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        Manual       Description                       ULPs
     acos        acos(3)      inverse trigonometric function    3
     acosh       acosh(3)     inverse hyperbolic function       3
     asin        asin(3)      inverse trigonometric function    3
     asinh       asinh(3)     inverse hyperbolic function       3
     atan        tan(3)       inverse trigonometric function    1
     atanh       atanh(3)     inverse hyperbolic function       3
     atan2       tan2(3)      inverse trigonometric function    2
     cabs        hypot(3)     complex absolute value            1
     cbrt        sqrt(3)      cube root                         1
     ceil        floor(3)     integer no less than              0
     copysign    ieee(3)      copy sign bit                     0
     cos         sin(3)       trigonometric function            1
     cosh        sinh(3)      hyperbolic function               3
     erf         erf(3)       error function                    ???
     erfc        erf(3)       complementary error function      ???
     exp         exp(3)       exponential                       1
     expm1       exp(3)       exp(x)-1                          1
     fabs        fabs(3)      absolute value                    0
     floor       floor(3)     integer no greater than           0
     hypot       hypot(3)     Euclidean distance                1
     ilogb       ieee(3)      exponent extraction               0
     isinf       isinf(3)     check exceptions
     isnan       isnan(3)     check exceptions
     j0          j0(3)        Bessel function                   ???
     j1          j0(3)        Bessel function                   ???
     jn          j0(3)        Bessel function                   ???
     lgamma      lgamma(3)    log gamma function                ???
     log         exp(3)       natural logarithm                 1
     log10       exp(3)       logarithm to base 10              3
     log1p       exp(3)       log(1+x)                          1
     pow         exp(3)       exponential x**y                  60-500
     remainder   ieee(3)      remainder                         0
     rint        rint(3)      round to nearest integer          0
     scalbn      ieee(3)      exponent adjustment               0
     sin         sin(3)       trigonometric function            1
     sinh        sinh(3)      hyperbolic function               3
     sqrt        sqrt(3)      square root                       1
     tan         tan(3)       trigonometric function            3
     tanh        tanh(3)      hyperbolic function               3
     y0          j0(3)        Bessel function                   ???
     y1          j0(3)        Bessel function                   ???
     yn          j0(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, on-
     ly G double-precision.  (Why? Why not?)

     Properties of D_floating-point:
           Wordsize:   64 bits, 8 bytes.
           Radix:      Binary.
           Precision:  56 sig. bits, roughly 17 sig. 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 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
                       accepted by the hardware, only the obvious representa-
                       tion 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, customar-
                       ily stops computation, so they are not much used.
           Exceptions:
                       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.
           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 com-
     puter 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 arith-
     metic operation that over/underflowed would behave.  Similarly, expres-
     sions 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 computa-
     tion!  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 implementa-
     tions 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 in-
     cluded in 4.3BSD is freely distributable; it may be copied freely provid-
     ed 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 stan-
     dard'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 ver-
     sions 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 sig. 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 sig. bits, roughly equivalent to 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 num-
                       ber.  Its sign transforms correctly through multiplica-
                       tion and division, and (finite)/+-Infinity  = +-0
                       (nonzero)/0 = +-Infinity.  But Infinity-Infinity, In-
                       finity*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 Num-
                       ber).  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 propa-
                       gate 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 compari-
                              son, rather than mere (in)equality, signal In-
                              valid 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 sig. bit is zero.  This kind of rounding is usu-
                       ally 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 Std 754-1985 can
                       do that.  But no single kind of rounding can be proved
                       best for every circumstance, so IEEE Std 754-1985 pro-
                       vides rounding towards zero or towards +Infinity or to-
                       wards -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 bad-
                       ly.  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 re-
                       sponse will serve most instances satisfactorily, the
                       unsatisfactory instances cannot justify aborting compu-
                       tation 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 unsatisfacto-
     ry:

     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 pro-
          gram 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 ze-
          ro 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 re-
          gardless of how tiny it may be.)  Products and quotients that under-
          flow 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 al-
          most 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 in-
          cident 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 follow-
          ing characteristics:

          -   No means is provided to substitute a value for the offending op-
              eration'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 ex-
              ception 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 environ-
          ment, is more for the programmer than the program.  It classifies an
          exception in advance as a symptom of a programmer's error; the ex-
          ception suspends execution as near as it can to the offending opera-
          tion so that the programmer can look around to see how it happened.
          Often times the first several exceptions turn out to be quite unex-
          ceptionable, 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 avail-
     able 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 func-
                 tion 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 in-
                 tentionally 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 amelio-
     rate 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 cancella-
                   tion 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 un-
                   derflow 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 sig. bits, roughly equivalent to 7 sig. 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 num-
                       ber.  Its sign transforms correctly through multiplica-
                       tion and division, and (finite)/+-Infinity  = +-0
                       (nonzero)/0 = +-Infinity.  But Infinity-Infinity, In-
                       finity*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 Num-
                       ber).  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 propa-
                       gate 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 compari-
                              son, rather than mere (in)equality, signal In-
                              valid 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 sig. bit is zero.  This kind of rounding is usu-
                       ally 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 Std 754-1985 can
                       do that.  But no single kind of rounding can be proved
                       best for every circumstance, so IEEE Std 754-1985 pro-
                       vides rounding towards zero or towards +Infinity or to-
                       wards -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 bad-
                       ly.  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 re-
                       sponse will serve most instances satisfactorily, the
                       unsatisfactory instances cannot justify aborting compu-
                       tation 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 help-
     ful 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 deci-
     sion to trap all divisions by zero could disrupt a function that would
     otherwise get a correct result despite division by zero.

                                  May 6, 1991                                7
