Average Error: 1.9 → 0.2
Time: 4.9s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 1.1028227832196924 \cdot 10^{+150}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a \cdot {k}^{m}} + \left(\frac{1}{a \cdot {k}^{m}} + \frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 1.1028227832196924 \cdot 10^{+150}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\

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

\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 1.1028227832196924e+150)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/
    1.0
    (+
     (* 10.0 (/ k (* a (pow k m))))
     (+ (/ 1.0 (* a (pow k m))) (* (/ k a) (/ k (pow k m))))))))
double code(double a, double k, double m) {
	return (((double) (a * ((double) pow(k, m)))) / ((double) (((double) (1.0 + ((double) (10.0 * k)))) + ((double) (k * k)))));
}
↓
double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.1028227832196924e+150)) {
		tmp = (((double) (a * ((double) pow(k, m)))) / ((double) (((double) (1.0 + ((double) (k * 10.0)))) + ((double) (k * k)))));
	} else {
		tmp = (1.0 / ((double) (((double) (10.0 * (k / ((double) (a * ((double) pow(k, m))))))) + ((double) ((1.0 / ((double) (a * ((double) pow(k, m))))) + ((double) ((k / a) * (k / ((double) pow(k, m))))))))));
	}
	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 < 1.10282278321969243e150

    1. Initial program Error: 0.1 bits

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

    if 1.10282278321969243e150 < k

    1. Initial program Error: 9.5 bits

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

    3. Applied clear-numError: 9.5 bits

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

    4. SimplifiedError: 9.5 bits

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

    5. Taylor expanded around inf Error: 9.5 bits

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

    6. SimplifiedError: 0.5 bits

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

  4. Final simplificationError: 0.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1028227832196924 \cdot 10^{+150}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a \cdot {k}^{m}} + \left(\frac{1}{a \cdot {k}^{m}} + \frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))