Abstract
The algorithms presented in the previous chapters for evaluating the elementary functions only give correct result if the argument is within a given bounded interval.
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Notes
- 1.
In the radix of the floating-point system being used —2 in general.
- 2.
Gal and Bachelis [199] use the same kind of range reduction, but slightly differently. Some accuracy is lost when subtracting \({{\mathrm{RN}}}(kC_2)\) from \({{\mathrm{RN}}}(x-{{\mathrm{RN}}}(kC_1))\). To avoid this, they keep these terms separate. This is also done by Tang (see section 6.2.1).
- 3.
Unless subtraction is not correctly rounded. To my knowledge, there is no current computer with such a poor arithmetic.
- 4.
- 5.
Important notice: one can easily find an even better approximation to C as a sum of two floating-point numbers, however, it may not satisfy the property presented in Theorem 18, that is, \(x - k\cdot {}(C_1+C_2)\) may not be computed exactly by Algorithm 23.
- 6.
Beware: \(RN(-k \cdot C_3)\) is not necessarily very small in front of \(v_\ell \). If we need that property—which is likely if we want to evaluate some approximating polynomial at the reduced argument—some further manipulations are needed.
- 7.
If \(Q_{j}\) is less than \(\beta ^{p-1}\), this value does not actually correspond to a floating-point number of exponent exponent. In such a case, the actual lowest value of \(\epsilon \) is larger. However, the value corresponding to \(Q_{j}\) is actually attained for another value of the exponent: let \(\ell \) be the integer such that \(\beta ^{p-1} \le \beta ^\ell {}Q_{j} < \beta ^p\); from
$$ CQ_{j}\left| \frac{P_{j}}{Q_{j}} - \frac{\beta ^{exponent-p+1}}{C}\right| = C\beta ^{\ell }Q_{j}\left| \frac{P_{j}}{\beta ^{\ell }Q_{j}} - \frac{\beta ^{exponent-\ell -p+1}}{C}\right| $$we deduce that the value of \(\epsilon \) corresponding to \(Q_{j}\) is actually attained when the exponent equals \(exponent-\ell \). Therefore the fact that \(Q_{j}\) may be less than \(\beta ^{p-1}\) has no influence on the final result, that is, the search for the lowest possible value of \(\epsilon \).
- 8.
This formula looks correct only for positive values of \(\nu \). It would be more correct, although maybe less clear, to write: \(r = \sum _{i=\nu }^{N-1}x_im_i + \sum _{i=-p}^{\nu -1}x_i2^i\).
- 9.
We denote \(\lfloor x \rceil \) the integer that is nearest to x.
- 10.
In all practical cases, C is a transcendental number, so that \(m_i^+\) and \(m_i^-\) are never equal to \(-C\), 0, or \(+C\).
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Muller, JM. (2016). Range Reduction. In: Elementary Functions. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-7983-4_11
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DOI: https://doi.org/10.1007/978-1-4899-7983-4_11
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