Average Error: 2.2 → 0.1
Time: 2.6min
Precision: binary64
Cost: 20673
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 2.350163294637587 \cdot 10^{+136}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}\\ \end{array}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
\mathbf{if}\;k \leq 2.350163294637587 \cdot 10^{+136}:\\
\;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}\\

\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 2.350163294637587e+136)
   (/ (* a (pow k m)) (+ 1.0 (* k (+ k 10.0))))
   (- (/ (* (pow k m) (/ a k)) k) (* 10.0 (/ (* a (pow k m)) (pow k 3.0))))))
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 <= 2.350163294637587e+136) {
		tmp = (a * pow(k, m)) / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = ((pow(k, m) * (a / k)) / k) - (10.0 * ((a * pow(k, m)) / pow(k, 3.0)));
	}
	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

Alternatives

Alternative 1
Error2.2
Cost7168
\[\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\]
Alternative 2
Error3.0
Cost7040
\[\frac{a \cdot {k}^{m}}{1 + k \cdot k}\]
Alternative 3
Error2.9
Cost6984
\[\begin{array}{l} \mathbf{if}\;m \leq -2.2363572764943224 \cdot 10^{-06} \lor \neg \left(m \leq 8.916671353359343 \cdot 10^{-09}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array}\]
Alternative 4
Error3.3
Cost1218
\[\begin{array}{l} \mathbf{if}\;m \leq -1.7590351739721123 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.4998279862905466:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 5
Error4.0
Cost1090
\[\begin{array}{l} \mathbf{if}\;m \leq -1.7590351739721123 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.6374213960110211:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 6
Error16.8
Cost1925
\[\begin{array}{l} \mathbf{if}\;m \leq -7.98549713398114 \cdot 10^{-35}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq -3.4141330431881928 \cdot 10^{-74}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq -5.701926045057698 \cdot 10^{-292}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;m \leq 4.1855111657774655 \cdot 10^{-128}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 0.5914995882010451:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 7
Error16.8
Cost1925
\[\begin{array}{l} \mathbf{if}\;m \leq -7.98549713398114 \cdot 10^{-35}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq -3.179614406887836 \cdot 10^{-71}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq -2.8984927067274725 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 1.1692655545065984 \cdot 10^{-127}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 0.889991338965889:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 8
Error18.0
Cost706
\[\begin{array}{l} \mathbf{if}\;m \leq -7.98549713398114 \cdot 10^{-35}:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 2.7459188292006855 \cdot 10^{-127}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 9
Error46.5
Cost64
\[a\]

Error

Derivation

  1. Split input into 2 regimes

  2. if k < 2.3501632946375871e136

    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. Simplified0.0

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

    if 2.3501632946375871e136 < k

    1. Initial program 10.3

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

    2. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{2}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}}\]

    3. Simplified10.3

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary64_24090.1

      \[\leadsto \color{blue}{\frac{\frac{a \cdot {k}^{m}}{k}}{k}} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}\]

    6. Simplified0.1

      \[\leadsto \frac{\color{blue}{\frac{a}{k} \cdot {k}^{m}}}{k} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}\]

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.350163294637587 \cdot 10^{+136}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k} - 10 \cdot \frac{a \cdot {k}^{m}}{{k}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))