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math(3m) [ultrix man page]

math(3m)																  math(3m)

       math - introduction to mathematical library functions

       These functions constitute the C math library libm.  There are two versions of the math library libm.a and libm43.a.

       The  first,  libm.a,  contains routines written in MIPS assembly language and tuned for best performance and includes many routines for the
       float data type. The routines in there are based on the algorithms of Cody and Waite or those in the 4.3 BSD  release,  whichever  provides
       the best performance with acceptable error bounds.  Those routines with Cody and Waite implementations are marked with a `*' in the list of
       functions below.

       The second version of the math library, libm43.a, contains routines all based on the original codes in the 4.3 BSD release.  The difference
       between	the two version's error bounds is typically around 1 unit in the last place, whereas the performance difference may be a factor of
       two or more.

       The link editor searches this library under the "-lm" (or "-lm43") 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
       The  cycle  counts  of all functions are approximate; cycle counts often depend on the value of argument. The error bound sometimes applies
       only to the primary range.

					  Error Bound (ULPs) Cycles
       Name	  Description		  libm.a   libm43.a   libm.a   libm43.a
       acos	  inverse trig function   3	   3	      ?        ?
       acosh	  inverse    hyperbolic   3	   3	      ?        ?
       asin	  inverse trig function   3	   3	      ?        ?
       asinh	  inverse    hyperbolic   3	   3	      ?        ?
       atan	  inverse trig function   1	   1	      152      260
       atanh	  inverse    hyperbolic   3	   3	      ?        ?
       atan2	  inverse trig function   2	   2	      ?        ?
       cabs	  complex      absolute   1	   1	      ?        ?
       cbrt	  cube root		  1	   1	      ?        ?
       ceil	  integer no less than	  0	   0	      ?        ?
       copysign   copy sign bit 	  0	   0	      ?        ?
       cos*	  trig function 	  2	   1	      128      243
       cosh*	  hyperbolic function	  ?	   3	      142      294
       drem	  remainder		  0	   0	      ?        ?
       erf	  error function	  ?	   ?	      ?        ?
       erfc	  complementary   error   ?	   ?	      ?        ?
       exp*	  exponential		  2	   1	      101      230
       expm1	  exp(x)-1		  1	   1	      281      281
       fabs	  absolute value	  0	   0	      ?        ?
       fatan*	  inverse trig function   3		      64
       fcos*	  trig function 	  1		      87
       fcosh*	  hyperbolic function	  ?		      105
       fexp*	  exponential		  1		      79
       flog*	  natural logarithm	  1		      100
       floor	  integer   no	greater   0	   0	      ?        ?
       fsin*	  trig function 	  1		      68
       fsinh*	  hyperbolic function	  ?		      44
       fsqrt	  square root		  1		      95
       ftan*	  trig function 	  ?		      61

       ftanh*	  hyperbolic function	  ?		      116
       hypot	  Euclidean distance	  1	   1	      ?        ?
       j0	  bessel function	  ?	   ?	      ?        ?
       j1	  bessel function	  ?	   ?	      ?        ?
       jn	  bessel function	  ?	   ?	      ?        ?
       lgamma	  log gamma function	  ?	   ?	      ?        ?
       log*	  natural logarithm	  2	   1	      119      217
       logb	  exponent extraction	  0	   0	      ?        ?
       log10*	  logarithm to base 10	  3	   3	      ?        ?
       log1p	  log(1+x)		  1	   1	      269      269
       pow	  exponential x**y	  60-500   60-500     ?        ?
       rint	  round   to	nearest   0	   0	      ?        ?
       scalb	  exponent adjustment	  0	   0	      ?        ?
       sin*	  trig function 	  2	   1	      101      222
       sinh*	  hyperbolic function	  ?	   3	      79       292
       sqrt	  square root		  1	   1	      133      133
       tan*	  trig function 	  ?	   3	      92       287
       tanh*	  hyperbolic function	  ?	   3	      156      293
       y0	  bessel function	  ?	   ?	      ?        ?
       y1	  bessel function	  ?	   ?	      ?        ?
       yn	  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
       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.

       RISC machines conform to the IEEE Standard 754 for Binary Floating-Point Arithmetic, to which only the notes for IEEE floating-point  apply
       and are included here.

   BIEEE STANDARD 754 Floating-Point Arithmetic:
       This standard is on its way to becoming more widely adopted than any other design for computer arithmetic.

       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 prove-
       nance 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 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  x  !=	x then x is NaN; every other predicate (x > y, x = y, x < y, ...) is FALSE if NaN is involved.	Trichotomy is vio-
		     lated 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
		     Divide by Zero	 +-Infinity

		     Underflow		 Gradual Underflow
		     Inexact		 Rounded value

		     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 satisfac-
		     torily, 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.	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 under-
		  flows 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 mid-
		  dle 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 classi-
		  fies 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 execu-
		  tion 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

       Ideally,  every	programmer should be able conveniently to turn a debugged subprogram into one that appears atomic to its users.  But simu-
       lating 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 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.

       Exceptions on RISC machines:
	      The  exception  enables  and  the  flags	that  are raised when an exception occurs (as well as the rounding mode) are in the float-
	      ing-point control and status register.  This register can be read or written by the routines described on the man page fpc(3).  This
	      register's layout is described in the file <mips/fpu.h> in UMIPS-BSD releases and in <sys/fpu.h> in UMIPS-SYSV releases.
	      What  is	currently  available  is only the raw interface which was 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 SIGFPE 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 indi-
	      cates this in it's implementation field) then the instruction that caused the exception is in the floating-point exception  instruc-
	      tion  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  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 and

       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
       fpc(3), signal(3), emulate_branch(3)
       R2010 Floating Point Coprocessor Architecture
       R2360 Floating Point Board Product Description

       An  explanation of IEEE 754 and its proposed extension p854 was published in the IEEE magazine MICRO in August 1984 under the title "A Pro-
       posed Radix- and Word-length-independent Standard for Floating-point Arithmetic" by W. J. Cody et al.

       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.

								       RISC								  math(3m)
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