Average Error: 2.0 → 0.2
Time: 4.4s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 3.408471258283922 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{k}{{k}^{m}} \cdot \left(\left(k + 10\right) \cdot \frac{1}{a}\right)}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 3.408471258283922 \cdot 10^{+144}:\\
\;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{k}{{k}^{m}} \cdot \left(\left(k + 10\right) \cdot \frac{1}{a}\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 3.408471258283922e+144)
   (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) (* a (pow k m))))
   (/
    1.0
    (+ (/ (pow k (- m)) a) (* (/ k (pow k m)) (* (+ k 10.0) (/ 1.0 a)))))))
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 <= 3.408471258283922e+144)) {
		tmp = (1.0 / (((double) (1.0 + ((double) (k * ((double) (k + 10.0)))))) / ((double) (a * ((double) pow(k, m))))));
	} else {
		tmp = (1.0 / ((double) ((((double) pow(k, ((double) -(m)))) / a) + ((double) ((k / ((double) pow(k, m))) * ((double) (((double) (k + 10.0)) * (1.0 / a))))))));
	}
	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 < 3.4084712582839217e144

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied clear-num_binary640.2

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

    if 3.4084712582839217e144 < k

    1. Initial program 9.5

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

    2. Simplified9.5

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

    4. Applied clear-num_binary649.5

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

    5. Taylor expanded around inf 9.5

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

    6. Simplified0.5

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

    8. Applied div-inv_binary640.5

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

    9. Applied div-inv_binary640.5

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

    10. Applied distribute-rgt-out_binary640.5

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

    11. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.408471258283922 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{k}{{k}^{m}} \cdot \left(\left(k + 10\right) \cdot \frac{1}{a}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))