Average Error: 1.9 → 0.2
Time: 11.5s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 1.6232515547158186 \cdot 10^{+149}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{1}{a \cdot {k}^{m}} + 10 \cdot \frac{k}{a \cdot {k}^{m}}\right) + k \cdot \frac{1}{\frac{a \cdot {k}^{m}}{k}}}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 1.6232515547158186 \cdot 10^{+149}:\\
\;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(\frac{1}{a \cdot {k}^{m}} + 10 \cdot \frac{k}{a \cdot {k}^{m}}\right) + k \cdot \frac{1}{\frac{a \cdot {k}^{m}}{k}}}\\

\end{array}
(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (if (<= k 1.6232515547158186e+149)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (/
    1.0
    (+
     (+ (/ 1.0 (* a (pow k m))) (* 10.0 (/ k (* a (pow k m)))))
     (* k (/ 1.0 (/ (* a (pow k m)) k)))))))
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 tmp;
	if (k <= 1.6232515547158186e+149) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = 1.0 / (((1.0 / (a * pow(k, m))) + (10.0 * (k / (a * pow(k, m))))) + (k * (1.0 / ((a * pow(k, m)) / k))));
	}
	return tmp;
}

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 < 1.6232515547158186e149

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied *-un-lft-identity_binary64_17830.1

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

    5. Applied times-frac_binary64_17890.1

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

    if 1.6232515547158186e149 < k

    1. Initial program 9.3

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

    2. Simplified9.3

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

    4. Applied clear-num_binary64_17829.4

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

    5. Taylor expanded around 0 9.4

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

    6. Simplified0.5

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

    8. Applied clear-num_binary64_17820.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6232515547158186 \cdot 10^{+149}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{1}{a \cdot {k}^{m}} + 10 \cdot \frac{k}{a \cdot {k}^{m}}\right) + k \cdot \frac{1}{\frac{a \cdot {k}^{m}}{k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021021 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))