{\frac{e^{m \cdot \log k}}{k} \cdot a}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))