Abstract
Multiple-precision arithmetic is a useful tool in many domains of contemporary science. Some numerical applications are known to sometimes require significantly more precision than provided by the usual binary32/single-precision, binary-64/double precision, and Intel extended-precision formats [22].
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Notes
- 1.
That is, there is no need to compute any of the previous digits. For instance [49], the \(2.5 \times 10^{14}\)th hexadecimal digit of \(\pi \) is an E.
- 2.
MPFUN2015 is available at http://www.davidhbailey.com/dhbsoftware/
- 3.
- 4.
Available at http://www.mpfr.org.
- 5.
- 6.
In fact, it is an elementary function. The functions we deal with in this book should be called elementary transcendental functions.
- 7.
I give this presentation assuming radix 2 is used. Generalization to other radices is straightforward.
- 8.
This is not exactly Katatsuba and Ofman’s presentation. I give here Knuth’s version of the algorithm [275], which is slightly better.
- 9.
To simplify we assume \(n = m\nu \) exactly. In practice, if needed, we add a few zero bits at the left of the binary representations of a and b, and we choose \(\nu = \lceil n/m \rceil \).
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- 11.
From n, compute \(\ln (n)\), then \(\ln (\ln (n))\), then \(\ln (\ln (\ln (n)))\), etc., until you get a result less than one. The number of iterations is \(\log ^* n\).
- 12.
It is assumed that there exist two constants \(\alpha \) and \(\beta \), \(0< \alpha , \beta < 1\), such that the delay M(n) of n-bit multiplication satisfies \(M(\alpha n) \le \beta M(n)\) if n is large enough [57]. This assumption is satisfied by the grammar school multiplication method, as well as by the Karatsuba-like, the Toom–Cook, and the FFT-based algorithms.
- 13.
\(n! \approx \sqrt{2n\pi }(n/e)^n\).
- 14.
According to Borwein and Borwein [51], the switchover is somewhere in the 100 to 1000 decimal digit range.
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Muller, JM. (2016). Multiple-Precision Evaluation of Functions. In: Elementary Functions. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-7983-4_7
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DOI: https://doi.org/10.1007/978-1-4899-7983-4_7
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