7 Branch Cuts for Complex Elementary Functions or Much Ado About Nothing's Sign Bit W. KAHAN ABSTRACT Zero has a usab Ie sign bit on some computers, but not on others. This accident of computer arithmetic influences the definition and use of familiar complex elementary functions like V, arctan and arccosh whose domains are the whole complex plane with a slit or two drawn in it. The Principal Values of those functions are defined in terms of the logarithm function from which they inherit discontinuities across the slit (s). These discontinuities are crucial for applications to conformal maps with corners. The behaviour of those functions on their slits can be read off immediately from defining Principal Expressions introduced in this paper for use by analysts. Also introduced herein are programs that implement the functions fairly accu- rately despite roundoff and other numerical exigencies. Except at logarithmic branch points, those functions can all be conti- nuous up to and onto their boundary slits when zero has a sign that behaves as specified by IEEE standards for floating-point arithmetic; but those functions must be discontinuous on one side of each slit when zero is unsigned. Thus does the sign of zero lay down a trail from computer hardware through programming language compilers, run-time support libraries and applications programmers to, finally, mathematical analysts. I. INTRODUCTION Conventions dictate the ways nine familiar multiple-valued complex elementary functions, namely V. ,Q,n, arcsin, arccos, arctan, arcsinh, arccosh, arctanh, Z UJ, shall be represented by single-valued functions called "Principal Values". These single-valued functions are defined and analytic 166 W. KAHAN BRANCH CUTS 167 throughout the complex plane except for discontinuities across The IEEE standards 754 and p854 take a different approach. certain straight lines called "slits" so situated as to maximize They prescribe representations for both +0 and -0 but do not the reign of continuity, conserving as many as possible of the distinguish between them during ordinary arithmetic operations, properties of these functions' familiar real restrictions to apt so the ambiguity is benign. Rather than think of + 0 and -0 as segments of the real axis. There can be no dispute about where distinct numerical values, think of their sign bit as an auxili- to put the slits; their locations are deducible. However, Princi- ary variable that conveys one bit of information (or misinforma- pal Values have too often been left ambiguous on the slits, tion) about any numerical variable that takes on 0 as its value. causing confusion and controversy insofar as computer programmers Usually this information is irrelevant; the value of 3 + x is no have had to agree upon their definitions. This paper's thesis different for x:= +0 than for x := -0, and the same goes for is that most of that ambiguity can and should be resolved; how- the functions signum(x) and sign(y,x) mentioned above. How- ever, on computers that conform to the IEEE standards 754 and ever, a few extraordinary arithmetic operations are affected by p854 for floating-point arithemetic the ambiguity should not be zero I s sign; for example 1/ (+0) +ca but 1/(-0) = -co .. To re- eliminated entirely because, paradoxically, what is left of it tain its usefulness, the sign bit must propagate through certain usually makes programs work better. arithmetic operations according to rules derived from continuity What has to be ambiguous is the sign of zero. In the past, considerations; for instance (-3)(+0):= -0, (-0)/(-5) =+0, most people and computers would assign no sign to zero except (-0) - (+0) = -0, etc. These rules are specified in the IEEE under auress, and then they would treat the sign as + rather standards along with the one rule that had to be chosen arbitra- than For example, the real function rily: x> signum(x) := +1 if 0 S -8 := +0 for every string s representing a finite real number. := 0 if x o Consequently when t=s, but O¢t¢co, then s-t and t-8 -I if x < 0 both produce + 0 instead of opposite signs. (That is why, in illustrates the traditional noncommittal attitude toward zero's IEEE style arithmetic, s - t and - (t-s) are numerically equal sign, whereas the Fortran function but not necessarily indistinguishable.) Implementations of ele- sign(l.O, := + 1.0 if x;;. 0 mentary transcendental functions like sin(z) and tan(z) and := -1.0 if x < 0 , their inverses and hyperbolic analogs, though not specified by must behave as if zero had a + sign in order that this function the IEEE standards, are expected to follow similar rules; if and its first argument have the same magnitude. Just as [(0) = 0 < 1 '(0), then the implementation of I(z) is expected sign ( I .0, x) is continuous at x 0+, i.e. as x approaches zero to reproduce the sign of z as well as its value at z = ±O .. from the right, so can each principal value above be continuous That does happen in several libraries of elementary transcenden- as its slit is reached from one side but not from the other. tal functions; for instance, it happens on the Motorola 68881 Sides can be chosen in a consistent way among all the elementary Floating-Point Coprocessor, on Apple computers in their Standard complex functions, as they have been chosen for the implementa- Apple Numerical Environment, in Intel's Common Elementary Function tions built into the Hewlett-Packard hp-15C calculator that will Libraries for the i8087 and i80287 floating-point coprocessors, be used to illustrate this approach. in analogous libraries now supplied with the Sun III, with the must be invariant under reflection in the origin z = O. Finally, ELXSI 6400 and with the IBM PC/RT, and in the C Math Library the slit(s) must begin and end at branch-points: these are currently distributed with 4.3 BSD UNIX for machines that con- singularities around which some branch of the function cannot be form to IEEE 754. With a few unintentional exceptions, it represented by a Taylor nor Laurent series expansion. A slit can happens also on the hp-71R,hand-held computer, whose arithmetic end at a branch point at infinity. was designed to conform to IEEE p854. Consequently the slit for v', £n and ZW turns out to be If a programmer does not find these rules helpful, or if the negative real axis. Then the slits for arcsin, arccos and he does not know about them, he can ignore them and, as has been arctanh turn out to be those parts of the real axis not between necessary in the past, insert explicit tests for zero in his pro- -I and + I; similarly those parts of the imaginary axis not be- gram wherever he must cope with a discontinuity at zero. On the tween -i and +i serve as slits for arctan and arcsinh. The other hand, if the standards' rules happen to produce the desired slit for arccosh, the only slit with a finite branch-point (-\) results without such tests, the tests may be omitted leaving the inside it, must be drawn along the real axis where z ~ +I. None programs simpler in appearance though perhaps more subtle. This of this is controversial, although a few other writers have at is just what happens to programs that implement or use the times drawn the slits elsewhere either for a special purpose or elementary functions named above, as will become evident below. by mistake; other tastes can be accommodated by substitutions sometimes so simple as writing, say, £n(-l)-£n(-I/z) in place 2. WHERE TO PUT THE SLITS of £n(z) to draw its slit along (and just under) the positive Each of our nine elementary complex functions f(z) has a real axis instead of the negative real axis. slit or slits that bound a region, called the principal domain, inside which f(z) has a principal value that is single valued 3. WHY DO SL ITSMATTER ? and analytic (representable locally by power series), though it A computer program that includes complex arithmetic opera- must be discontinuous across the slit(s). That principal value tions must be a product of a deductive process. One stage in is an extension, with maximal principal domain, of a real elemen- that process might have been a model formulated in terms of ana- tary function f(x) analytic at every interior point of its lytic expressions that constrain physically meaningful variables domain, which is a segment of the real x-axis. To conserve the without telling explicitly how to compute them. From those power series' validity, points strictly inside that segment must expressions somebody had to deduce other complex analytic expres- also lie strictly inside the principal domain; therefore the sions that the computer will evaluate to solve the given physical slit(s) cannot intersect the segment's interior. Let z* =x-iy problem. The deductive process entails transformations among denote the complex conjugate of z =x + iy; the power series for which some may resemble algebraic manipulations of real expres- f(x) satisfy the identity f(z*) =f(z)* within some complex sions, but with a crucial difference: neighbourhood of the segment's interior, so the identity should Certain transformations, generally val id for real persevere throughout the principal domain' s interior too. expressions, are valid for complex expressions only while their variables remain within suitable regions Consequently complex conjugation must map the slit(s) to itself/ in the complex plane. themselves. The slit(s) of an odd function f(z) == - f(-z) Moreover, those regions of validity can depend disconcertingly In general, singularities can be upon the computer that will be used to evaluate the expres- regions of convergence, etc. the singularities are obviously hard to find; in our examples sions in question. For example, simplifying the expression the pole at 2'" I, the branch-point 2 = 0, and respective slits ";(21 (4 I») ,/( II (4 - I») to ";(4) I (4 I) seems legitimate in so 0< 2 < I, 2 < I and 2 < 0 whereon the quantities under square far as they both describe the same complex function, one that is root signs are negative real. continuous everywhere except for a pole at 4 = I and a jump- Taken together, the singularities partition the complex discontinuity along the negative real axis 2 < O. And when those 2. plane into a collection of disjoint connected components. two expressions are evaluated upon a variety of computers includ- Inside each such component locate a sma?? aontinuum upon which the equivalence of the given two expressions ing the ELXSI 6400, the Sun III, the IBM PC/RT, the IBM pclAT, can be decided; that decision is valid throughout the PCIXT and PC using i80287 or" i8087, and the hp-71B, they component's interior. agree everywhere within a rounding error or two. But when the The "small continuum" might be a small disk inside which same expressions are evaluated upon a different collection of both expressions are represented by the same Taylor series; or computers including CRAYs, the IBM 370 family, the DEC VAX it could be a curvilinear arc within which both expressions take line, and the hp-15C, those expressions take opposite signs values that can be proved equal by the laws of real algebra. along the negative real axis! An experience like this could Other possibilities exist; some will be suggested by whatever undermine one's faith in some computers. motivated the attempt to prove that the given expressions are What deserves to be undermined is blind faith in the power equivalent. In our example, the two expressions are easily of Algebra. We should not believe that the equivalence class of on that part of the real axis where 2 > 1 , which proven equal expressions that all describe the same complex analytic function happens to lie inside the one connected component into which the can be recognized by algebraic means alone, not even if rela- slits along the rest of the real axis divide the complex plane. tively uncomplicated expressions are the only ones considered. Therefore the two expressions must be equivalent everywhere in To locate the domain upon which two analytic expressions take the complex plane except possibly for real 2 < 1 • equal values generally requires a combination of algebraic, The singularities cons titute loci in the plane upon 3. which the processes in steps 1 and 2 above can be analytical and topological techniques. The paradigm is familiar repeated, finally leaving isolated singular points to to complex analysts, but it will be summarized here for the sake be handled individually. End of paradigm. of other readers, using the two expressions given above for In our example, the slit along 2 < 1 is partitioned into concrete illustration. two connected components by the branch-point at Z = O. Each Whether the two expres- How to decide where two analytic expressions describe component has to be handled separately. the same function. sions are equivalent on a component must depend upon the defini- I. Locate the singularities of each constituent subexpression tion of complex ";2 on its slit where Z < 0; there diverse of the given expressions. That is what this paper is about. computers appear to disagree. The singularities of an analytic function are the boundary More generally, programmers who compose complex analytic points of its domain of analyticity. These will consist of poles, expressions out of the nine elementary functions listed at this branch-points and slits in this paper; but more generally they paper's beginning will have to verify whether their expressions would include certain contours of integration, boundaries of deliver the functions that they intend to compute. In principle, 4. PRINCIPAL VALUES ON THE SLITS, IEEE STYLE Since all the slits in question lie on either the real or that verification could proceed without prior agreements about the imaginary axis, every point z on a slit is represented in the functions' values on their slits if instead analysts and pro- at least two ways, at least once with a +0 and at least once grammers were obliged to supply an explicit expression to handle with a -0 for whichever of the real and imaginary parts of z every boundary situation as they intend. Such a policy seems vanishes. Benignly, ambiguity in z at a discontinuity of f(z) inconsiderate (not to say unconscionable) considering how hard permits fez) to be defined formally continuously, except possi- some singu~arities are to find, and how easy to overlook; but bly at the ends of some slits, by continuation from inside the that policy is not entirely heartless since verifying correctness principal domain. This continuity goes beyond mere formality. along a boundary costs the intellect nearly as much as writing By analytic continuation, the domain of each of our nine elemen- down a statement of intent about that boundary. The trouble with those statements is that they generally contain inequalities and tary functions fez) extends until it fills out a Riemann Surface; think of this surface as a multiple covering wrapped tests and diverse cases, and as they accumulate they burden like a bandage around the Riemann Sphere and mapped onto it proofs and programs with a dangerously enlarged capture cross- section for errors. And almost all of those statements become continuously by f. To construct f's principal domain, cut the bandage along the slit(s) and discard all but one layer covering superfluous in programs after we agree upon reasonable defini- the sphere. That layer is a closed surface mapped by f con- tions for the functions in question on their slits. tinuously onto a subset of the sphere. The shadow of that layer For instance, in our exampIe above we had to discover whether the two expressions agreed on an interval 0 < z < 1 that projected down upon the sphere is the principal domain; it con- lies strictly inside the dom~in of the desired function's analy- sists of the whole sphere, but with slit(s) covered twice. That ticity, not on its boundary. That interval turns out to be a is why we wish to represent slits ambiguously. removable singularity, and it does remove itself from all the Here are some illustrative examples, the first of a real computers mentioned above because they evaluate both expressions function that is recommended for any implementation of IEEE correctly on that interval; diverse computers disagree only on standard 754 or p854. the boundary where the desired function is discontinuous. Per- copysign(x, y) := ± x where the sign bit is that of y, so } copysign(1,+O) =+1 = lim copysign(l, y) at y= 0+, and (4.1) haps that's just luck. (Unlucky examples do exist and one will be presented later.) Let us accept good luck with gratitude copysign(l,-O) -I = lim copysign(l, y) at y = 0-. whenever it simplifies our programs. v(-I+iO) =+O+i=lim,/(-I +iy) at y= 0+; } (4.2) Complex analytic expressions that involve slits and other v(-I-iO) =+0 -i =lim,j(-I +iy) at y =c- . singularities are intrinsically complicated, and they get more Consequently, v(z*) =v(z)* for every z, and v(llz) = I/v(z) too. compl icated when rounding errors are taken into account. Our These identities persist within roundoff provided the programs objective cannot be to make complicated things simple but rather, used for square root and reciprocal are those, supplied in this by choosing reasonable values for our nine elementary functions paper, that would have been chosen anyway for their efficiency on their slits, to make them no worse than necessary. and accuracy. 174 W. KAHAN BRANCH CUTS 175 arccos (2 + iO) == +0 - i arccosh(2) directly, not by first coercing the real to complex, lest the = lim arccos(2 + iy) sign of zero be rendered uninformative; the same goes for com- at y 0+;) (4.3) binations of pure imaginary quantities with complex variables. arccos(2 - iO) == +0 + i arccosh(2) And doing arithmetic directly this way saves execution time that lim arccos(2+iy) at y == 0- • would otherwise be squandered manipulating zeros.) When z is An implementation of arccos that preserves full accuracy in the near ±I the expression a(z) nearly vanishes and loses its rela- I imaginary part of arccos (2 + i y) when Iy is very tiny can be tive accuracy to roundoff. Although this loss could be avoided expected to get its sign right when y = ±0 too without extra by rewriting a (z) := v' (z - I) (z + I»), doing so would obscure tests in the code; such a program is supplied later in this the discontinuity on the imaginary axis in a cloud of roundoff paper. which obliterates Re (z) whenever it is very tiny compared wi th But the foregoing examples make it all seem too simple. 1 as well as when it is ±O. The next example presents a more balanced picture. Also obscure is what happens at the ends of some slits. Let function a(x) := V(x2 -\) for real x with x 2 ;> I, ie, Take for example £n(z)=£n(p)+ where p= Izi and e=arg(z) and let b(x):= a(x) for real x;> \ j note that b(x) is not are the polar coordinates of z = x + iy and satisfy yet defined when x ~ \. The principal values of the complex x =p cos e, y == p sin e, p;> 0 and -11" ~ e ~ 1f extensions of a and b following the principles enunciated Evidently p:= +v(x2 + y2), and when 0 < p < +co then above turn out to be e := 2 arctan (y/(p+x») if x;> 0, or a(z) = v'(z2- I) = a(-z}, and :=2arctan(p-x)/y) if x~O. b(z) v'(z-I)v(z+ I) =-b(-z) . At the end of the slit where z = x = y p == 0 (and tn = -co) Both a and b are defined throughout the complex plane and both the value of e may seem arbitrary, but in fact it must cohere have a slit on the real axis running from I to + I, but a has with other almost arbitrary choices concerning division by zero another slit that runs along the entire imaginary axis separat- and arithmetic with infinity. A reasonable choice is to inter- ing the right half-plane where a b from the left half-plane pose the reassignment where a=-b. The functions are different because generally if p == 0 then x := copysign (\, x) v'(~) v'(n) v'(~ n) when largO;) + arg(n) I < 1f between the computations of p and e above. More about that v(~ n) when larg(O + arg(n) I> 11" later. =± v'( ~ n) (hard to say which) when i; n ~ O. The foregoing examples provide an unsettling glimpse of Both functions a and b are continuous up to and onto ambiguous the complexities that have daunted implementers of compilers and boundary points in IEEE style arithmetic, as described above, run-time libraries who would otherwise extend to complex arith- only if that arithmetic is implemented carefully; in particular, metic the facilities they have supplied for real floating-point the expression z + I should not be replaced by the ostensibly computation. These complexities are attributable to failures, equivalent z + (I + iO) lest the sign of zero in the imaginary in complex floating-point arithmetic, of familiar relationships part of z be reversed wrongly. (Generally, mixed-roode arith- like algebraic identities that we have come to take for granted metic combining real and complex variables should be performed in the arena of real variables. Three classes of failures can be discerned: investigation of an analytic function f(z) whose domain is slit (i) The domain of an analytic expression can enclose singu- along some segment of the real or imaginary axis; say the real larities that have no counterparts inside the domain of its real restriction. That is why. for example. axis. When 2 = X + iy crosses the slit, f(z) jumps discontin- '/(z2 - I) '# '/(2 I) -Ie2 + I) • uously as y reverses sign although f(z) is continuous as z (ii) Rounding errors can obscure the singularities. That is 2 approaches one side of the slit or the other. Consequently the why, for example, -1(2 --,1) =-1(2 1)(z+I») fails so badly when either z2 = I very nearly or when z2 < 0 two limits very nearly. To avoid this problem, the programmer may f(x+iO):= limf(x+iy) as y-+O+ and have to decompose complex arithmetic expressions into separate computations of real and imaginary parts. f(x - iO):= limf(x +iy) as y -+ 0- thereby forgoing Some of the advantages of a compact notation. both exist, but they are different when x has a real value inside the slit. Ideally. a subroutine P(z) programmed to (iii) Careless handling can turn infinity or the sign of zero into misinformation that subsequently disappears leaving compute f(z) should match these values; P(x±iO)=f(x±iO) behind only a plausible but incorrect result. That is respectively should be satisfied within a small tolerance for why compilers must not transform z - I into z - (I + iO) • as we have seen above, nor - (-x x 2) into x +x2 • as roundoff. This normally happens in IEEE style arithmetic as a we shall see below, lest a subsequent logarithm or square by-product of whatever steps have been taken to ensure that root produce a nonzero imaginary part whose sign is oppo- site to what was intended. P(x +iy) = f(x + iy), within a similarly small tolerance, for The first two classes are hazards to all kinds of arith- all sufficiently small but nonzero Iyl. To generate a discon- metic; only the third kind of failure is peculiar to IEEE style tinuity, the subroutine P must contain constructions similar to arithmetic with its signed zero. Yet all three kinds, must be copysign ( ••• ,y) or arctan(1/y) possibly with "y" replaced by linked together esoterically because the third kind is not some other expression that either vanishes or tends to infinity usually found in an applications program unless that program as y -+ O. That expression cannot normally be a sum or difference suffers also from the second kind. The link is fragile, easily like arctan(y-l) +'IT/4 or exp(y) 1 that vanishes by cancella- broken if the rational operations or elementary functions, from tion, because roundoff can give such expressions values (typic- which applications programs are composed. contain either of the ally 0) that have the wrong sign when iyl is tiny enough. last two kinds of failures. Therefore. implementers of compilers Instead, to preserve accuracy when Iyl is tiny, that expression and run-time libraries bear a heavy burden of attention to detail must normally be a real product or quotient involving a power of if applications programmers are to realize the full benefit of the y or sin (y) or some other bui It-in function that vanishes with IEEE style of complex arithmetic. That benefit deserves Some y and therefore should inherit its sign at y = ± O. Thus does discussion here if only to reassure implementers that their careful implementation of compiler and library combine with care- assiduity will be appreciated. ful applications programming to yield correct behaviour on and The first benefit that users of IEEE style complex arith- near the slit. And if two such carefully programmed subroutines metic notice is that familiar identities tend to be preserved P(z), though based upon different formulas, agree within roundoff more often than when other styles of arithmetic are used. everywhere near the slit, then the foregoing reasoning implies The mechanism that preserves identities can be revealed by an that normally they have to agree on the slit too; this is the way IEEE style arithmetic preserves identities like -I(z*) = (-12)* andv'(IIZ)=lh/z that would have to fail on slits if zero lower free boundary of the jet, then from left to right along no sign. the jet's,upper free boundary and back along the upper wall. Of course, applications programmers generally have things This is just what happens when fez) is plotted from a one-line more important than the preservation of identi ties on their program on the hp-71 B calculator, which imp lements the proposed minds. Figure I shows a more typical and realistic example. IEEE standard p854. But when fez) is programmed onto the hp- Here I(z) :=I+z2+zv'(I+z2)+R-n(z2+ zv'(I+z2)), and we con- 15C, whose zero is unsigned, the lower wall disappears. Its pre- strue the, equation 1;;:== fez) as a conformal map, from the plane image, the lower part of the imaginary axis where ali < -I, is of z = x + iy to the plane of I;; =I;; + in, that maps the right half- mapped during the computation of fez) into the slit that belongs plane x;;' + 0 onto the space occupied by a liquid that is forced to V and R-n; the upper part zli > 1 gets mapped onto the same by high preSSure to Jet into a slot. The walls of the slot, slit. For lack of a signed zero, that slit gets attached to a side where I;; < 0 and n== ±'TT, should be the images of those parts of that is right for the upper wan but wrong for the lower wall, the imaginary axi s z2 < - 1 lying beyond ± i. The free surfaces thereby throwing the pre-image of the lower wall away into a tiny of the jet, curving forward from I;; = ±i'TT and then back to segment of the upper wall. To put the lower wall back, x must be <: = - 00 ±i'TT/2, should be the image of that segment of the increased from 0 to a tiny positive value while y runs from-oo imaginary axis -1 < < 0 between ±i. to-I. (How tiny should x be? That's a nontrivial question.) I The misbehaviour revealed in the foregoing example fez) liquid flow ) may appear to be deserved because fez) has slits on the imagin- ----Wall in .i, ary axis z2 < -I beyond ± i. Should mapping a slit to the wrong air ~ I --------------------;'" "" place be blamed upon the discontinuity there rather than upon II I liquid arithmetic with an unsigned zero? No. Arithmetic with an un- I signed zero can also cause other programs to misbehave similarly - liquid jet _ - liquid flow o~ +-­ I at places where the functions being implemented are otherwise ----------------------, I liquid well behaved. For example consider a(z) := z -iv'(iz + 1 )V(iz - I), I . "­ whose slit lies in the imaginary axis l<z2<0 between ±i. Now Wail air 'I 1-' '-in J air I;; := a(z) maps the slit z plane onto the I;; plane outside the • liquid flow circle 1<:1;;. I; vertical lines in the z plane map to stream ';-plane z-planc lines in the vertical flow of a fluid around the circle. Imple- FIG. 1 Conformal map I;; ;=f(z) of half-plane menting a(z), the programmer notices that he can reduce two to jet with free boundary expensive square roots to one by rewriting The picture of fez) should be symmetrical about the real a(z) := z + v(z:i.+ I) copysign(l, Re(z»). *. axis because f(z*) = fez) As z runs up the imaginary axis, The two expressions for a(z) match everywhere in IEEE style wi th x == +0 and y running from -GO through -I toward -0 and arithmetic; but when zero has only one sign, say +, the second then from + 0 through + I toward +00, its image I;; = fez) should expression maps the lower part of the imaginary axis, where run from left to right along the lower wall and back along the zli < -I, into the inside instead of the outside of the circle, although a(z) should be continuous there. dispatched properly; and then familiar identities will persist, The ease with which IEEE style arithmetic handled the despite roundoff, more often than in other styles of arithmetic. important singularities near z = ±i in the examples above should This thought would comfort us more if the aberrations were not be allowed to persuade the reader that all singularities can easier to uncover. Locating potential aberrations remains an be dispatched so easily. The singularities f(O) and f(oo) and onerous task for an applications programmer, regardless of the the overflows near z = would have to be handled in the usual 00 style of arithmetic; however that style can affect the locus of ways if they did not lie so far off the left-hand side of the aberration fundamentally. In IEEE style arithmetic, a program- picture that nobody cares. Another kind of singularity that did med implementation of a complex analytic function can take not matter here, but might matter elsewhere, insinuated weasel aberrant boundary values, different from what would be produced words like "not usually", "tends to be" and "normally" into the by continuation from the interior, because of roundoff or similar earlier discussion of Sums and differences that normally vanish phenomena. In arithmetic without a signed zero, such an aberra- by cancellation. Sums and differences can vanish without cancel~ . tion can be caused as well by an unfortunate choice of analytic lation if they combine terms that have already vanished; an expression, though the programmer has implemented it faithfully. 2 example is h(x) ;=x+x when x=O. Evaluating h(±O) in IEEE The fact that an analytic expression determines the values of an style real arithemetic yields +0 instead of ±0 respectively, analytic function correctly inside its domain is no reason to losing the sign of zero. h(x) has other troubles: it signals expect the boundary values to be determined correctly too when Underflow when x is very tiny, suffers inaccuracy when x is zero is unsigned. very near -I, and becomes Invalid at x=-oo. Simply rewriting h(x) :=x(l +x) dispels all these troubles, but is slightly less 5. PRINCIPAL VALUES ON THE SLITS, hp-J5C STYLE accurate for very tiny Ixl than is h(x) :=-(-x - x 2). which Of course, the hp-15C is not the only machine with an preserves accuracy and the sign of zero for all tiny real x. unsigned zero; a DEC VAX 1 J model is similar but lacks so far a Complex arithmetic complicates this situation. Both expressions careful software implementation of some of the functions under z+z2 and z(l+z) produce zeros with the wrong sign for Im(h(z») discussion - in time that lack will be remedied. Many other on various segments of the real z-axis; to get the correct sign machines, the IBM 370 series among them, have a signed zero in and better accuracy requires an expression like their hardware but no provision for propagating its sign in a h(x+iy) := x(l+x)_y2+ 2iy(x+0.5) coherent and useful way, so they are customarily programmed as regardless of arithmetic style. For similar reasons. the if zero were unsigned. All these machines discourage attempts expression for f(z) used above for the conformal map would to distinguish one side of a slit from the other on the slit have to be rewritten if the interesting part of its domain were itself• the left instead of right half-plane. What we have to do is attach each slit to one of its sides IEEE style complex arithmetic appears to burden the imp le- in accordance with some reasonable rule, thereby obtaining a menters of compilers and run-time libraries with a host of principal value which is continuous up to the slit from that complicated details that need rarely bother the user if they are side but not from the other. In other words, we have to assign a sign to zero on each slit and then compute the same principal value as would have been computed using IEEE-style arithmetic. a.rcsin(z) 'Ir/2 (power series in l-z)y( I-z) for z near I, The assignment cannot be arbitrary; for instance we cannot arccosh(z~ = 9,n(2z) - (power series in J /z) when z I is huge, change sides in the middle of the slit lest a gratuitous singu- arctanh(z) =-0.5 9,n(l-z) + (power series in I-z) for z near I. larity be insinuated by the change. On the other hand, some degree of arbitrariness is obligatory. For instance, the two In each case the power series is determined uniquely. In general, functi~ns if S is a finite branch-point at the end of a slit belonging to one of our nine functions I(z), and if the function is analytic b(z) := Y(z-I) y(z+ I) and -b(-z) are indistinguishable everywhere except in the slit _I < z < 1 inside some circular disk Iz- 61 < p except on the slit, then I(z) can be represented inside that slit disk by one of the for- across which they are discontinuous, but in hp-15C style arith- metic one function must be continuous onto the top of the slit mulas I(z) = P(z-S)+p(z-S)y«z-6)/e) , or and the other onto the bottom. Evidently no general rule attach- I(z) P(z-S)+p(z-S)R.n «z-6)/e) , or ing a slit to one of its sides can depend solely upon the slit's shape nOr solely upon the function's values off the slit. And I(z) = P (some nonintegral power of y((z - 6) /e») , where e=lim(S-z)/!S-z! as z-+6 along the slit, so 101=1 yet, paradoxically, the hp-15C appears to follow just such a rule, namely and (z-S)/e < 0 in the slit, and P(t) and p(t) are represent- Counter>-CloakUJiBe Continuity (CCC) : able by power series around t= O. Given 13 and 1 and its slit, o and P and pare oanonioal (determined uniquely). Formulas Attach each slit to whichever side is approached when the finite branch-point at its end is circled counter- slightly more general than these, but still essentially unique, clockwise. cope with more general elementary functions or with isolated Thus when z is real and negative CCC defines YZ =iY!z! branch-points at I I 00. and R.n (z) = R.n z + in. Actually CCC is merely a mnemonic summary The dominant terms of these canonical formulas provide of the implications, for the nine functions that are the subject approximations useful near branch-points, and are therefore pre- of this note, of the following more general convention applica- cious to analysts and programmers who have to exploit or compen- ble also to b(z) above, as CCC is not. sate for singularities, so these formulas should not be violated The Pr>ineipal Expr>eBBion: unnecessarily on the slits. Programs that handle singularities Assign to each elementary function in question not are complicated enough without the additional burden of treating merely a Principal Value but also a ~ineipal Expr>eB­ Bion in terms of 9;n(z) and YZ, using the simplest specially those slits that need no special care so long as pro- formula that manifests its behaviour at finite grams remain as valid on the slits as off them near their ends. branch-points without gratuitous singularities elsewhere. Then programmers can predict from Principal Expressions how their What makes this convention effective is a canonical asso- programs will behave on slits. The Principal Expressions for all ciation between the archetypal branch-points of 9,n(z) and YZ nine of our elementary functions are determined by convention on the one hand, and on the other any isolated branch-point at and tabulated nearby. For other functions the choice of Principal the end of a slit belonging to any other elementary function. Expression is forced by the choice of slits except when a slit For example, contains just two singularities, both finite branch points at its