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MATH(3M)							      MATH(3M)

NAME
       math - introduction to mathematical library functions

DESCRIPTION
       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>.  The Fortran
       math library is described in ``man 3f intro''.

LIST OF FUNCTIONS
       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
       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
       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			   ???
       yn	   j0.3m	bessel function			   ???

NOTES
       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
       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 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
       developed.  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.

       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.
		     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.
	      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 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 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 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
       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 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
       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 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 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.
	      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 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 Operation when NaN is involved.
	      Rounding:
		     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
		     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		   ±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
		     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 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's
       libm.   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  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 inappropriate exception except possibly ...
	      Over/Underflow
		     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
		     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.

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 correct results despite division by zero.

SEE ALSO
       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.

AUTHOR
       W.  Kahan,  with	 the  help  of	Z-S. Alex Liu, Stuart I. McDonald, Dr.
       Kwok-Choi Ng, Peter Tang.

4th Berkeley Distribution	 May 27, 1986			      MATH(3M)
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