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MATH Mathematical Library Procedures MATH NNAAMMEE math - introduction to mathematical library functions DDEESSCCRRIIPPTTIIOONN These functions constitute the C math library, _l_i_b_m. The link editor searches this library under the "-lm" option. Declarations for these functions may be obtained from the include file <_m_a_t_h._h>. The Fortran math library is described in ``man 3f intro''. LLIISSTT OOFF FFUUNNCCTTIIOONNSS _N_a_m_e _A_p_p_e_a_r_s _o_n _P_a_g_e _D_e_s_c_r_i_p_t_i_o_n _E_r_r_o_r _B_o_u_n_d (_U_L_P_s) acos sin.3m inverse trigonometric function 3 acosh asinh.3m inverse hyperbolic function 3 asin sin.3m inverse trigonometric function 3 asinh asinh.3m inverse hyperbolic function 3 atan sin.3m inverse trigonometric function 1 atanh asinh.3m inverse hyperbolic function 3 atan2 sin.3m inverse trigonometric function 2 cabs hypot.3m complex absolute value 1 cbrt sqrt.3m cube root 1 ceil floor.3m integer no less than 0 copysign ieee.3m copy sign bit 0 cos sin.3m trigonometric function 1 cosh sinh.3m hyperbolic function 3 drem ieee.3m remainder 0 erf erf.3m error function ??? erfc erf.3m complementary error function ??? exp exp.3m exponential 1 expm1 exp.3m exp(x)-1 1 fabs floor.3m absolute value 0 floor floor.3m integer no greater than 0 hypot hypot.3m Euclidean distance 1 infnan infnan.3m signals exceptions j0 j0.3m bessel function ??? j1 j0.3m bessel function ??? jn j0.3m bessel function ??? lgamma lgamma.3m log gamma function; (formerly gamma.3m) log exp.3m natural logarithm 1 logb ieee.3m exponent extraction 0 log10 exp.3m logarithm to base 10 3 log1p exp.3m log(1+x) 1 pow exp.3m exponential x**y 60-500 rint floor.3m round to nearest integer 0 scalb ieee.3m exponent adjustment 0 sin sin.3m trigonometric function 1 sinh sinh.3m hyperbolic function 3 sqrt sqrt.3m square root 1 tan sin.3m trigonometric function 3 tanh sinh.3m hyperbolic function 3 y0 j0.3m bessel function ??? y1 j0.3m bessel function ??? Sprite v1.0 May 27, 1986 1 MATH Mathematical Library Procedures MATH yn j0.3m bessel function ??? NNOOTTEESS In 4.3 BSD, distributed from the University of California in late 1985, most of the foregoing functions come in two ver- sions, one for the double-precision "D" format in the DEC VAX-11 family of computers, another for double-precision arithmetic conforming to the IEEE Standard 754 for Binary Floating-Point Arithmetic. The two versions behave very similarly, as should be expected from programs more accurate and robust than was the norm when UNIX was born. For instance, the programs are accurate to within the numbers of _u_l_ps tabulated above; an _u_l_p is one _Unit in the _Last _Place. And the programs have been cured of anomalies that afflicted the older math library _l_i_b_m in which incidents like the fol- lowing had been reported: sqrt(-1.0) = 0.0 and log(-1.0) = -1.7e38. cos(1.0e-11) > cos(0.0) > 1.0. pow(x,1.0) != x when x = 2.0, 3.0, 4.0, ..., 9.0. pow(-1.0,1.0e10) trapped on Integer Overflow. sqrt(1.0e30) and sqrt(1.0e-30) were very slow. However the two versions do differ in ways that have to be explained, to which end the following notes are provided. DDEECC VVAAXX--1111 DD__ffllooaattiinngg--ppooiinntt:: This is the format for which the original math library _l_i_b_m was developed, and to which this manual is still principally dedicated. It is _t_h_e double-precision format for the PDP-11 and the earlier VAX-11 machines; VAX-11s after 1983 were provided with an optional "G" format closer to the IEEE double-precision format. The earlier DEC MicroVAXs have no D format, only G double-precision. (Why? Why not?) Properties of D_floating-point: Wordsize: 64 bits, 8 bytes. Radix: Binary. Precision: 56 sig. bits, roughly like 17 sig. decimals. If x and x' are consecutive positive D_floating-point numbers (they differ by 1 _u_l_p), then 1.3e-17 < 0.5**56 < (x'-x)/x <_ 0.5**55 < 2.8e-17. Range: Overflow threshold = 2.0**127 = 1.7e38. Underflow threshold = 0.5**128 = 2.9e-39. NOTE: THIS RANGE IS COMPARATIVELY NARROW. Overflow customarily stops computation. Underflow is customarily flushed quietly to zero. CAUTION: It is possible to have x != y and yet x-y = 0 because of underflow. Similarly x > y > 0 cannot prevent either x*y = 0 or y/x = 0 from happening without warning. Sprite v1.0 May 27, 1986 2 MATH Mathematical Library Procedures MATH Zero is represented ambiguously. Although 2**55 different representations of zero are accepted by the hardware, only the obvious representation is ever produced. There is no -0 on a VAX. Infinity is not part of the VAX architecture. Reserved operands: of the 2**55 that the hardware recognizes, only one of them is ever produced. Any floating-point operation upon a reserved operand, even a MOVF or MOVD, customarily stops computation, so they are not much used. Exceptions: Divisions by zero and operations that overflow are invalid operations that customarily stop computa- tion or, in earlier machines, produce reserved operands that will stop computation. Rounding: Every rational operation (+, -, *, /) on a VAX (but not necessarily on a PDP-11), if not an over/underflow nor division by zero, is rounded to within half an _u_l_p, and when the rounding error is exactly half an _u_l_p then rounding is away from 0. Except for its narrow range, D_floating-point is one of the better computer arithmetics designed in the 1960's. Its properties are reflected fairly faithfully in the elementary functions for a VAX distributed in 4.3 BSD. They over/underflow only if their results have to lie out of range or very nearly so, and then they behave much as any rational arithmetic operation that over/underflowed would behave. Similarly, expressions like log(0) and atanh(1) behave like 1/0; and sqrt(-3) and acos(3) behave like 0/0; they all produce reserved operands and/or stop computation! The situation is described in more detail in manual pages. _T_h_i_s _r_e_s_p_o_n_s_e _s_e_e_m_s _e_x_c_e_s_s_i_v_e_l_y _p_u_n_i_t_i_v_e, _s_o _i_t _i_s _d_e_s_t_i_n_e_d _t_o _b_e _r_e_p_l_a_c_e_d _a_t _s_o_m_e _t_i_m_e _i_n _t_h_e _f_o_r_e_- _s_e_e_a_b_l_e _f_u_t_u_r_e _b_y _a _m_o_r_e _f_l_e_x_i_b_l_e _b_u_t _s_t_i_l_l _u_n_i_- _f_o_r_m _s_c_h_e_m_e _b_e_i_n_g _d_e_v_e_l_o_p_e_d _t_o _h_a_n_d_l_e _a_l_l _f_l_o_a_t_i_n_g-_p_o_i_n_t _a_r_i_t_h_m_e_t_i_c _e_x_c_e_p_t_i_o_n_s _n_e_a_t_l_y. _S_e_e _i_n_f_n_a_n(_3_M) _f_o_r _t_h_e _p_r_e_s_e_n_t _s_t_a_t_e _o_f _a_f_f_a_i_r_s. How do the functions in 4.3 BSD's new _l_i_b_m for UNIX compare with their counterparts in DEC's VAX/VMS library? Some of the VMS functions are a little faster, some are a little more accurate, some are more puritanical about exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)), and most occupy much more memory than their counterparts in _l_i_b_m. The VMS codes interpolate in large table to achieve speed and accuracy; the _l_i_b_m codes use tricky formulas compact enough that all of them may some day fit into a ROM. Sprite v1.0 May 27, 1986 3 MATH Mathematical Library Procedures MATH More important, DEC regards the VMS codes as proprietary and guards them zealously against unauthorized use. But the _l_i_b_m codes in 4.3 BSD are intended for the public domain; they may be copied freely provided their provenance is always acknowledged, and provided users assist the authors in their researches by reporting experience with the codes. Therefore no user of UNIX on a machine whose arithmetic resembles VAX D_floating-point need use anything worse than the new _l_i_b_m. IIEEEEEE SSTTAANNDDAARRDD 775544 FFllooaattiinngg--PPooiinntt AArriitthhmmeettiicc:: This standard is on its way to becoming more widely adopted than any other design for computer arithmetic. VLSI chips that conform to some version of that standard have been pro- duced by a host of manufacturers, among them ... Intel i8087, i80287 National Semiconductor 32081 Motorola 68881 Weitek WTL-1032, ... , -1165 Zilog Z8070 Western Electric (AT&T) WE32106. Other implementations range from software, done thoroughly in the Apple Macintosh, through VLSI in the Hewlett-Packard 9000 series, to the ELXSI 6400 running ECL at 3 Megaflops. Several other companies have adopted the formats of IEEE 754 without, alas, adhering to the standard's way of handling rounding and exceptions like over/underflow. The DEC VAX G_floating-point format is very similar to the IEEE 754 Dou- ble format, so similar that the C programs for the IEEE ver- sions of most of the elementary functions listed above could easily be converted to run on a MicroVAX, though nobody has volunteered to do that yet. The codes in 4.3 BSD's _l_i_b_m for machines that conform to IEEE 754 are intended primarily for the National Semi. 32081 and WTL 1164/65. To use these codes with the Intel or Zilog chips, or with the Apple Macintosh or ELXSI 6400, is to forego the use of better codes provided (perhaps freely) by those companies and designed by some of the authors of the codes above. Except for _a_t_a_n, _c_a_b_s, _c_b_r_t, _e_r_f, _e_r_f_c, _h_y_p_o_t, _j_0-_j_n, _l_g_a_m_m_a, _p_o_w and _y_0-_y_n, the Motorola 68881 has all the functions in _l_i_b_m on chip, and faster and more accurate; it, Apple, the i8087, Z8070 and WE32106 all use 64 sig. bits. The main virtue of 4.3 BSD's _l_i_b_m codes is that they are intended for the public domain; they may be copied freely provided their provenance is always acknowledged, and pro- vided users assist the authors in their researches by reporting experience with the codes. Therefore no user of UNIX on a machine that conforms to IEEE 754 need use any- thing worse than the new _l_i_b_m. Properties of IEEE 754 Double-Precision: Wordsize: 64 bits, 8 bytes. Radix: Binary. Precision: 53 sig. bits, roughly like 16 sig. Sprite v1.0 May 27, 1986 4 MATH Mathematical Library Procedures MATH decimals. If x and x' are consecutive positive Double-Precision numbers (they differ by 1 _u_l_p), then 1.1e-16 < 0.5**53 < (x'-x)/x <_ 0.5**52 < 2.3e-16. Range: Overflow threshold = 2.0**1024 = 1.8e308 Underflow threshold = 0.5**1022 = 2.2e-308 Overflow goes by default to a signed Infinity. Underflow is _G_r_a_d_u_a_l, rounding to the nearest integer multiple of 0.5**1074 = 4.9e-324. Zero is represented ambiguously as +0 or -0. Its sign transforms correctly through multiplica- tion or division, and is preserved by addition of zeros with like signs; but x-x yields +0 for every finite x. The only operations that reveal zero's sign are division by zero and copysign(x,+_0). In particular, comparison (x > y, x >_ y, etc.) cannot be affected by the sign of zero; but if finite x = y then Infinity = 1/(x-y) != -1/(y-x) = -Infinity. Infinity is signed. it persists when added to itself or to any finite number. Its sign transforms correctly through multiplication and division, and (finite)/+_Infinity = +_0 (nonzero)/0 = +_Infinity. But Infinity-Infinity, Infinity*0 and Infinity/Infinity are, like 0/0 and sqrt(-3), invalid operations that produce _N_a_N. ... Reserved operands: there are 2**53-2 of them, all called _N_a_N (_Not _a _Number). Some, called Signaling _N_a_Ns, trap any floating-point operation performed upon them; they are used to mark missing or uninitialized values, or nonexistent elements of arrays. The rest are Quiet _N_a_Ns; they are the default results of Invalid Operations, and propagate through subse- quent arithmetic operations. If x != x then x is _N_a_N; every other predicate (x > y, x = y, x < y, ...) is FALSE if _N_a_N is involved. NOTE: Trichotomy is violated by _N_a_N. Besides being FALSE, predicates that entail ordered comparison, rather than mere (in)equality, signal Invalid Operation when _N_a_N is involved. Rounding: Every algebraic operation (+, -, *, /, sqrt) is rounded by default to within half an _u_l_p, and when the rounding error is exactly half an _u_l_p then the rounded value's least significant bit is zero. This kind of rounding is usually the best kind, sometimes provably so; for instance, for every x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find (x/3.0)*3.0 == x and (x/10.0)*10.0 == x and ... Sprite v1.0 May 27, 1986 5 MATH Mathematical Library Procedures MATH despite that both the quotients and the products have been rounded. Only rounding like IEEE 754 can do that. But no single kind of rounding can be proved best for every circumstance, so IEEE 754 provides rounding towards zero or towards +Infin- ity or towards -Infinity at the programmer's option. And the same kinds of rounding are speci- fied for Binary-Decimal Conversions, at least for magnitudes between roughly 1.0e-10 and 1.0e37. Exceptions: IEEE 754 recognizes five kinds of floating-point exceptions, listed below in declining order of probable importance. Exception Default Result __________________________________________ Invalid Operation _N_a_N, or FALSE Overflow +_Infinity Divide by Zero +_Infinity Underflow Gradual Underflow Inexact Rounded value NOTE: An Exception is not an Error unless handled badly. What makes a class of exceptions excep- tional is that no single default response can be satisfactory in every instance. On the other hand, if a default response will serve most instances satisfactorily, the unsatisfactory instances cannot justify aborting computation every time the exception occurs. For each kind of floating-point exception, IEEE 754 provides a Flag that is raised each time its exception is signaled, and stays raised until the program resets it. Programs may also test, save and restore a flag. Thus, IEEE 754 provides three ways by which programs may cope with exceptions for which the default result might be unsatisfactory: 1) Test for a condition that might cause an exception later, and branch to avoid the exception. 2) Test a flag to see whether an exception has occurred since the program last reset its flag. 3) Test a result to see whether it is a value that only an exception could have produced. CAUTION: The only reliable ways to discover whether Underflow has occurred are to test whether products or quotients lie closer to zero than the underflow threshold, or to test the Underflow flag. (Sums and differences cannot underflow in IEEE 754; if x != y then x-y is correct to full precision and cer- tainly nonzero regardless of how tiny it may be.) Sprite v1.0 May 27, 1986 6 MATH Mathematical Library Procedures MATH Products and quotients that underflow gradually can lose accuracy gradually without vanishing, so com- paring them with zero (as one might on a VAX) will not reveal the loss. Fortunately, if a gradually underflowed value is destined to be added to some- thing bigger than the underflow threshold, as is almost always the case, digits lost to gradual underflow will not be missed because they would have been rounded off anyway. So gradual under- flows are usually _p_r_o_v_a_b_l_y ignorable. The same cannot be said of underflows flushed to 0. At the option of an implementor conforming to IEEE 754, other ways to cope with exceptions may be provided: 4) ABORT. This mechanism classifies an exception in advance as an incident to be handled by means trad- itionally associated with error-handling statements like "ON ERROR GO TO ...". Different languages offer different forms of this statement, but most share the following characteristics: - No means is provided to substitute a value for the offending operation's result and resume computation from what may be the middle of an expression. An exceptional result is abandoned. - In a subprogram that lacks an error-handling state- ment, an exception causes the subprogram to abort within whatever program called it, and so on back up the chain of calling subprograms until an error-handling statement is encountered or the whole task is aborted and memory is dumped. 5) STOP. This mechanism, requiring an interactive debugging environment, is more for the programmer than the program. It classifies an exception in advance as a symptom of a programmer's error; the exception suspends execution as near as it can to the offending operation so that the programmer can look around to see how it happened. Quite often the first several exceptions turn out to be quite unexceptionable, so the programmer ought ideally to be able to resume execution after each one as if execution had not been stopped. 6) ... Other ways lie beyond the scope of this docu- ment. The crucial problem for exception handling is the problem of Scope, and the problem's solution is understood, but not enough manpower was available to implement it fully in time Sprite v1.0 May 27, 1986 7 MATH Mathematical Library Procedures MATH to be distributed in 4.3 BSD's _l_i_b_m. Ideally, each elemen- tary function should act as if it were indivisible, or atomic, in the sense that ... i) No exception should be signaled that is not deserved by the data supplied to that function. ii) Any exception signaled should be identified with that function rather than with one of its subroutines. iii) The internal behavior of an atomic function should not be disrupted when a calling program changes from one to another of the five or so ways of handling excep- tions listed above, although the definition of the function may be correlated intentionally with excep- tion handling. Ideally, every programmer should be able _c_o_n_v_e_n_i_e_n_t_l_y to turn a debugged subprogram into one that appears atomic to its users. But simulating all three characteristics of an atomic function is still a tedious affair, entailing hosts of tests and saves-restores; work is under way to ameliorate the inconvenience. Meanwhile, the functions in _l_i_b_m are only approximately atomic. They signal no inappropriate exception except pos- sibly ... Over/Underflow when a result, if properly computed, might have lain barely within range, and Inexact in _c_a_b_s, _c_b_r_t, _h_y_p_o_t, _l_o_g_1_0 and _p_o_w when it happens to be exact, thanks to fortuitous cancellation of errors. Otherwise, ... Invalid Operation is signaled only when any result but _N_a_N would probably be misleading. Overflow is signaled only when the exact result would be finite but beyond the overflow threshold. Divide-by-Zero is signaled only when a function takes exactly infinite values at finite operands. Underflow is signaled only when the exact result would be nonzero but tinier than the underflow threshold. Inexact is signaled only when greater range or precision would be needed to represent the exact result. BBUUGGSS When signals are appropriate, they are emitted by certain operations within the codes, so a subroutine-trace may be Sprite v1.0 May 27, 1986 8 MATH Mathematical Library Procedures MATH needed to identify the function with its signal in case method 5) above is in use. And the codes all take the IEEE 754 defaults for granted; this means that a decision to trap all divisions by zero could disrupt a code that would other- wise get correct results despite division by zero. SSEEEE AALLSSOO An explanation of IEEE 754 and its proposed extension p854 was published in the IEEE magazine MICRO in August 1984 under the title "A Proposed Radix- and Word-length-independent Standard for Floating-point Arith- metic" by W. J. Cody et al. The manuals for Pascal, C and BASIC on the Apple Macintosh document the features of IEEE 754 pretty well. Articles in the IEEE magazine COMPUTER vol. 14 no. 3 (Mar. 1981), and in the ACM SIGNUM Newsletter Special Issue of Oct. 1979, may be helpful although they pertain to superseded drafts of the standard. AAUUTTHHOORR W. Kahan, with the help of Z-S. Alex Liu, Stuart I. McDonald, Dr. Kwok-Choi Ng, Peter Tang. Sprite v1.0 May 27, 1986 9