MATH(3)                                                                MATH(3)

       math - introduction to mathematical library functions

       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>.

       Name      Appears on Page   Description               Error Bound (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        atan.3      inverse trigonometric function      1
       atanh       atanh.3     inverse hyperbolic function         3
       atan2       atan2.3     inverse trigonometric function      2
       cbrt        sqrt.3      cube root                           1
       ceil        ceil.3      integer no less than                0
       copysign    ieee.3      copy sign bit                       0
       cos         cos.3       trigonometric function              1
       cosh        cosh.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
       finite      ieee.3      test for finity                     0
       floor       floor.3     integer no greater than             0
       fmod        fmod.3      remainder                          ???
       hypot       hypot.3     Euclidean distance                  1
       ilogb       ieee.3      exponent extraction                 0
       isinf       isinf.3     test for infinity                   0
       isnan       isnan.3     test for not-a-number               0
       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
       nan         nan.3       return quiet NaN                    0
       nextafter   ieee.3      next representable number           0
       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
       trunc       trunc.3     nearest integral value              3
       y0          j0.3        Bessel function                    ???
       y1          j0.3        Bessel function                    ???
       yn          j0.3        Bessel function                    ???

       Name        Value                     Description
       M_E         2.7182818284590452354     e
       M_LOG2E     1.4426950408889634074     log 2e
       M_LOG10E    0.43429448190325182765    log 10e
       M_LN2       0.69314718055994530942    log e2
       M_LN10      2.30258509299404568402    log e10
       M_PI        3.14159265358979323846    pi
       M_PI_2      1.57079632679489661923    pi/2
       M_PI_4      0.78539816339744830962    pi/4
       M_1_PI      0.31830988618379067154    1/pi
       M_2_PI      0.63661977236758134308    2/pi
       M_2_SQRTPI  1.12837916709551257390    2/sqrt(pi)
       M_SQRT2     1.41421356237309504880    sqrt(2)
       M_SQRT1_2   0.70710678118654752440    1/sqrt(2)

       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  dou-
       ble-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 simi-
       larly, 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  devel-
       oped,  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 com-
       puter 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 puritani-
       cal 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  stan-
       dard'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 Macin-
       tosh or ELXSI 6400, is to forego the use of better codes provided (per-
       haps  freely) by those companies and designed by some of the authors of
       the codes above.  Except for  atan,  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 Infin-
                     ity/Infinity  are,  like 0/0 and sqrt(-3), invalid opera-
                     tions 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 sub-
                     sequent 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  excep-
                     tions, listed below in declining order of probable impor-
                            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  Under-
                  flow  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.  For-
                  tunately, 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 ignor-
                  able.  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 aban-

              --  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 pro-
                  grammer'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 indi-
       visible, 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  dis-
             rupted  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 cbrt, hypot, log10 and pow
                     when it happens to be exact, thanks to fortuitous cancel-
                     lation 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  oper-
              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.

       An  explanation  of  IEEE  754 and its proposed extension p854 was pub-
       lished 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.

       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.

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