math man page on OpenBSD

Man page or keyword search:  
man Server   11362 pages
apropos Keyword Search (all sections)
Output format
OpenBSD logo
[printable version]

MATH(3)			  OpenBSD Programmer's Manual		       MATH(3)

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>.

LIST OF FUNCTIONS
     Name	    Description			      ULPs
     acos(3)	    inverse trigonometric function    3
     acosh(3)	    inverse hyperbolic function	      3
     asin(3)	    inverse trigonometric function    3
     asinh(3)	    inverse hyperbolic function	      3
     atan(3)	    inverse trigonometric function    1
     atan2(3)	    inverse trigonometric function    2
     atanh(3)	    inverse hyperbolic function	      3
     cabs(3)	    complex absolute value	      1
     cbrt(3)	    cube root			      1
     ceil(3)	    integer no less than	      0
     copysign(3)    copy sign bit		      0
     cos(3)	    trigonometric function	      1
     cosh(3)	    hyperbolic function		      3
     erf(3)	    error function		      1
     erfc(3)	    complementary error function      1
     exp(3)	    exponential			      1
     expm1(3)	    exp(x)-1			      1
     fabs(3)	    absolute value		      0
     floor(3)	    integer no greater than	      0
     fmod(3)	    remainder			      0
     fpclassify(3)  classify real floating type	      0
     hypot(3)	    Euclidean distance		      1
     ilogb(3)	    exponent extraction		      0
     isfinite(3)    test for finite value	      0
     isinf(3)	    check for infinity		      0
     isnan(3)	    check for not-a-number	      0
     isnormal(3)    test for normal value	      0
     j0(3)	    Bessel function		      ???
     j1(3)	    Bessel function		      ???
     jn(3)	    Bessel function		      ???
     lgamma(3)	    log gamma function		      1
     log(3)	    natural logarithm		      1
     log10(3)	    logarithm to base 10	      3
     log1p(3)	    log(1+x)			      1
     nan(3)	    generate NaN		      0
     nextafter(3)   next representable number	      0
     pow(3)	    exponential x**y		      60-500
     remainder(3)   remainder			      0
     remquo(3)	    remainder			      0
     rint(3)	    round to nearest integer	      0
     round(3)	    round to nearest integer	      0
     scalbn(3)	    exponent adjustment		      0
     signbit(3)	    test sign			      0
     sin(3)	    trigonometric function	      1
     sinh(3)	    hyperbolic function		      3
     sqrt(3)	    square root			      1
     tan(3)	    trigonometric function	      3
     tanh(3)	    hyperbolic function		      3
     tgamma(3)	    gamma function		      4
     trunc(3)	    nearest integral value	      3
     y0(3)	    Bessel function		      ???
     y1(3)	    Bessel function		      ???
     yn(3)	    Bessel function		      ???

NOTES
     In 4.3BSD, 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 IEEE Std 754-1985.  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 number of ulps tabulated above; a
     ulp is one Unit in the Last Place.	 The functions 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 following 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 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 significant bits, roughly 17 significant decimal
		       digits.	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 a ulp, and
		       when the rounding error is exactly half a 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 4.3BSD.  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.	See infnan(3) for the present
	   state of affairs.

     How do the functions in 4.3BSD-'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
     implementations interpolate in large table to achieve speed and accuracy;
     the libm implementations use tricky formulas compact enough that all of
     them may some day fit into a ROM.

     More importantly, DEC considers the VMS implementation proprietary and
     guards it zealously against unauthorized use.  In contrast, the libm
     included in 4.3BSD is freely distributable; it may be copied freely
     provided their provenance is always acknowledged.	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 is the most widely adopted standard 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 for 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 Std 754-1985 without, alas, adhering to the
     standard's method of handling rounding and exceptions such as
     over/underflow.  The DEC VAX G_floating-point format is very similar to
     IEEE Std 754-1985 Double format.  It is 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 code in 4.3BSD-'s libm for machines that conform to IEEE Std 754-1985
     is intended primarily for the National Semi. 32081 and WTL 1164/65.  To
     use this code with the Intel or Zilog chips, or with the Apple Macintosh
     or ELXSI 6400, is to forego the use of better code provided (perhaps for
     free) by those companies and designed by some of the authors of the code
     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 is faster and more accurate to boot; it,
     Apple, the i8087, Z8070 and WE32106 all use 64 significant bits.  The
     main virtue of 4.3BSD-'s libm is that it is freely distributable; it may
     be copied freely provided its provenance is always acknowledged.
     Therefore no user of UNIX on a machine that conforms to IEEE Std 754-1985
     need use anything worse than the new libm.

     Properties of IEEE Std 754-1985 Double-Precision:
	   Wordsize:   64 bits, 8 bytes.
	   Radix:      Binary.
	   Precision:  53 significant bits, roughly equivalent to 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, 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 a ulp, and when the rounding
		       error is exactly half a 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
		       Std 754-1985 can do that.  But no single kind of
		       rounding can be proved best for every circumstance, so
		       IEEE Std 754-1985 provides rounding towards zero or
		       towards +infinity or towards -infinity at the
		       programmer's discretion.	 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 Std 754-1985 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 Std 754-1985 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 Std 754-1985 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 Std 754-1985; 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 Std 754-1985, 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.  Often times 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 4.3BSD-'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 to conveniently 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.

     Properties of IEEE Std 754-1985 Single-Precision:
	   Wordsize:   32 bits, 4 bytes.
	   Radix:      Binary.
	   Precision:  24 significant bits, roughly equivalent to 7
		       significant decimals.
		       If x and x' are consecutive positive Double-Precision
		       numbers (they differ by 1 ulp, then
		       6.0e-8 < 0.5**24 < (x'-x)/x <= 0.5**23 < 1.2e-7.
	   Range:      Overflow threshold = 2.0**128 = 3.4e38.
		       Underflow threshold = 0.5**126 = 1.2e-38
		       Overflow goes by default to a signed infinity.
		       Underflow is Gradual, rounding to the nearest integer
		       multiple of 0.5**149 = 1.4e-45.
	   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**24-2 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 a ulp, and when the rounding
		       error is exactly half a 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
		       Std 754-1985 can do that.  But no single kind of
		       rounding can be proved best for every circumstance, so
		       IEEE Std 754-1985 provides rounding towards zero or
		       towards +infinity or towards -infinity at the
		       programmer's discretion.	 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 Std 754-1985 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.

SEE ALSO
     An explanation of IEEE Std 754-1985 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 Std 754-1985 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.

BUGS
     When signals are appropriate, they are emitted by certain operations
     within libm, so a subroutine-trace may be needed to identify the function
     with its signal in case method 5) above is in use.	 All the code in libm
     takes the IEEE Std 754-1985 defaults for granted; this means that a
     decision to trap all divisions by zero could disrupt a function that
     would otherwise get a correct result despite division by zero.

OpenBSD 4.9		       February 20, 2010		   OpenBSD 4.9
[top]

List of man pages available for OpenBSD

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]
Tweet
Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
Vote for polarhome
Free Shell Accounts :: the biggest list on the net