Average Error: 2.0 → 0.1
Time: 22.0s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 5.99520752198366 \cdot 10^{+89}:\\ \;\;\;\;\frac{a}{\frac{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}{\frac{{k}^{m}}{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{e^{m \cdot \log k}}{k}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 5.99520752198366 \cdot 10^{+89}:\\
\;\;\;\;\frac{a}{\frac{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}{\frac{{k}^{m}}{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{a}{k} \cdot \frac{e^{m \cdot \log k}}{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 5.99520752198366e+89)
   (/
    a
    (/
     (sqrt (+ 1.0 (+ (* k 10.0) (* k k))))
     (/ (pow k m) (sqrt (+ 1.0 (+ (* k 10.0) (* k k)))))))
   (* (/ a k) (/ (exp (* m (log k))) 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 <= 5.99520752198366e+89) {
		tmp = a / (sqrt(1.0 + ((k * 10.0) + (k * k))) / (pow(k, m) / sqrt(1.0 + ((k * 10.0) + (k * k)))));
	} else {
		tmp = (a / k) * (exp(m * log(k)) / 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 < 5.99520752198365953e89

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt_binary64_18050.2

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

    5. Applied associate-/l*_binary64_17280.2

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

    if 5.99520752198365953e89 < k

    1. Initial program 7.3

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

    2. Simplified7.3

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

    3. Taylor expanded around inf 7.3

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.99520752198366 \cdot 10^{+89}:\\ \;\;\;\;\frac{a}{\frac{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}{\frac{{k}^{m}}{\sqrt{1 + \left(k \cdot 10 + k \cdot k\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{e^{m \cdot \log k}}{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))