Average Error: 1.9 → 0.3
Time: 10.1s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 2.0989642968183564 \cdot 10^{-27}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot {k}^{m}} + \frac{k}{a \cdot {k}^{m}} \cdot \left(k + 10\right)}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 2.0989642968183564 \cdot 10^{-27}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1}{a \cdot {k}^{m}} + \frac{k}{a \cdot {k}^{m}} \cdot \left(k + 10\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 2.0989642968183564e-27)
   (/ (* a (pow k m)) (+ 1.0 (* k 10.0)))
   (/ 1.0 (+ (/ 1.0 (* a (pow k m))) (* (/ k (* a (pow k m))) (+ k 10.0))))))
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 <= 2.0989642968183564e-27) {
		tmp = (a * pow(k, m)) / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / ((1.0 / (a * pow(k, m))) + ((k / (a * pow(k, m))) * (k + 10.0)));
	}
	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 < 2.0989642968183564e-27

    1. Initial program 0.1

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

    2. Simplified0.0

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

    3. Taylor expanded around 0 0.0

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

    if 2.0989642968183564e-27 < k

    1. Initial program 4.6

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

    2. Simplified4.6

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

    4. Applied clear-num_binary64_14414.9

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

    5. Taylor expanded around 0 4.9

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

    6. Simplified0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.0989642968183564 \cdot 10^{-27}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot {k}^{m}} + \frac{k}{a \cdot {k}^{m}} \cdot \left(k + 10\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))