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MATH(3) 			   BSD Library Functions Manual 			  MATH(3)

NAME
     math -- introduction to mathematical library functions

LIBRARY
     Math Library (libm, -lm)

SYNOPSIS
     #include <math.h>

DESCRIPTION
     These functions constitute the C Math Library (libm, -lm).  Declarations for these functions
     may be obtained from the include file <math.h>.

   List of Functions
     Name	  Man page     Description		      Error Bound (ULPs)
     acos	  acos(3)      inverse trigonometric function 3
     acosh	  acosh(3)     inverse hyperbolic function    3
     asin	  asin(3)      inverse trigonometric function 3
     asinh	  asinh(3)     inverse hyperbolic function    3
     atan	  atan(3)      inverse trigonometric function 1
     atanh	  atanh(3)     inverse hyperbolic function    3
     atan2	  atan2(3)     inverse trigonometric function 2
     cbrt	  sqrt(3)      cube root		      1
     ceil	  ceil(3)      integer no less than	      0
     copysign	  copysign(3)  copy sign bit		      0
     cos	  cos(3)       trigonometric function	      1
     cosh	  cosh(3)      hyperbolic function	      3
     erf	  erf(3)       error function		      ???
     erfc	  erf(3)       complementary error function   ???
     exp	  exp(3)       exponential		      1
     expm1	  exp(3)       exp(x)-1 		      1
     fabs	  fabs(3)      absolute value		      0
     finite	  finite(3)    test for finity		      0
     floor	  floor(3)     integer no greater than	      0
     fmod	  fmod(3)      remainder		      ???
     hypot	  hypot(3)     Euclidean distance	      1
     ilogb	  ilogb(3)     exponent extraction	      0
     isinf	  isinf(3)     test for infinity	      0
     isnan	  isnan(3)     test for not-a-number	      0
     j0 	  j0(3)        Bessel function		      ???
     j1 	  j0(3)        Bessel function		      ???
     jn 	  j0(3)        Bessel function		      ???
     lgamma	  lgamma(3)    log gamma function	      ???
     log	  log(3)       natural logarithm	      1
     log10	  log(3)       logarithm to base 10	      3
     log1p	  log(3)       log(1+x) 		      1
     nan	  nan(3)       return quiet NaN 	      0
     nextafter	  nextafter(3) next representable number      0
     pow	  pow(3)       exponential x**y 	      60-500
     remainder	  remainder(3) remainder		      0
     rint	  rint(3)      round to nearest integer       0
     scalbn	  scalbn(3)    exponent adjustment	      0
     sin	  sin(3)       trigonometric function	      1
     sinh	  sinh(3)      hyperbolic function	      3
     sqrt	  sqrt(3)      square root		      1
     tan	  tan(3)       trigonometric function	      3
     tanh	  tanh(3)      hyperbolic function	      3
     trunc	  trunc(3)     nearest integral value	      3
     y0 	  j0(3)        Bessel function		      ???
     y1 	  j0(3)        Bessel function		      ???
     yn 	  j0(3)        Bessel function		      ???

   List of Defined Values
     Name	     Value			 Description
     M_E	     2.7182818284590452354	 e
     M_LOG2E	     1.4426950408889634074	 log 2e
     M_LOG10E	     0.43429448190325182765	 log 10e
     M_LN2	     0.69314718055994530942	 log e2
     M_LN10	     2.30258509299404568402	 log e10
     M_PI	     3.14159265358979323846	 pi
     M_PI_2	     1.57079632679489661923	 pi/2
     M_PI_4	     0.78539816339744830962	 pi/4
     M_1_PI	     0.31830988618379067154	 1/pi
     M_2_PI	     0.63661977236758134308	 2/pi
     M_2_SQRTPI      1.12837916709551257390	 2/sqrt(pi)
     M_SQRT2	     1.41421356237309504880	 sqrt(2)
     M_SQRT1_2	     0.70710678118654752440	 1/sqrt(2)

NOTES
     In 4.3 BSD, distributed from the University of California in late 1985, most of the forego-
     ing functions come in two versions, 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 ULPs tabulated
     above; an ULP is one Unit in the Last Place.  And the programs have been cured of anomalies
     that afflicted the older math library in which incidents like the following 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 fol-
     lowing notes are provided.

   DEC VAX-11 D_floating-point
     This is the format for which the original math library was developed, and to which this man-
     ual is still principally dedicated.  It is the 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 significant bits, roughly like 17 significant decimals.  If x and x' are
		   consecutive positive D_floating-point numbers (they differ by 1 ULP), 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 qui-
		   etly 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.

	   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 computation or, in earlier machines, produce reserved oper-
		   ands 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 ULP, and when the rounding error is exactly half an ULP 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.

     This response seems excessively punitive, so it is destined to be replaced at some time in
     the foreseeable future by a more flexible but still uniform scheme being developed to handle
     all floating-point arithmetic exceptions neatly.

     How do the functions in 4.3 BSD's new math library 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 libm.  The VMS
     codes interpolate in large table to achieve speed and accuracy; the libm codes use tricky
     formulas compact enough that all of them may some day fit into a ROM.

     More important, DEC regards the VMS codes as proprietary and guards them zealously against
     unauthorized use.	But the libm 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 libm.

   IEEE STANDARD 754 Floating-Point Arithmetic
     This standard is on its way to becoming more widely adopted than any other design for com-
     puter 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
     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.  Sev-
     eral 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 Double format, so similar that the C
     programs for the IEEE versions 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 libm for machines that conform to IEEE 754 are intended primarily for
     the National Semiconductor 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 atan(), cbrt(), erf(), erfc(), hypot(), j0-jn(), lgamma(), pow(),
     and y0-yn(), the Motorola 68881 has all the functions in libm on chip, and faster and more
     accurate; it, Apple, the i8087, Z8070 and WE32106 all use 64 significant bits.  The main
     virtue of 4.3 BSD's libm codes is that they 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 that conforms to IEEE 754 need use anything worse than the new libm.

     Properties of IEEE 754 Double-Precision:

	   Wordsize: 64 bits, 8 bytes.

	   Radix:  Binary.

	   Precision: 53 significant bits, roughly like 16 significant decimals.  If x and x' are
		   consecutive positive Double-Precision numbers (they differ by 1 ULP), 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 Gradual, 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, compari-
		   son (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 infinity
		   (finite)/+-	= +-0 (nonzero)/0 = +- infinity.  But oo-oo, oo*0 and oo/oo are,
		   like 0/0 and sqrt(-3), invalid operations that produce NaN.

	   Reserved operands: there are 2**53-2 of them, all called NaN (Not A Number).  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 Oper-
		   ation when NaN is involved.

	   Rounding: Every algebraic operation (+, -, *, /, \/) is rounded by default to within
		   half an ULP, and when the rounding error is exactly half an ULP then the
		   rounded value's least significant bit is zero.  This kind of rounding is usu-
		   ally 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 +infinity or towards -infinity 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		 +-oo
		   Divide by Zero	 +-oo
		   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 satisfac-
		   tory 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 usually provably 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 excep-
	  tions 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 solu-
     tion is understood, but not enough manpower was available to implement it fully in time to
     be distributed in 4.3 BSD's libm.	Ideally, each elementary function should act as if it
     were indivisible, or atomic, in the sense that ...

     1.   No exception should be signaled that is not deserved by the data supplied to that func-
	  tion.

     2.   Any exception signaled should be identified with that function rather than with one of
	  its subroutines.

     3.   The internal behavior of an atomic function should not be disrupted when a calling pro-
	  gram 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 conveniently 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 libm are only approximately atomic.  They signal no inappropri-
     ate exception except possibly ...

	   Over/Underflow
	   when a result, if properly computed, might have lain barely within range, and

	   Inexact in cbrt(), hypot(), log10(and) pow()
	   when 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.

SEE ALSO
     An explanation of IEEE 754 and its proposed extension p854 was published in the IEEE maga-
     zine 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 stan-
     dard.

BUGS
     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 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 cor-
     rect results despite division by zero.

BSD					February 23, 2007				      BSD
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