IRIX Base Documentation 2002 November

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MATH(3M) Last changed: 2-2-99 NNAAMMEE mmaatthh - introduction to mathematical library functions IIMMPPLLEEMMEENNTTAATTIIOONN IRIX systems DDEESSCCRRIIPPTTIIOONN These functions constitute the C math library lliibbmm. There are four versions of the math library: lliibbmm..aa,, lliibbmmxx..aa,, lliibbmm4433..aa, and lliibbffaassttmm..aa The first, lliibbmm..aa,, contains routines implemented in 1994 that use algorithms which take advantage of the MIPS architecture and include many routines for the ffllooaatt data type. For the --6644 and --nn3322 versions of lliibbmm..aa,, a second version of the math library, lliibbmmxx..aa,, contains functions which give identical results to those in lliibbmm..aa, but which use System V error handling. See mmaatthheerrrr(3M) for a description of error handling for lliibbmmxx..aa functions. The third version of the math library, lliibbmm4433..aa, contains routines all based on the original codes in the 4.3BSD release. The difference between the error bounds for lliibbmm..aa and lliibbmm4433..aa is typically around 1 unit in the last place (ULP), whereas the performance difference may be a factor of two or more. The link editor searches this library under the --llmm, --llmmxx, or --llmm4433 option. Declarations for these functions may be obtained from the include file <<mmaatthh..hh>>. The fourth library, lliibbffaassttmm..aa,, contains faster, lower-precision versions of various routines from lliibbmm..aa. LLiisstt ooff ffuunnccttiioonnss Error bounds (in ULPs) in the final two columns in this table apply only to the --6644 and --nn3322 versions of lliibbmm..aa and lliibbmmxx..aa, The error bound sometimes applies only to the primary range. ------------------------------------------------------------------------ Name Man page Description lliibbmm..aa lliibbmm4433..aa ------------------------------------------------------------------------ aaccooss ssiinn(3M) inverse trigonometric function 2 3 aaccoossff ssiinn(3M) inverse trigonometric function 1 aaccoosshh aassiinnhh(3M) inverse hyperbolic function 3 3 aassiinn ssiinn(3M) inverse trigonometric function 2 3 aassiinnff ssiinn(3M) inverse trigonometric function 1 aassiinnhh aassiinnhh(3M) inverse hyperbolic function 3 3 aattaann ssiinn(3M) inverse trigonometric function 1.5 1 aattaannff ssiinn(3M) inverse trigonometric function 1 aattaannhh aassiinnhh(3M) inverse hyperbolic function 3 3 aattaann22 ssiinn(3M) inverse trigonometric function 2 2 aattaann22ff ssiinn(3M) inverse trigonometric function 1 ccaabbss hhyyppoott(3M) complex absolute value 1 1 aabbssff hhyyppoott(3M) complex absolute value 1 ccbbrrtt ssqqrrtt(3M) cube root 1 1 cceeiill fflloooorr(3M) integer no less than 0 0 cceeiillff fflloooorr(3M) integer no less than 0 0 ccooppyyssiiggnn iieeeeee(3M) copy sign bit 0 0 ccooss ssiinn(3M) trigonometric function 2 1 ccoossff ssiinn(3M) trigonometric function 1 ccoosshh ssiinnhh(3M) hyperbolic function 2 3 ccoosshhff ssiinnhh(3M) hyperbolic function 1 ddrreemm iieeeeee(3M) remainder 0 0 eerrff eerrff(3M) error function 1 ? eerrffcc eerrff(3M) complementary error function 4 ? eexxpp eexxpp(3M) exponential 1 1 eexxppff eexxpp(3M) exponential 1 eexxppmm11 eexxpp(3M) exp(x)-1 1 1 eexxppmm11ff eexxpp(3M) exp(x) - 1 1 ffaabbss fflloooorr(3M) absolute value 0 0 ffaabbssff fflloooorr(3M) absolute value 0 0 ffiinniittee iieeeeee(3M) floating point arithmetic (N/A) fflloooorr fflloooorr(3M) integer no greater than 0 0 fflloooorrff fflloooorr(3M) integer no greater than 0 0 ffmmoodd fflloooorr(3M) remainder function 0 ffmmooddff fflloooorr(3M) remainder function 0 hhyyppoott hhyyppoott(3M) Euclidean distance 1 1 hhyyppoottff hhyyppoott(3M) Euclidean distance 1 1 jj00 jj00(3M) bessel function ? ? jj11 jj00(3M) bessel function ? ? jjnn jj00(3M) bessel function ? ? llggaammmmaa llggaammmmaa(3M) log gamma function ? ? lloogg eexxpp(3M) natural logarithm 1 1 llooggff eexxpp(3M) natural logarithm 1 llooggbb iieeeeee(3M) exponent extraction 0 0 lloogg1100 eexxpp(3M) logarithm to base 10 2 3 lloogg1100ff eexxpp(3M) logarithm to base 10 1.5 lloogg11pp eexxpp(3M) log(1+x) 1 1 lloogg11ppff eexxpp(3M) log(1+x) 1 1 ppooww eexxpp(3M) exponential x**y 2 60-500 ppoowwff eexxpp(3M) exponential x**y 1 rriinntt fflloooorr(3M) round to nearest integer 0 0 ssiinn ssiinn(3M) trigonometric function 2 1 ssiinnff ssiinn(3M) trigonometric function 1 ssiinnhh ssiinnhh(3M) hyperbolic function 2 3 ssiinnhhff ssiinnhh(3M) hyperbolic function 1 ssqqrrtt ssqqrrtt(3M) square root 1 1 ssqqrrttff ssqqrrtt(3M) square root 1 ttaann ssiinn(3M) trigonometric function 2 3 ttaannff ssiinn(3M) trigonometric function 1 ttaannhh ssiinnhh(3M) hyperbolic function 2 3 ttaannhhff ssiinnhh(3M) hyperbolic function 1 ttrruunncc fflloooorr(3M) truncate to whole number 0 0 ttrruunnccff fflloooorr(3M) truncate to whole number 0 0 yy00 jj00(3M) bessel function ? ? yy11 jj00(3M) bessel function ? ? yynn jj00(3M) bessel function ? ? ------------------------------------------------------------------------ VVeeccttoorr IInnttrriinnssiiccss Beginning with IRIX 6.2, lliibbmm now supports the following vector intrinsics: Single precision vector routines: vacosf( float *x, float *y, long count, long stridex, long stridey ) vasinf( float *x, float *y, long count, long stridex, long stridey ) vatanf( float *x, float *y, long count, long stridex, long stridey ) vcosf( float *x, float *y, long count, long stridex, long stridey ) vexpf( float *x, float *y, long count, long stridex, long stridey ) vlogf( float *x, float *y, long count, long stridex, long stridey ) vlog10f( float *x, float *y, long count, long stridex, long stridey ) vsinf( float *x, float *y, long count, long stridex, long stridey ) vsqrtf( float *x, float *y, long count, long stridex, long stridey ) vtanf( float *x, float *y, long count, long stridex, long stridey ) Double precision vector routines: vacos( double *x, double *y, long count, long stridex, long stridey ) vasin( double *x, double *y, long count, long stridex, long stridey ) vatan( double *x, double *y, long count, long stridex, long stridey ) vcos( double *x, double *y, long count, long stridex, long stridey ) vexp( double *x, double *y, long count, long stridex, long stridey ) vlog( double *x, double *y, long count, long stridex, long stridey ) vlog10( double *x, double *y, long count, long stridex, long stridey ) vsin( double *x, double *y, long count, long stridex, long stridey ) vsqrt( double *x, double *y, long count, long stridex, long stridey ) vtan( double *x, double *y, long count, long stridex, long stridey ) Input and output arrays for the above routines should either be identical or non-overlapping. On MIPS4 processors, these routines are software-pipelined to take advantage of the multiple execution units. On that machine, throughput is up to several times greater than by calling the scalar intrinsics repeatedly. On processors other than the MIPS4, these routines are still available; although not software-pipelined on those processors, they still eliminate considerable call overhead when they can be used. The vector routines do not support denormals on the MIPS4 processors. The single precision vector routines can also be called by the names vvffaaccooss, vvffaassiinn, etc. SSeemmaannttiiccss ooff tthheessee rroouuttiinneess i=0, 1, ..., count-1: y[i*stridey] = f(x[i*stridex]) Example: ddoouubbllee xx[[1100000000]],, yy[[1100000000]];; ...... ffoorr ((ii==00;; ii<<11000000;; ii++++ )) yy[[22**ii]] == ssiinn((xx[[33**ii]]));; Transform (by hand) into vvssiinn((xx,, yy,, 11000000,, 33,, 22));; Vector and scalar routines may differ slightly; however, none of the results differ from the mathematically correct result by more than 2 ULPs. The vector square root routines are less accurate than the hardware versions; vvssqqrrtt and vvssqqrrttff use the reciprocal square root instruction and lose up to about 2 bits of accuracy. vvssqqrrtt and vvffssqqrrtt give correct answers for zero and infinite arguments. LLoonngg DDoouubbllee AArriitthhmmeettiicc Long double arithmetic is supported by the compiler. The representation used is not IEEE compliant; long doubles are represented on this system as the sum or difference of two doubles, normalized so that the smaller double is <= .5 ULP of the larger. This is equivalent to a 107 bit mantissa with an 11 bit biased exponent (bias = 1023), and 1 sign bit. In terms of decimal precision, this is approximately 34 decimal digits. Long double constants are coded as double precision constants followed by the letter 'l' (upper or lower case). The largest (finite) long double constant is 1.797693134862315807937289714053023e308L. The smallest long double precision constant is 4.940656458412465441765687928682213e-324L. Long doubles less than 1.805194375864829576069262081173746e-276L may require a double denormal in their representation and therefore contain less than 107 bits precision. Long double NaNs and (signed) infinities are supported by the compiler. Long double infinity is represented as the sum of a double infinity and a double zero; similarly for NaNs. In Fortran, long doubles are denoted by the term REAL *16. In general, long double arithmetic operations (++,, --,, **,, //) are not precisely rounded, but are accurate to approximately 3 ULPs. Long double arithmetic operations are done in software by MIPSpro compilers; results of these operations may vary slightly from release to release due to improvements in the algorithms which implement them. Long double operations on this system are only supported in _r_o_u_n_d- _t_o-_n_e_a_r_e_s_t rounding mode (the default). The system must be in round- to-nearest rounding mode when issuing long double arithmetic operations or calling any of the long double functions; otherwise, incorrect answers result. DDiiffffeerreenncceess bbeettwweeeenn --oo3322, --nn3322, --6644 At the IRIX 6.2 release, faster and more accurate algorithms were implemented, and vector functions were added to the math library. In order to maintain numerical compatibility with older releases, these changes were made only in the --nn3322 and --6644 versions of the library and not in the --oo3322 version. If there are differences in accuracy, this document describes the behavior of the --nn3322 and --6644 versions of the library. To take advantage of the new functions and algorithms, you need to compile and link using either the --nn3322 or the --6644 option. The --oo3322 version of lliibbmmxx contains all routines present in the --nn3322 and --6644 versions of lliibbmmxx except the quad precision and vector routines, and gives results identical to the --nn3322 and --6644 versions. NNOOTTEESS Users concerned with portability to other computer systems should note that the long double and float versions of these functions are optional according to the ANSI C Programming Language Specification ISO/IEC 9899 : 1990 (E). Long double functions have been renamed to be compliant with the ANSI-C standard; to be backward compatible, however, they may still be called with the double precision function name prefixed with a qq. The exceptions are functions ffaabbssll and ffmmooddll, which may be called with names qqaabbss and qqmmoodd. In 4.3BSD, distributed from the University of California in late 1985, most of the foregoing functions come in two versions, one for the double-precision "D" format in the DDEECC, VVAAXX--1111 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. For instance, the programs are accurate to within the numbers of ULPs tabulated above. And the programs no longer have the anomalies found in the older math library lliibbmm in which incidents like the following had been reported: * ssqqrrtt((--11..00)) == 00..00 and lloogg((--11..00)) == --11..77ee3388 ** ((ccooss((11..00ee--1111)) >> ccooss((00..00)) >> 11..00 ** ppooww((xx,,11..00)) !!== xx when xx == 22..00,, 33..00,, 44..00,, ......,, 99..00 * ppooww((--11..00,,11..00ee1100)) trapped on Integer Overflow * ssqqrrtt((11..00ee3300)) and ssqqrrtt((11..00ee--3300)) were very slow This machine conforms to the IEEE Standard 754 for Binary Floating- point Arithmetic, to which only the notes for IEEE floating-point apply and are included here. See the notes regarding long double precision below. IIEEEEEE SSTTAANNDDAARRDD 775544 FFllooaattiinngg--ppooiinntt AArriitthhmmeettiicc This standard has become more widely adopted than any other design for computer arithmetic. Properties of IEEE 754 Double-precision: * Wordsize: 64 bits, 8 bytes. Radix: Binary. * Precision: 53 significant bits, roughly 16 significant decimals. If xx and xx'' are consecutive positive double-precision numbers (they differ by 1 ULP), then 11..11ee --1166 << 00..55****5533 << ((xx''--xx))//xx <<== 00..55****5522 << 22..33ee--1166 * Range: Overflow threshold = 22..00****11002244 == 11..88ee330088 Underflow threshold = 00..55****11002222 == 22..22ee --330088 Overflow goes by default to a signed Infinity. Underflow is _G_r_a_d_u_a_l, rounding to the nearest integer multiple of 00..55****11007744 == 44..99ee --332244. * Zero is represented ambiguously as ++00, or --00. Its sign transforms correctly through multiplication or division, and is preserved by addition of zeros with like signs; but xx--xx yields ++00 for every finite _x. The only operations that reveal zero's sign are division by zero and ccooppyyssiiggnn((xx,,++__00)). In particular, comparison ((xx >> yy,, xx >>== yy, etc.) cannot be affected by the sign of zero; but if finite xx == yy then IInnffiinniittyy == 11//((xx--yy)) !!== 11//((yy--xx)) == --IInnffiinniittyy. * 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 IInnffiinniittyy -- IInnffiinniittyy, IInnffiinniittyy**00 and IInnffiinniittyy//IInnffiinniittyy are, like 0/0 and ssqqrrtt((--33)), invalid operations that produce NaN. * Reserved operands: there are 22****5533--22 of them, all called NaN (_Not _a _Number). Some, called Signaling NaNs, trap any floating-point operation performed upon them; they could be 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 xx !!== xx then x is NaN; every other predicate ((xx >> yy,, xx == yy,, xx << yy,, ......)) 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, (++,,--,, **,, //,, ssqqrrtt)) is rounded by default to within half a ULP, and when the rounding error is exactly half a ULP 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 xx == 11..00,, 22..00,, 33..00,, 44..00,, ......,, 22..00****5522, we find ((xx//33..00))**33..00 ==== xx and ((xx//1100..00))**1100..00 ==== xx 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 ++IInnffiinniittyy or towards --IInnffiinniittyy at the programmer's option. * Exceptions: IEEE 754 recognizes five kinds of floating-point exceptions, listed below in declining order of probable importance. EExxcceeppttiioonn DDeeffaauulltt RReessuulltt Invalid Operation NaN or FALSE Overflow +_Infinity Divide by Zero +_Infinity 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 can handle 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 if an exception has occurred since the program last reset its flag. 3. Test a result to see if it is a value that only an exception could have produced. CAUTION: The only reliable ways to discover if underflow has occurred are to test if 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 xx !!== yy, then xx--yy 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 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: 1. ABORT. This mechanism classifies an exception in advance as an incident to be handled by means traditionally associated with error-handling statements like OONN EERRRROORR GGOO TTOO ....... 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. 2. 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 should be able to resume execution after each one as if execution had not been stopped. Ideally, each elementary function should act as if it were indivisible, or atomic, in the following sense: * No exception should be signaled that is not deserved by the data supplied to that function. * Any exception signaled should be identified with that function rather than with one of its subroutines. * 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 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. Meanwhile, the functions in lliibbmm are only approximately atomic. They signal no inappropriate exception except (possibly) oovveerrffllooww or uunnddeerrffllooww when a result, if properly computed, might have lain barely within range, and inexact in ccaabbss, ccbbrrtt, hhyyppoott, lloogg1100 and ppooww when it happens to be exact, thanks to cancellation of errors. Otherwise, IInnvvaalliidd OOppeerraattiioonn is signaled only when any result but NaN would probably be misleading. OOvveerrffllooww is signaled only when the exact result would be finite but beyond the overflow threshold. DDiivviiddee--bbyy aa ffuunnccttiioonn takes exactly infinite values at finite operands. UUnnddeerrffllooww is signaled only when the exact result would be nonzero but tinier than the underflow threshold. IInneexxaacctt is signaled only when greater range or precision would be needed to represent the exact result. EExxcceeppttiioonnss The exception enables and the flags that are raised when an exception occurs (as well as the rounding mode), are in the floating-point control and status register. This register can be read or written by the routines described on ffppcc(3C). This register's layout is described in the file <<ssyyss//ffppuu..hh>>. A useful set of ``user trap handlers'' is available. See ssiiggffppee(3C). The raw interface to the hardware registers is only intended to be used by the code to implement IEEE user trap handlers. IEEE floating-point exceptions are enabled by setting the enable bit for that exception in the floating-point control and status register. If an exception then occurs the UNIX signal SSIIGGFFPPEE is sent to the process. It is up to the signal handler to determine the instruction that caused the exception and to take the action specified by the user. The instruction that caused the exception is in one of two places. If the floating-point board is used (the floating-point implementation revision register indicates this in its implementation field) then the instruction that caused the exception is in the floating-point exception instruction register. In all other implementations the instruction that caused the exception is at the address of the program counter as modified by the branch delay bit in the cause register. Both the program counter and cause register are in the sigcontext structure passed to the signal handler (see ssiiggnnaall(2)). If the program is to be continued past the instruction that caused the exception, the program counter in the signal context must be advanced. If the instruction is in a branch delay slot then the branch must be emulated to determine if the branch is taken and then the resulting program counter can be calculated (see eemmuullaattee__bbrraanncchh(3X) and ssiiggnnaall(2)). On systems using the R8000 processor, floating point exceptions are generally fatal when trapped unless the process is being run in precise exception mode. PPLLAATTFFOORRMM SSPPEECCIIFFIICC LLIIBBRRAARRIIEESS When compiling with --nn3322 or --6644, each processor has specially tuned, hardware-specific versions of lliibbmm and lliibbffaassttmm that the run-time linker will use, by default, whenever available. The R10000 tuned libraries are found in the directories: /usr/lib32/mips4/r10000/ /usr/lib64/mips4/r10000/ The R8000 tuned libraries are found in the directories: /usr/lib32/mips4/r8000/ /usr/lib64/mips4/r8000/ The R5000 tuned libraries are found in the directories: /usr/lib32/mips4/ /usr/lib64/mips4/ And the R4000 tuned libraries are found in the directories: /usr/lib32/mips3/ /usr/lib64/mips3/ At runtime, each program automatically uses the "best" library for the system on which it is executing. For example, if the executing program is a MIPS3 program designed to run on an r4000 processor, it will still use the MIPS4 R10000-tuned math library when running on an r10000 system. BBUUGGSS When 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. 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. SSEEEE AALLSSOO ssiiggnnaall(2), ffppcc(3C), eemmuullaattee__bbrraanncchh(3X), ssiiggffppee(3C), mmaatthheerrrr(3M) This man page is available only online.