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MATH(3M)							      MATH(3M)

       math - introduction to mathematical library functions

       These  functions	 constitute the	C math library,	libm.  The link	editor
       searches	this library under the "-lm" option.  Declarations  for	 these
       functions may be	obtained from the include file <math.h>.

       Each of the following double functions has a float counterpart with the
       name ending in f,  as  an  example  the	float  counterpart  of	double
       acos(double x) is float acosf(float x).

       Name	 Appears on Page    Description		      Error Bound (ULPs)
       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
       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
       ilogb	   ieee.3m	exponent extraction		    0
       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
       log10	   exp.3m	logarithm to base 10		    3
       log1p	   exp.3m	log(1+x)			    1
       pow	   exp.3m	exponential x**y		 60-500
       remainder   ieee.3m	remainder			    0
       rint	   floor.3m	round to nearest integer	    0
       scalbn	   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			   ???
       yn	   j0.3m	bessel function			   ???

       In 4.3 BSD, distributed from the	University of California in late 1985,
       most of the foregoing functions come in two versions, one for the  dou-
       ble-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 simi-
       larly, 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 libm 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	following notes	are provided.

       DEC VAX-11 D_floating-point:

       This is the format for which the	original math library libm was	devel-
       oped,  and  to which this manual	is still principally dedicated.	 It is
       the double-precision format for the PDP-11 and the earlier  VAX-11  ma-
       chines;	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 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.
		     Overflow customarily stops	computation.
		     Underflow is customarily flushed quietly to zero.
			    It is possible to have x !=	y and yet x-y =	0  be-
			    cause  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 ac-
		     cepted 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.
		     Divisions by zero and operations that  overflow  are  in-
		     valid operations that customarily stop computation	or, in
		     earlier machines, produce	reserved  operands  that  will
		     stop computation.
		     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 com-
       puter 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
	      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 libm 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 puritani-
       cal 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 computer arithmetic.  VLSI chips that conform to some
       version of that standard	have been produced 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	 stan-
       dard'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 Semi. 32081 and  WTL	 1164/65.   To
       use these codes with the	Intel or Zilog chips, or with the Apple	Macin-
       tosh or ELXSI 6400, is to forego	the use	of better codes	provided (per-
       haps  freely) by	those companies	and designed by	some of	the authors of
       the codes above.	 Except	for atan, cabs,	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 sig.	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 sig.  bits,	roughly	like 16	sig.  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, 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 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	Infin-
		     ity/Infinity  are,	 like 0/0 and sqrt(-3),	invalid	opera-
		     tions 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 op-
		     eration performed upon them; they are used	to mark	 miss-
		     ing  or  uninitialized values, or nonexistent elements of
		     arrays.  The rest are Quiet NaNs; they  are  the  default
		     results of	Invalid	Operations, and	propagate through sub-
		     sequent 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  or-
			    dered  comparison,	rather than mere (in)equality,
			    signal Invalid Operation when NaN is involved.
		     Every algebraic operation (+, -, *, /, sqrt)  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
		     usually the best kind, sometimes  provably	 so;  for  in-
		     stance,  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 ev-
		     ery 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.
		     IEEE 754 recognizes five kinds of	floating-point	excep-
		     tions, listed below in declining order of probable	impor-
			    Exception		   Default Result

			    Invalid Operation	   NaN,	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	exceptional is that no
		     single default response can be satisfactory in every  in-
		     stance.   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 ex-
		  ception could	have produced.
		  CAUTION: The only reliable ways to discover  whether	Under-
		  flow	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 ac-
		  curacy  gradually  without vanishing,	so comparing them with
		  zero (as one might on	a VAX) will not	reveal the loss.  For-
		  tunately, 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	ignor-
		  able.	 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 aban-

	      --  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 pro-
		  grammer'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's
       libm.  Ideally, each elementary function	should act as if it were indi-
       visible,	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  dis-
	     rupted  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	conveniently  to  turn	a  de-
       bugged subprogram into one that appears atomic to its users.  But simu-
       lating all three	characteristics	of an atomic function is still	a  te-
       dious  affair, entailing	hosts of tests and saves-restores; work	is un-
       der way to ameliorate the inconvenience.

       Meanwhile, the functions	in libm	are only approximately	atomic.	  They
       signal no inappropriate exception except	possibly ...
		     when  a  result,  if  properly  computed, might have lain
		     barely within range, and
	      Inexact in cabs, cbrt, hypot, log10 and pow
		     when it happens to	be exact, thanks to fortuitous cancel-
		     lation 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
	      Divide-by-Zero is	signaled only when
		     a function	takes exactly infinite values at finite	 oper-
	      Underflow	is signaled only when
		     the exact result would be nonzero but tinier than the un-
		     derflow threshold.
	      Inexact is signaled only when
		     greater range or precision	would be needed	 to  represent
		     the exact result.

       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 de-
       cision  to  trap	 all divisions by zero could disrupt a code that would
       otherwise get correct results despite division by zero.

       fpgetround(3), fpsetround(3), fpgetprec(3), fpsetprec(3), fpgetmask(3),
       fpsetmask(3),  fpgetsticky(3),  fpresetsticky(3)	 - IEEE	floating point

       An explanation of IEEE 754 and its proposed  extension  p854  was  pub-
       lished  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.

4th Berkeley Distribution	  May 6, 1991			      MATH(3M)


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