NAMEmath - introduction to mathematical library functionsDESCRIPTIONThese functions constitute the C math library,libm. This library is based Sun Microsystems FDLIBM (Freely Distributable LIBM), a C math library for machines that support IEEE 754 floating-point arithmetic. Declarations for the functions in this library may be obtained from the include file <math.h>.LISTOFFUNCTIONSNameAppearsonPageDescriptionacossin(3)inverse trigonometric function acoshasinh(3)inverse hyperbolic function asinsin(3)inverse trigonometric function asinhasinh(3)inverse hyperbolic function atansin(3)inverse trigonometric function atanhasinh(3)inverse hyperbolic function atan2sin(3)inverse trigonometric function cbrtsqrt(3)cube root ceilfloor(3)integer no less than copysignieee(3)copy sign bit cossin(3)trigonometric function coshsinh(3)hyperbolic function remainderieee(3)remainder erferf(3)error function erfcerf(3)complementary error function expexp(3)exponential expm1exp(3)exp(x)-1 fabsfloor(3)absolute value floorfloor(3)integer no greater than hypothypot(3)Euclidean distance ilogbieee(3)exponent extraction j0j0(3)bessel function j1j0(3)bessel function jnj0(3)bessel function logexp(3)natural logarithm logbieee(3)exponent extraction log10exp(3)logarithm to base 10 log1pexp(3)log(1+x) powexp(3)exponential x**y rintfloor(3)round to nearest integer scalbieee(3)exponent adjustment scalbnieee(3)exponent adjustment sinsin(3)trigonometric function sinhsinh(3)hyperbolic function sqrtsqrt(3)square root tansin(3)trigonometric function tanhsinh(3)hyperbolic function y0j0(3)bessel function y1j0(3)bessel function ynj0(3)bessel functionNOTESFromtheFDLIBMREADMEFDLIBM (double precision version) assumes: a. IEEE 754 style (if not precise compliance) arithmetic; b. 32 bit 2's complement integer arithmetic; c. Each double precision floating-point number must be in IEEE 754 double format, and that each number can be retrieved as two 32-bit integers; Example: let y = 2.0 double fp number y: 2.0 IEEE double format: 0x4000000000000000 Referencing y as two integers: *(int*)&y,*(1+(int*)&y) = {0x40000000,0x0} (on sparc) {0x0,0x40000000} (on 386) Note: FDLIBM will detect, at run time, the correct ordering of the high and low part of a floating-point number. d. IEEE exceptions may trigger "signals" as is common in Unix implementations. ------------------- 2. EXCEPTION CASES ------------------- All exception cases in the FDLIBM functions will be mapped to one of the following four exceptions:+huge*huge,+tiny*tiny,+1.0/0.0,+0.0/0.0 (overflow) (underflow) (divided-by-zero) (invalid) For example,log(0)is a singularity and is thus mapped to -1.0/0.0 = -infinity. That is, FDLIBM's log will compute -one/zero and return the computed value. On an IEEE machine, this will trigger the divided-by-zero exception and a negative infinity is returned by default. Similarly, exp(-huge) will be mapped to tiny*tiny to generate an underflow signal.IEEESTANDARD754Floating-PointArithmetic:Properties of IEEE 754 Double-Precision: Wordsize: 64 bits, 8 bytes. Radix: Binary. Precision: 53 significant bits, roughly 16 significant decimals. If x and x' are consecutive positive Double-Precision numbers (they differ by 1ulp), 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 Inf. Underflow isGradual, 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 multiplication 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 Inf = 1/(x-y) != -1/(y-x) = -Inf. Inf is signed. it persists when added to itself or to any finite number. Its sign transforms correctly through multiplication and division, and (finite)/+Inf =+0 (nonzero)/0 =+Inf. But Inf-Inf, Inf*0 and Inf/Inf are, like 0/0 and sqrt(-3), invalid operations that produce NaN. ... Reserved operands: there are 2**53-2 of them, all called NaN (NotaNumber). Some, called Signaling NaNs, 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 NaNs; they are the default results of Invalid Operations, and propagate through subsequent arithmetic operations. If x != x then x is NaN; every other predicate (x > y, x = y, x < y, ...) is FALSE if NaN is involved. NOTE: Trichotomy is violated by NaN. Besides being FALSE, predicates that entail ordered comparison, rather than mere (in)equality, signal Invalid Operation when NaN is involved. Rounding: Every algebraic operation (+, -, *, /, sqrt) is rounded by default to within half anulp, and when the rounding error is exactly half anulpthen 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 ... 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 +Inf or towards -Inf at the programmer's option. And the same kinds of rounding are specified 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 NaN, or FALSE Overflow+Inf Divide by Zero+Inf Underflow Gradual Underflow Inexact Rounded value NOTE: An Exception is not an Error unless handled badly. What makes a class of exceptions exceptional 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 certainly nonzero regardless of how tiny it may be.) Products and quotients that underflow gradually can lose accuracy gradually without vanishing, so comparing 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 something 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 underflows are usuallyprovablyignorable. 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 traditionally 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 statement, 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 document. 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 to be distributed in 4.3 BSD'slibm. Ideally, each elementary 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 exceptions listed above, although the definition of the function may be correlated intentionally with exception handling. Ideally, every programmer should be ableconvenientlyto 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 inlibmare only approximately atomic. They signal no inappropriate exception except possibly ... Over/Underflow when a result, if properly computed, might have lain barely within range, and Inexact incbrt,hypot,log10andpowwhen it happens to be exact, thanks to fortuitous cancellation of errors. Otherwise, ... Invalid Operation is signaled only when any result but NaN 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.BUGSWhen signals are appropriate, they are emitted by certain operations within the codes, so a subroutine-trace may be 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 otherwise get correct results despite division by zero. The math manual pages have been adapted from the 4.3BSD 3M manual pages for FDLIBM by Kees J. Bot <kjb@cs.vu.nl> who normally avoids floating point like the plague. Some text may not apply to FDLIBM, but KJB didn't know whether to remove it or not. Don't blame the original authors mentioned on these pages for inaccuracies introduced.SEEALSOasinh(3),erf(3),exp(3),floor(3),hypot(3),ieee(3),j0(3),sin(3),sinh(3),sqrt(3). 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 Arithmetic" 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.AUTHORW. Kahan, with the help of Z-S. Alex Liu, Stuart I. McDonald, Dr. Kwok-Choi Ng, Peter Tang.