Abstract
We present the simplest shift-and-add algorithms and the notions that will be useful for designing more sophisticated algorithms.
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Notes
- 1.
In this estimation, we do not take the rounding errors into account. They will depend on the precision used for representing the intermediate variables of the algorithm.
- 2.
In this estimation, we do not take the rounding errors into account. They will depend on the precision used for representing the intermediate variables of the algorithm.
- 3.
If \(d_n\) was chosen by means of a comparison, computation would be slowed down — the time saved by performing fully parallel additions would be lost — and huge tables would be required if \(d_n\) was given by a table lookup — and, incidentally, if we were able to implement such huge tables efficiently, we would rather directly tabulate the exponential function.
- 4.
Remember, \(L_{n}^{*}\) is a multiple of 1 / 2; therefore \(L_{n}^{*} > -1\) implies \(L_{n}^{*} \ge -1/2\).
- 5.
Since \(L_n^*\) is a multiple of 1 / 2, “\(-1/2 \le L_n^* \le 0\)” is equivalent to “\(L_n^* \in \{-1/2,0\}\).”
- 6.
Using a modified CORDIC algorithm; CORDIC is presented in the next chapter.
- 7.
This can be viewed as the possible use of \(d_{i}=2\) for a few values of i, or as the use of a new discrete base, obtained by repeating a few terms of the sequence \((w_{i})\).
- 8.
That is, \(0 \le x_0 \le \sum _{k=0}^{\infty }w_k\).
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Muller, JM. (2016). Introduction to Shift-and-Add Algorithms. In: Elementary Functions. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-7983-4_8
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DOI: https://doi.org/10.1007/978-1-4899-7983-4_8
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Publisher Name: Birkhäuser, Boston, MA
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