Acommon practice for computing elementary transcendental

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1 678 IEEE TRANSACTIONS ON COMPUTERS, VOL. 53, NO. 6, JUNE 004 Near Optimality of Chebyshev Interpolation for Elementary Function Computations Ren-Cang Li Abstract A common practice for computing an elementary transcendental function in an libm implementation nowadays has two phases: reductions of input arguments to fall into a tiny interval and polynomial approimations for the function within the interval. Typically, the interval is made tiny enough so that polynomials of very high degree aren t required for accurate approimations. Often, approimating polynomials as such are taken to be the polynomials or any others such as the Chebyshev interpolating polynomials. The polynomial of degree n has the property that the biggest difference between it and the function is smallest among all possible polynomials of degrees no higher than n. Thus, it is natural to choose the polynomials over others. In this paper, it is proven that the polynomial can only be more accurate by at most a fractional bit than the Chebyshev interpolating polynomial of the same degree in computing elementary functions or, in other words, the Chebyshev interpolating polynomials will do just as well as the polynomials. Similar results were obtained in 967 by Powell who, however, did not target elementary function computations in particular and placed no assumption on the function and, remarkably, whose results imply accuracy differences of no more than to 3 bits in the contet of this paper. Inde Terms Elementary function computation, libm, Chebyshev Interpolation, Remez, polynomial, accuracy. æ INTRODUCTION Acommon practice for computing elementary transcendental functions fðþ has two phases (see [], [], [3], [4], [5], [7], [8], [9], [0], and references therein):. An argument reduction that typically reduces to fall into a very narrow interval ½ ; Š, e.g., 5 or smaller.. An efficient polynomial approimation to fðþ for ½ ; Š. Often, approimating polynomials are forced to match fðþ at ¼ 0 for computational advantages. This is especially true when fð0þ ¼0. Small ensures the use of approimating polynomials of low degrees and, consequently, little time to evaluate them. Often, is very tiny, especially for table-driven algorithms [3], [4], [5], [7], [8], [9], [0]. There are many ways to find approimating polynomials. Given the degree n of the polynomials, common ones are ) finding the polynomial p n ðþ that minimizes ma ½ ;Š jfðþ p n ðþj (or a variation of this if fð0þ is to be preserved; see below for details) and ) finding the Chebyshev interpolating polynomial. Rational approimation is another alternative, but it is seldom used in elementary function computations because division is much slower than addition/multiplication and, moreover, the division can only be initiated after the availability of the denominator and/or the numerator.. There may be eceptions to this, e.g., computing sin may require polynomial approimations to both sin and cos, depending on algorithms.. The author is with the Department of Mathematics, University of Kentucky, Leington, KY rcli@ms.uky.edu. Manuscript received 3 Apr. 003; revised 6 Nov. 003; accepted 7 Nov For information on obtaining reprints of this article, please send to: tc@computer.org, and reference IEEECS Log Number 853. This paper compares the approimation accuracy achieved by the two polynomial approimation techniques. Earlier results along this line are due to Powell [6]. However, Powell s paper went largely unnoticed in the literature of elementary function computations. Even so, through practice it has become folklore [3], [4] that the Chebyshev interpolating polynomial will be just as effective as the polynomial with the same degree. In fact, most error analysis for the polynomial solution part in Markstein [3] is based on the Chebyshev polynomial interpolation. In this paper, we will show that the Chebyshev interpolating polynomial may be less accurate, if indeed, than the polynomial of the same degree by only a fractional bit. The polynomials are typically obtained through a slowly convergent iterative procedure the Remez procedure [], while we do have a close formula for the Chebyshev interpolating polynomials. This does not, however, necessarily place the polynomial approach at a disadvantage since all approimating polynomials are precomputed in any libm implementations [], [] and, thus, the cost in computing approimating polynomials becomes somewhat irrelevant. Even so, we feel that our contribution here has its significance, that is, that simple Chebyshev interpolation is sufficient and that the common practice of feeding Chebyshev interpolation polynomials into the Remez procedure offers negligible improvement, if any, and thus may be considered unnecessary. Our results compare favorably to Powell s results which, applied to the contet here, say the difference in accuracy is no more than to 3 bits. It is worth mentioning that Powell s results were obtained, remarkably, without any assumption on fðþ. The rest of this paper is organized as follows: Section briefly introduces the general interpolation theory, enough to serve our purpose later on. Chebyshev interpolation and /04/$0.00 ß 004 IEEE Published by the IEEE Computer Society

2 LI: NEAR OPTIMALITY OF CHEBYSHEV INTERPOLATION FOR ELEMENTARY FUNCTION COMPUTATIONS 679 polynomial approimations are discussed in Sections 3 and 4 with emphasis on error epansions on these approimations. Our main theorems that compare the accuracies by the two approaches are given in Section 5, where a theorem of Powell s will also be discussed. An asymptotic analysis in Section 6 on the main theorems confirms our claim made at the beginning of this paper and concludes that the approimation accuracies between the polynomials and the Chebyshev interpolating polynomials differ by Oð Þ bits for odd and even functions and by OðÞ for others. Finally, in Section 8, the main theorems are applied to the most common elementary transcendental functions, such as epðþ, ln, sin, and so on.. Notation Throughout the remainder of this paper, the interval of interest is ½; Š on which fðþ is defined. It is assumed that fðþ is continuous and has continuous derivatives up to ðn þ Þth (or more as deemed necessary), where n is the degree of polynomials sought to approimate fðþ. If needed, 0 ð; Þ is also implied. Define f 0 ðþ :¼ fðþ fð0þ m ; ðþ where m is the multiplicity of ¼ 0 as a zero of fðþ fð0þ, and := is the assignment operator. Usually, m ¼, but may not be, e.g., for cos it is. Unknown points ; ½; Š in error equations later may be different from one occurrence to another. GENERAL POLYNOMIAL INTERPOLATION Polynomial interpolation is an ancient idea that is still widely in use today to approimate a function. Let 0 ; ;...; n ðþ be interpolating points in the interval. The interpolating polynomial p n ðþ is uniquely determined by p n ðþ and fðþ coinciding at those points in the sense that, if i is repeated k times, p ðjþ n ð iþ¼f ðjþ ð i Þ for j ¼ 0;...k : (So, if k ¼, only the function value matches at i.) If none of the points i collapse, this is called Lagrange Interpolation. In the etreme case when all i s collapse to one point, p n ðþ is just the truncated Taylor series of fðþ at the collapsed point. In any case, the difference of fðþ and p n ðþ is given as follows: If f ðnþþ ðþ eists and is continuous, then fðþ p n ðþ ¼ fðnþþ ðþ ð i Þ; ð3þ ðn þ Þ! for some minf; 0 ;...; n g maf; 0 ;...; n g [3, p. 60]. This equation offers one way to find a good polynomial to make ma jfðþ p n ðþj small. Since f ðnþþ ðþ is function-dependent and, often, is unknown, it is natural to look for i s so that ma j Q n ð iþj is minimized. It turns out this minimization problem can be solved elegantly: Pick i s as the translated zeros of the ðn þ Þth Chebyshev polynomial (see Theorem 3. below). At times, polynomials that preserve certain properties of fðþ are called for so that simple but important mathematical identities, such as log ¼ 0, epð0þ ¼, sin 0 ¼ 0, and cosð0þ ¼, are numerically conserved. In elementary function computations, this typically happens when 0 ½; Š (and ¼ as well, but we shall not assume this for now). To make sure that identities as such remain valid in the face of roundoffs, we need to place ¼ 0 to the list of interpolating points to ensure fð0þ ¼p n ð0þ eactly! One way to implement this is to interpolate f 0 ðþ at n m þ other points, ~ 0 ; ~ ;...; ~ n m ; ð4þ by a polynomial h n m ðþ of degree n m. Then, set ~p n ðþ :¼ fð0þþ m h n m ðþ; a polynomial of degree n that can be seen to coincide with fðþ at ¼ 0, as well as at ~ i s. An application of (3) gives for some fðþ ~p n ðþ ¼ fðnþþ ðþ ðn þ Þ! m ð ~ i Þ minf; 0; ~ 0 ;...; ~ n m g maf; 0; ~ 0 ;...; ~ n m g: 3 CHEBYSHEV INTERPOLATION The standard Chebyshev polynomial interpolation for a function uðtþ is on the closed interval ½ ; Š. The idea is to seek a polynomial q n ðtþ of degree n that matches eactly uðtþ at the zeros t i :¼ cos i þ ; i ¼ 0; ;...;n; ð6þ n þ of T nþ ðtþ cosðn þ Þ arccos t, the ðn þ Þth Chebyshev polynomial, i.e., uðt i Þ¼q n ðt i Þ for i ¼ 0; ;...;n. It can be proven that (see, e.g., [3, p. 99]) where q n ðtþ ¼ Xn k¼0 k T k ðtþ; P n k ¼ nþ uðt iþ; if k ¼ 0; P n nþ T kðt i Þuðt i Þ; otherwise: Theorem 3.. If u is odd (even), then q n is also odd (even). Proof. This is a well-known property of Chebyshev interpolation. But, we shall give a proof here for completeness. Notice that T nþ ð tþ ¼cosðn þ Þ arccosð tþ ¼ cos½ðn þ Þð arccos tþš ¼ð Þ nþ T nþ ðtþ:. Mathematically, it does not have to be 0 and, in fact, we can preserve the function value at any given floating-point number so long as the function value is also a floating-point number. Such a generalization, however, is no more general than 0 itself because shifting of origin can always make it 0! ð5þ ð7þ ð8þ

3 680 IEEE TRANSACTIONS ON COMPUTERS, VOL. 53, NO. 6, JUNE 004 T nþ is an even function if n þ is even and an odd function otherwise. Therefore, t i and t i appear in pairs in the list (6). Suppose u is odd, i.e., uð tþ ¼ uðtþ. Let s n ðtþ :¼ q n ð tþ, also a polynomial of degree n. We have s n ðt i Þ¼ q n ð t i Þ¼ uð t i Þ¼uðt i Þ; which says that s n also interpolates u at the same points (6) as q n does. Since the interpolating polynomial is unique, we conclude that q n ðtþ s n ðtþ ¼ q n ð tþ, i.e., q n is odd. Now, suppose u is even, i.e., uð tþ ¼uðtÞ. Let s n ðtþ :¼ q n ð tþ, a polynomial of degree n. We have s n ðt i Þ¼q n ð t i Þ¼uð t i Þ¼uðt i Þ; which says that s n also interpolates u at the same points (6) as q n does. The uniqueness of the interpolating polynomial leads to q n ðtþ s n ðtþ ¼q n ð tþ, i.e., q n is even. tu Now, we turn to the continuous function fðþ defined on an arbitrary closed interval ½; Š. The idea is first to devise a transformation ¼ ðtþ such that gðtþ :¼ fððtþþ is defined on ½ ; Š. One such commonly used transformation is : ½ ; Š! ½; Š t! ðtþ :¼ t þ þ : ð9þ The inverse transformation of ðtþ is tðþ :¼ Write, for i ¼ 0; ;...;n, i :¼ ðt i Þ :¼ þ t i þ þ ; : ð0þ called the translated zeros in ½; Š of T nþ. The polynomial c n ðþ of degree n such that fð i Þ¼c n ð i Þ for all i is called the Chebyshev interpolating polynomial of degree n for fðþ. The general theory described in Section still applies here, so the error equation (7) holds with i assigned here and for some ½; Š. Theorem 3.. The translated zeros i soft nþ satisfy ma ð i Þ ma ð y i Þ ðþ for any y 0 ;y ;...;y n ½; Š. Proof. This is also well-known. Again, a proof is given for completeness; also, a key equation (3) is useful later. We have i ¼ t i þ ¼ þ ðþ t i ; ð i Þ¼ nþ þ t i ¼ nþ n T nþðtþ; ð3þ where the last equality holds because t i s are all the zeros of T nþ whose leading coefficient is n, t is as defined by (0). Similarly, ð y i Þ¼ nþ q nþðtþ; where q nþ ðtþ is polynomial in t of degree n with leading coefficient. It is known [3, p. 6] that, on ½ ; Š, T nþ ðtþ= n takes the least maimum absolute value among all polynomials of degree n þ with leading coefficient. Therefore, () holds. tu It follows from (7) and (8) that the Chebyshev interpolating polynomial c n ðþ of degree n for fðþ on ½; Š is given by c n ðþ ¼ Xn k T k þ ; ð4þ k¼0 P n nþ k ¼ fð iþ; if k ¼ 0; P n nþ T kðt i Þfð i Þ; otherwise: The proof of Theorem 3. also yields fðþ c n ðþ ¼ fðnþþ ðþ ð i Þ ðn þ Þ! ð5þ ð6þ ¼ nþ f ðnþþ ðþ n ðn þ Þ! T nþðtþ; ð7þ for some ½; Š and t is defined as in (0). Theorem 3.3. Suppose ¼. Then, if f is odd (even), c n is odd (even). Proof. ¼ implies ðtþ ¼t. IffðÞ is odd (even) in, fðtþ is odd (even) in t. Hence, by Theorem 3., c n ðtþ is odd (even) in t and, thus, c n ðþ is odd (even) in. tu T nþ ðtþ has a zero at t ¼ 0 for even n. Thus, if ¼ and n is even, the Chebyshev interpolating polynomial c n ðþ automatically satisfies c n ð0þ ¼fð0Þ. But, it is a different story when n is odd. As we pointed out in Section, we can ensure that fð0þ is always recovered eactly by looking for a Chebyshev interpolating polynomial h n m ðþ of degree n m for f 0 ðþ and then setting ~c n ðþ :¼ fð0þþ m h n m ðþ: Notice that, now, h n m ðþ coincides with f 0 ðþ at the translated zeros ~ i soft n mþ. It is interesting to notice that T n mþ ð0þ ¼0 also for odd n m þ ; this implies ¼ 0 appears m þ times in the interpolating points for fðþ when ¼ ; so, for odd n m þ and ¼, we have

4 LI: NEAR OPTIMALITY OF CHEBYSHEV INTERPOLATION FOR ELEMENTARY FUNCTION COMPUTATIONS 68 fð0þ ¼~c n ð0þ; f 0 ð0þ ¼~c 0 n ð0þ;...;fðmþ ð0þ ¼~c ðmþ n ð0þ: With the arguments similar to those which lead to the error equation (7), we can prove fðþ ~c n ðþ fðþ ½fð0Þþ m h n m ðþš ¼ fðnþþ ðþ m ð ~ i Þ ðn þ Þ! ¼ n mþ n m fðnþþ ðþ ðn þ Þ! m T n mþ ðtþ; ð8þ ð9þ for some ½; Š and t is defined as in (0). It can be seen that Theorem 3.3 holds with c n replaced by ~c n. There is an important implication to this fact and Theorem 3.3. Assume ¼ and n is odd (or even), depending on whether f is odd (or even). For better error bounds, we shall always think of interpolating f (or f 0 ) at the translated zeros of T nþ (or of T n mþ ), in contrast to T nþ (or T n mþ ) for the general f. In general, this will produce a Chebyshev interpolating polynomial of degree n þ, but, since it has to be odd (or even), its leading coefficient must be zero! and thus degenerates to a polynomial of degree n. Theorem 3.4. Assume ¼ and f is odd (even). If n is odd (even), then fðþ c n ðþ ¼ nþ nþ fðnþþ ðþ ðn þ Þ! T nþðtþ; fðþ ~c n ðþ ¼ n mþ n mþ fðnþþ ðþ ðn þ Þ! m T n mþ ðtþ; where ; ½ ; Š and t is defined as in (6). 4 BEST APPROXIMATION ð0þ ðþ Given a continuous function fðþ defined on a closed interval ½; Š, the polynomial approimation b n ðþ of degree n for fðþ is the one such that n :¼ ma jfðþ b nðþj :¼ min ma jfðþ p nðþj; p n where the min is taken over all possible polynomials p n ðþ of degree n. It is known [3], [4, p. 58], [5] that b n ðþ eists and is unique and is characterized by the following theorem, traditionally called the Chebyshev Theorem, although he did not prove it (see [4, p. 58] for a detailed history of it). Theorem 4. (Chebyshev). Assume f is not a polynomial of degree less than or equal to n. Then, there are at least n þ points y 0 <y <...<y nþ at which fðy i Þ b n ðy i Þ¼sð Þ i n for 0 i n þ, where s ¼, independent of i. In other words, fðþ p n ðþ travels from one etreme point to another of the opposite sign for at least n þ times. By the continuity of the functions, we conclude that, between every two consecutive y i s, fðþ b n ðþ has a zero, i.e., there are y 0 <z 0 <...<y i <z i <y iþ <...<z n <y nþ such that fðz i Þ¼b n ðz i Þ for i ¼ 0; ;...;n. So, the polynomial may well be thought of as a Lagrange interpolation polynomial with interpolating points not known a priori and, therefore, fðþ b n ðþ ¼ fðnþþ ðþ ð z i Þ; ðþ ðn þ Þ! for some ½; Š. Such reinterpretation is usually not useful from a numerical point of view, but it is a key step in deriving our main theorems in the net section. For the same reason as eplained in Sections and 3, it may also be advantageous to work with f 0 ðþ for the sake of simple but important mathematical identities. Let h n m ðþ be the polynomial for f 0 ðþ in the sense that ma jf 0ðÞ h n m ðþj ¼ min ma jf 0ðÞ p n m ðþj p n m taken over all possible polynomials p n m ðþ of degree n m, and set ~b n ðþ :¼ fð0þþ m h n m ðþ; which approimates fðþ over the interval. Of course, then ~b n ðþ may do worse than 3 b n ðþ, but the trade off is that now fð0þ ¼ ~ b n ð0þ always! Similarly, we have the following error equation fðþ ~ b n ðþ ¼ fðnþþ ð ~ Þ ðn þ Þ! m ð ~z i Þ; ð3þ for some ~ ½; Š and ~z 0 < ~z <...< ~z n m. Even though, now, bn ~ ðþ is no longer the polynomial for fðþ, we still call it the polynomial at places where no confusion may rise. It can be seen that the uniqueness of the polynomial implies that b n and b ~ n are odd (even) if f is odd (even) and ¼. This implies: Theorem 4.. Assume ¼ and f is odd (even). If n is odd (even), then Consequently, b n ¼ b nþ ; ~ bn ¼ ~ b nþ : fðþ b n ðþ ¼ fðnþþ ðþ Ynþ ð z i Þ; ðn þ Þ! fðþ ~ b n ðþ ¼ fðnþþ ð ~ Þ ðn þ Þ! m n mþ Y ð ~z i Þ; ð4þ ð5þ where z i ;;~z i ; ~ ½ ; Š. Proof. Consider the case where n is odd and f is odd only. The other case can be proven in the same way. All the 3. It may be true the other way. See Remark 5..

5 68 IEEE TRANSACTIONS ON COMPUTERS, VOL. 53, NO. 6, JUNE 004 polynomials of f are odd, so b nþ is odd. But, since n þ is even, a necessary condition for b nþ to be odd is its leading coefficient is zero, i.e., b nþ s degree is no higher than n. Now, the uniqueness of the polynomials implies b n ¼ b nþ. b ~ n ¼ b ~ nþ can be proven in the same way. tu 5 MAIN THEOREMS We shall use BestAppro n ðfþ : approimation polynomial; ChebAppro n ðfþ :Chebyshev interpolating polynomial; of degree n, as building blocks to construct polynomials. Specifically, we will study two sets of approimation polynomials of degree n defined either as b n ðþ :¼ BestAppro n ðfþðþ; c n ðþ :¼ ChebAppro n ðfþðþ; or (when 0 ð; Þ of course) ð5þ ð6þ ~b n ðþ :¼ fð0þþ m BestAppro n m ðf 0 ÞðÞ; ð7þ ~c n ðþ :¼ fð0þþ m ChebAppro n m ðf 0 ÞðÞ; ð8þ which preserve fð0þ eactly. Define absolute approimation errors :¼ ma jfðþ b nðþj; cheb :¼ ma jfðþ c nðþj; and relative approimation errors fðþ b n ðþ :¼ ma fðþ ; fðþ c n ðþ cheb :¼ ma fðþ ; and similarly define errors ~, ~ cheb, ~, and ~ cheb when b n ðþ and c n ðþ are replaced by b ~ n ðþ and ~c n ðþ, respectively. For any function hðþ on ½; Š, define ðhþ :¼ ma jhðþj min jhðþj: Our theorems below are only useful when ðhþ is around or at least not too big for the h of interest. The worst case is when h has a zero in ½; Š and then ðhþ ¼þ. Fortunately, in elementary function computations, the interval ½; Š is kept relatively narrow by choice and h varies little and stays away from zero and, in fact, often ðhþ :, as will be detailed in Section 7. Our theorems show the ratio of cheb over lies in a tiny neighborhood of unless fð0þ ¼0, for which case the ratio of ~ cheb over ~ enjoys the same property. This is significant because, as we mentioned in the introduction, it shows that simple Chebyshev interpolation is sufficient. It is also argued that, in the case of ~ b n or ~c n, good approimations to f 0 ðþ will generally yield good approimations to fðþ. 5. Approimations by b n ðþ and c n ðþ Theorem 5.. Let b n ðþ and c n ðþ be defined by (5) and (6). We have cheb ðf ðnþþ Þ; ðfþ cheb ðfþðf ðnþþ Þ: ð9þ ð30þ Proof. cheb holds since b n ðþ is the among all polynomials of degree n. This proves the leftmost inequality in (9). We now turn to the rightmost inequality in (9). Our proof relies on (3) with translated zeros i soft nþ, (), and Theorem 3.. We have cheb ma jfðnþþ ðþj ðn þ Þ! ma ð i Þ ; min jfðnþþ ðþj ðn þ Þ! ma ð z i Þ ; where the second inequality holds because of ) ma jwðþjjhðþj ðmin jwðþjþjhðþj jwðþjjhðþj min jwðþj ma jhðþj for two arbitrary functions w and h. The rightmost inequality in (9) is a consequence of the above two inequalities and Theorem 3.. The inequalities in (30) are consequences of those in (9) and cheb ma jfðþj cheb cheb min jfðþj ; min jfðþj ma jfðþj : Remark 5.. For (9) to be of any use at all, ðf ðnþþ Þ must not be too big and, for (30) to be useful, both ðf ðnþþ Þ and ðfþ must not be too big, which, if <0 <, ecludes the case fð0þ ¼0. Although cheb always by (9), there is no guarantee that cheb by (30), ecept the two will be close under certain circumstances. For eample, epðþ on ½ ; Š with ¼ 5 and n ¼ 5, cheb 44:453 < 44:447 ; ~ cheb 43:453 < ~ 43:447 : Odd (or even) functions are special. By Theorems 3.4 and 4., along the same lines of the above proof, we have the following theorem: Theorem 5.. Let b n ðþ and c n ðþ be defined by (5) and (6). Suppose ¼, f is even (odd), and n is even (odd). Then, cheb ðf ðnþþ Þ: ut ð3þ

6 LI: NEAR OPTIMALITY OF CHEBYSHEV INTERPOLATION FOR ELEMENTARY FUNCTION COMPUTATIONS 683 For even f, only and fð0þ 6¼ 0, ðfþ cheb ðfþðf ðnþþ Þ: ð3þ 5. Approimations by ~ b n ðþ and ~c n ðþ It is natural to develop some version of Theorem 5. for ~b n ðþ and ~c n ðþ defined by (7) and (8). It is actually very hard to bound one of ~ cheb and ~ by the other or one of ~ cheb and ~ by the other. The reason for this is that, now, approimations to fðþ are obtained indirectly through f 0 ðþ. First, we will still have ma jf 0ðÞ Best n m ðf 0 ÞðÞj ma jf 0ðÞ Cheb n m ðf 0 ÞðÞj; but this does not imply that one of ~ cheb and ~ is always smaller than the other. Second, we also still have ma ð ~ i Þ ma ð ~z i Þ ; but this does not imply that one of ma m ð ~ i Þ and ma m ð ~z i Þ is always smaller than the other, either, where the ~ i s are the translated zeros of T n mþ. Nevertheless, the following analysis indicates that a good approimation to f 0 ðþ would yield a good approimation to fðþ. Let s say that we have an approimation hðþ to f 0 ðþ and hðþ ¼f 0 ðþ½ þ ðþš; where ðþ is the relative error (and is bounded by a tiny number). Then, fð0þþ m hðþ ¼fð0Þþ m f 0 ðþð þ ðþþ ¼ fð0þþ m f 0 ðþþ m f 0 ðþ ðþ ¼ fðþ þ m f 0 ðþ ðþ : fðþ In the case fð0þ ¼0, the induced relative error to fðþ is also ðþ and, thus, an approimation to f 0 ðþ with a smaller relative error ma jðþj leads to one with a smaller relative error for fðþ as well. However, when fð0þ 6¼ 0, a smaller ma jðþj does not necessarily imply a smaller m f 0 ðþ ma ðþ fðþ : But, since m f 0 ðþ ma ðþ fðþ ma m f 0 ðþ fðþ ma jðþj; unless the two sides of the above inequality differ dramatically, we still epect that a small relative error ma jðþj would ultimately indicate a good approimation to fðþ. Set and 0 :¼ ma jf 0ðÞ Best n m ðf 0 ÞðÞj; cheb0 :¼ ma jf 0ðÞ Cheb n m ðf 0 ÞðÞj; fðþ b 0 :¼ ma ~ n ðþ m f 0 ðþ ; fðþ ~c n ðþ cheb0 :¼ ma m f 0 ðþ : Theorem 5.3. Let ~ b n ðþ and ~c n ðþ be defined by (7) and (8). We have cheb0 0 ðf ðnþþ Þ; ðf 0 Þ cheb0 ðf 0 Þðf ðnþþ Þ: 0 If fð0þ ¼0, then (34) becomes ð33þ ð34þ ðf 0 Þ ~ cheb ~ ðf 0 Þðf ðnþþ Þ: ð35þ Proof. Equations (33) and (34) can be proven along the same lines as for Theorem 5., using (5) and (3). For (35), we notice that ~ ¼ 0 and ~ cheb ¼ cheb0 for both the and the Chebyshev polynomials when fð0þ ¼0. tu As the counterpart to Theorem 5., we have: Theorem 5.4. Let b ~ n ðþ and ~c n ðþ be defined by (7) and (8). Suppose ¼, f is odd (even), and n is odd (even). Then, cheb0 0 ðf ðnþþ Þ; ðf 0 Þ cheb0 ðf 0 Þðf ðnþþ Þ: 0 For the case fð0þ ¼0, (37) becomes ð36þ ð37þ ðf 0 Þ ~ cheb ~ ðf 0 Þðf ðnþþ Þ: ð38þ 5.3 Powell s Theorem Using an entirely different technique, Powell [6] showed that cheb n :¼ þ X n ði þ =Þ tan : ð39þ n þ ðn þ Þ Powell s technique is quite elegant and needs no assumption on f being differentiable as ours does. Powell also showed that n grows rather slowly as n grows. In fact, :44 n 4:037 for n 5: Analogously to (30), (39) yields ðfþ cheb n ðfþ: ð40þ

7 684 IEEE TRANSACTIONS ON COMPUTERS, VOL. 53, NO. 6, JUNE 004 Since :7 log ð n Þ:0 for n 5, (40) is translated into differences of no more than to 3 bits in accuracy if ðfþ is nearly ; see Section 6 for more analysis. Similar conclusions on approimations to f 0 ðþ can be derived, but we omit the detail. 6 AN ASYMPTOTIC ANALYSIS We shall now perform an asymptotic analysis to eplain the implications of our main theorems in the contet of elementary function computations, which is the original motivation for our work. In this application, usually ¼ ¼ >0; < is tiny; e:g:; 5 or smaller: ð4þ This is especially true for table-based methods. Notice that log jðrel: err:þj ¼ ðno: of accurate bits in an appro:þ: We are interested in knowing how big the differences, :¼ log cheb log ; 0 :¼ log cheb0 log 0 ; ð4þ ð43þ ~ :¼ log ~ cheb log ~ ; ð44þ can get. As we shall see later, these differences are on the order of OðÞ, in general, and Oð Þ for even or odd functions. It follows from Theorem 5. and Theorem 5.3 that f 0 ð0þ ln fð0þ < < f ðmþþ ð0þ ðm þ Þ ln f ðmþ ð0þ < 0 < f 0 ð0þ ln fð0þ þ f ðnþþ ð0þ ln f ðnþþ ð0þ ; ð48þ f ðmþþ ð0þ ðm þ Þ ln f ðmþ ð0þ þ f ðnþþ ð0þ ln f ðnþþ ð0þ ; ð49þ and, if fð0þ ¼0, f ðmþþ ð0þ ðm þ Þ ln f ðmþ ð0þ < ~ < f ðmþþ ð0þ ðm þ Þ ln f ðmþ ð0þ þ f ðnþþ ð0þ ln f ðnþþ ð0þ : ð50þ In elementary function computations, usually these upper bounds are less than and lower bounds bigger than and, thus, the differences in the numbers of correct bits are under bit! The bounds in (48)-(50) are of order OðÞ. This can be improved to order Oð Þ for odd (or even) function f. To see this, it suffices to show, under appropriate conditions, the s in both Theorems 5. and 5.4 are in a neighborhood of width Oð Þ of. For the purpose of showing this idea, we take Theorem 5.4 as an eample. Under the conditions of Theorem 5.4, we have log ðfþ log ðfþþlog ðf ðnþþ Þ; log ðf 0 Þ 0 log ðf 0 Þþlog ðf ðnþþ Þ; and, if fð0þ ¼0, log ðf 0 Þ~ log ðf 0 Þþlog ðf ðnþþ Þ: ð45þ ð46þ ð47þ What do these inequalities tell us? Suppose fðþ is wellbehaved in ½ ; Š, as are most elementary mathematical functions. Since is very small, we have 4 fðþ ¼fð0Þþf 0 ð0þþoð Þ; f 0 ðþ ¼ fðmþ ð0þ þ fðmþþ ð0þ m! ðm þ Þ! þoð Þ; f ðnþþ ðþ ¼f ðnþþ ð0þþf ðnþþ ð0þþoð Þ; which imply ðfþ þ f0 ð0þ fð0þ þoð Þ; ðf 0 Þþ f ðmþþ ð0þ m þ f ðmþ ð0þ þoð Þ; ðf ðnþþ Þþ fðnþþ ð0þ f ðnþþ ð0þ þoð Þ: Therefore, ignoring the terms of order Oð Þ or higher, we have that 4. For f 0 ðþ, we notice that m is the algebraic multiplicity of ¼ 0 as a zero of fðþ fð0þ. f 0 ðþ ¼ fðmþ ð0þ þ fðmþþ ð0þ m! ðm þ Þ! þoð4 Þ; f ðnþþ ðþ ¼f ðnþþ ð0þþ fðnþ4þ ð0þ þoð 4 Þ;! since, for odd (even) functions, even (odd) derivatives at ¼ 0 are zero. Therefore, f ðmþþ ð0þ ðf 0 Þþ ðm þ Þðm þ Þ f ðmþ ð0þ þoð4 Þ; ðf ðnþþ Þþ fðnþ4þ ð0þ f ðnþþ ð0þ þoð4 Þ: 7 APPLICATIONS TO ELEMENTARY FUNCTION COMPUTATIONS In this section, we shall see the implications of the bounds we developed in the previous sections on the most common elementary functions eponentials, logarithms, and trigonometric functions. Assume (4). 7. Functions a, a for a> Here, fðþ ¼a or a, where a>, and, then, f 0 ðþ ¼ða Þ=. This includes all natural eponential functions as special cases. We have f 0 ðþ ¼a ln a, f ðnþþ ðþ ¼a ðln aþ nþ, and f0 0 ðþ ln a ða Þ ¼a :

8 LI: NEAR OPTIMALITY OF CHEBYSHEV INTERPOLATION FOR ELEMENTARY FUNCTION COMPUTATIONS 685 TABLE Bounds on Differences in Numbers of Accurate Bits to a It can be proven that f 0 ðþ > 0 and f0 0 ðþ > 0, using their Maclaurin series. Therefore, for fðþ ¼a ðfþ ¼a ¼ ðf ðnþþ Þ; ðf 0 Þ¼f 0 ðþ=f 0 ð Þ ¼a : We have, by (45) and (46) log a 4 log a; log a 0 3 log a; ð5þ ð5þ and, if fðþ ¼a for which f 0 ðþ is the same as above, polynomials ~ b n ðþ or ~c n ðþ should be used, by (47) log a ~ 3 log a: ð53þ Table displays 4 log a as an upper bound on the differences in accurate bits by the two approimations as a and ¼ N vary. 7. Function log a ð þ Þ for a> Here, fðþ ¼log a ð þ Þ for a>. This includes the natural logarithm as a special case. Since fð0þ ¼0, f 0 ðþ ¼ log a ð þ Þ= and approimations b ~ n ðþ or ~c n ðþ should be used. We have f 0 ðþ ¼ðþÞ = ln a, and f ðnþþ ðþ ¼ð Þ n n! ln a ð þ Þ n ; f0 0 þ Þ lnð þ Þ ðþ ¼=ð : ln a It can be seen that 5 for jj <, f 0 ðþ > 0 and f0 0 ðþ < 0, using their Maclaurin series. Therefore, ðf 0 Þ¼ f 0ð Þ f 0 ðþ ¼ log að Þ Þ lnð log a ð þ Þ lnð þ Þ ; ðf ðnþþ Þ¼ þ nþ : We have bounds lnð Þ lnð Þ log ~ log lnð þ Þ lnð þ Þ ð54þ þ þðnþþlog : Table displays the last quantity in (54) as an upper bound on the differences in accurate bits by the two approimations as n and ¼ N vary. More than bit differences may occur only for ¼ 5 with n ¼ 0 or higher. Usually, is made much smaller than this. Current HP-UX implementation [] and 5. They are actually true for any >. Intel s implementation [] for log a for a ¼ ;e;0 on Itanium processors [6] uses Function sin Here, fðþ ¼sin. Assume jj <=4or smaller and n is odd. Since fð0þ ¼0, f 0 ðþ ¼sin = and approimations ~b n ðþ or ~c n ðþ should be used. We have f 0 ðþ ¼cos, f ðkþ ðþ ¼ð Þ k sin, f ðkþþ ðþ ¼ð Þ k cos, and f0 0 cos sin ðþ ¼ : It can be seen that for f 0 ðþ > 0, and that f0 0 ðþ > 0 for <0 and f0 0 ðþ < 0 for 0 <, using their Maclaurin series. Therefore, ðf 0 Þ¼ f 0ð0Þ f 0 ðþ ¼ sin ; ðfðnþþ Þ¼ cos : By (47), we have bounds log ~ log sin log sin ðcos Þ: ð55þ The bounds in (55), as well as those in (59) and (60) for cos, are independent of n. Table 3 displays largest bounds :¼ log log sin ðcos Þ; ð56þ :¼ log cos ; ð57þ 3 :¼ log ð cos Þ log cos ð58þ in (55), and (59) and (60) below as upper bounds on the difference in accurate bits by the two approimations as ¼ N varies for both sin and cos. 7.4 Function cos Now, fðþ ¼cos and then f 0 ðþ ¼ðcos Þ=. Assume jj <=4 or smaller, and n is even. We have f 0 ðþ ¼ sin, f ðk Þ ðþ ¼ð Þ k sin, f ðkþ ðþ ¼ð Þ k cos, and f0 0 sin cos þ ðþ ¼ 3 : It can be proven that f 0 ðþ < 0 and f0 0 ðþ 0; so, f 0ðÞ is an increasing function for >0and a decreasing one for <0. Therefore,

9 686 IEEE TRANSACTIONS ON COMPUTERS, VOL. 53, NO. 6, JUNE 004 TABLE Bounds on Differences in Numbers of Accurate Bits to log a TABLE 3 Bounds on Differences in Numbers of Accurate Bits to sin and cos ðfþ ¼ cos ¼ ðfðnþþ Þ; We have, by (45) and (46), log cos log cos ; ðf 0 Þ¼ f 0ð0Þ f 0 ðþ ¼ ð cos Þ : ð59þ log ð cos Þ 0 log ð cos Þ log cos : ð60þ Table 3 displays the upper bounds in (59) and (60) as and Function arcsin Here, fðþ ¼ arcsin for jj = and, then, f 0 ðþ ¼arcsin =. Assume n is odd. The Maclaurin series of fðþ is [7, p. 8] arcsin ¼ þ 3 3 þ þþðk Þ!! ðkþ!! Thus, ma jj jf ðnþþ ðþj ¼ f ðnþþ ðþ and kþ k þ þ: min jj jfðnþþ ðþj ¼ f ðnþþ ð0þ n!! ¼ðnþÞ! ðn þ Þ!! ðn þ Þ ¼ðn!!Þ ; min jj f 0 ðþ ¼f 0 ð0þ ¼, and ma jj f 0 ðþ ¼f 0 ðþ. Therefore, ðf ðnþþ Þ¼ ðn!!þ fðnþþ ðþ; which imply log arcsin ðf 0 Þ¼ f 0ðÞ f 0 ð0þ ¼ arcsin ; ~ log arcsin þ log ðn!!þ fðnþþ ðþ: ð6þ Table 4 displays the last quantity in (6) as an upper bound on the differences in accurate bits by the two approimations as n and ¼ N vary.

10 LI: NEAR OPTIMALITY OF CHEBYSHEV INTERPOLATION FOR ELEMENTARY FUNCTION COMPUTATIONS 687 TABLE 4 Bounds on Differences in Numbers of Accurate Bits to arcsin 8 CONCLUSIONS This paper shows that the Chebyshev interpolating polynomials are just as good as the polynomials in the contet of table-driven methods for elementary function computations and, thus, the common practice of feeding Chebyshev interpolating polynomials into a Remez procedure offers negligible improvement, if any, and thus may be considered unnecessary. Our results improve the earlier results due to Powell [6] which, however, have more general applications. ACKNOWLEDGMENTS The author is grateful to Prof. W. Kahan of the University of California at Berkeley who brought Powell [6] to his attention and to Professor M.J.D. Powell of the University of Cambridge for sending the paper to him. He thanks the referees for their careful reading of the manuscript and constructive suggestions that improved the presentation. Part of this work was done while the author was on leave at Hewlett-Packard Company. He is grateful for help received from Jim Thomas, Jon Okada, and Peter Markstein of the HP Itanium floating-point and elementary math library team at Cupertino, California. This work was supported in part by the US National Science Foundation CAREER award under Grant No. CCR REFERENCES [] J. Harrison, T. Kubaska, S. Story, and P.T.P. Tang, The Computation of Transcendental Functions on the IA-64 Architecture, Intel Technology J., no. Q4, pp. -7, Nov [] R.-C. Li, S. Boldo, and M. Daumas, Theorems on Efficient Argument Reductions, Proc. 6th IEEE Symp. Computer Arithmetic, June 003. [3] P. Markstein, IA-64 and Elementary Functions: Speed and Precision, 000. [4] J.-M. Muller, Elementary Functions: Algorithms and Implementation. Boston, Basel, Berlin: Birkhåuser, 997. [5] K.C. Ng, Argument Reduction for Huge Arguments: Good to the Last Bit, SunPro, technical report, 99, com/. [6] M.J.D. Powell, On the Maimum Errors of Polynomial Approimations Defined by Interpolation and by Least Squares Criteria, The Computer J., vol. 9, no. 4, pp , Feb [7] P.T.P. Tang, Table-Driven Implementation of the Eponential Function in IEEE Floating-Point Arithmetic, ACM Trans. Math. Software, vol. 5, no., pp , June 989. [8] P.T.P. Tang, Table-Driven Implementation of the Logarithm Function in IEEE Floating-Point Arithmetic, ACM Trans. Math. Software, vol. 6, no. 4, pp , Dec [9] P.T.P. Tang, Table Lookup Algorithms for Elementary Functions and Their Error Analysis, Proc. 0th IEEE Symp. Computer Arithmetic, P. Kornerup and D.W. Matula, eds., pp. 3-36, June 99. [0] P.T.P. Tang, Table-Driven Implementation of the epm Function in IEEE Floating-Point Arithmetic, ACM Trans. Math. Software, vol. 8, no., pp. -, June 99. [] E. Remez, Sur un Procédé Convergent d Approimations Successives pour Déterminer les Polynômes d Approimation, C.R. Académie des Sciences, Paris, vol. 98, 934. [] R.-C. Li, P. Markstein, J. Okada, and J. Thomas, The libm Library and Floating-Point Arithmetic in HP-UX for Itanium II, June 000, Itanium/FP_White_Paper_v.pdf. [3] E.W. Cheney, Approimation Theory, second ed. Providence, R.I.: AMS, 98. [4] R.A. DeVore and G.G. Lorentz, Constructive Approimation. Berlin, Heidelberg, New York: Springer-Verlag, 993. [5] G.G. Lorentz, Approimation of Functions, second ed. New York: Chelsea Publishing, 986. [6] INTEL, Intel IA-64 Architecture Software Developer s Manual. Intel Corp., 00, vol. 3: Instruction Set Reference, document No , manuals/inde.htm. [7] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I.A Stegun, eds., ninth ed. New York: Dover Publications, 970. Ren-Cang Li received the BS degree in computational mathematics from Xiamen University, People s Republic of China, in 985, the MS degree, also in computational mathematics, from the Chinese Academy of Science in 988, and the PhD degree in applied mathematics from the University of California at Berkeley in 995. He is currently an associate professor in the Department of Mathematics, University of Kentucky, Leington. He was awarded the 995 Householder Fellowship in Scientific Computing by Oak Ridge National Laboratory, a Friedman memorial prize in Applied Mathematics from the University of California at Berkeley in 996, and a CAREER award from the US National Science Foundation in 999. His research interest includes elementary function computations, floating-point support for scientific computing, large and sparse linear systems, eigenvalue problems, and model reduction, and unconventional schemes for differential equations.