Average Error: 1.9 → 0.1
Time: 3.5s
Precision: 64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.2205424189704093 \cdot 10^{149}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(\frac{\frac{m}{2}}{2}\right)}}{\frac{10 \cdot k + \left({k}^{2} + 1\right)}{{k}^{\left(\frac{\frac{m}{2}}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(99 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{4}} - 10 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{3}}\right) + \frac{\frac{a}{k} \cdot {\left({\left(\frac{1}{k}\right)}^{m}\right)}^{-1}}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \le 3.2205424189704093 \cdot 10^{149}:\\
\;\;\;\;\frac{a \cdot {k}^{\left(\frac{\frac{m}{2}}{2}\right)}}{\frac{10 \cdot k + \left({k}^{2} + 1\right)}{{k}^{\left(\frac{\frac{m}{2}}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\left(99 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{4}} - 10 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{3}}\right) + \frac{\frac{a}{k} \cdot {\left({\left(\frac{1}{k}\right)}^{m}\right)}^{-1}}{k}\\

\end{array}
double code(double a, double k, double m) {
	return ((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k)));
}
↓
double code(double a, double k, double m) {
	double VAR;
	if ((k <= 3.2205424189704093e+149)) {
		VAR = ((a * pow(k, ((m / 2.0) / 2.0))) / (((10.0 * k) + (pow(k, 2.0) + 1.0)) / (pow(k, ((m / 2.0) / 2.0)) * pow(k, (m / 2.0)))));
	} else {
		VAR = (((99.0 * ((a * pow(exp((-0.5 * (m * log((1.0 / k))))), 2.0)) / pow(k, 4.0))) - (10.0 * ((a * pow(exp((-0.5 * (m * log((1.0 / k))))), 2.0)) / pow(k, 3.0)))) + (((a / k) * pow(pow((1.0 / k), m), -1.0)) / k));
	}
	return VAR;
}

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 < 3.2205424189704093e+149

    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 +-commutative0.1

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

    4. Applied associate-+l+0.1

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

    5. Simplified0.1

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

    7. Applied sqr-pow0.1

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

    8. Applied associate-*r*0.1

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

    10. Applied sqr-pow0.1

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

    11. Applied associate-*r*0.1

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

    12. Applied associate-*l*0.1

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

    13. Applied associate-/l*0.1

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

    if 3.2205424189704093e+149 < k

    1. Initial program 9.1

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

    3. Applied +-commutative9.1

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

    4. Applied associate-+l+9.1

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

    5. Simplified9.1

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

    7. Applied sqr-pow9.1

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

    8. Applied associate-*r*9.1

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

    9. Taylor expanded around inf 9.1

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

    10. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3.2205424189704093 \cdot 10^{149}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(\frac{\frac{m}{2}}{2}\right)}}{\frac{10 \cdot k + \left({k}^{2} + 1\right)}{{k}^{\left(\frac{\frac{m}{2}}{2}\right)} \cdot {k}^{\left(\frac{m}{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(99 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{4}} - 10 \cdot \frac{a \cdot {\left(e^{\frac{-1}{2} \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}^{2}}{{k}^{3}}\right) + \frac{\frac{a}{k} \cdot {\left({\left(\frac{1}{k}\right)}^{m}\right)}^{-1}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))