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NAME    [Toc]    [Back]

       math - introduction to mathematical library functions

DESCRIPTION    [Toc]    [Back]

       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>.

LIST OF FUNCTIONS    [Toc]    [Back]

       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 		   ???

NOTES    [Toc]    [Back]

       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,
  and  to which this manual 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 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
			    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.
		     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.
		     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
	      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 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.
		     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.
		     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 ...
		     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
	      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    [Toc]    [Back]

       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    [Toc]    [Back]

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

NOTES    [Toc]    [Back]

       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.

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