math - introduction to mathematical library functions

     These functions constitute the C math  library,  libm.  This  library  is
     based  Sun  Microsystems  FDLIBM  (Freely  Distributable  LIBM), a C math
     library for machines that support  IEEE  754  floating-point  arithmetic.
     Declarations  for  the functions in this library may be obtained from the
     include file <math.h>.


     Name       Appears on Page   Description

     acos         sin(3)      inverse trigonometric function
     acosh        asinh(3)    inverse hyperbolic function
     asin         sin(3)      inverse trigonometric function
     asinh        asinh(3)    inverse hyperbolic function
     atan         sin(3)      inverse trigonometric function
     atanh        asinh(3)    inverse hyperbolic function
     atan2        sin(3)      inverse trigonometric function
     cbrt         sqrt(3)     cube root
     ceil         floor(3)    integer no less than
     copysign     ieee(3)     copy sign bit
     cos          sin(3)      trigonometric function
     cosh         sinh(3)     hyperbolic function
     remainder    ieee(3)     remainder
     erf          erf(3)      error function
     erfc         erf(3)      complementary error function
     exp          exp(3)      exponential
     expm1        exp(3)      exp(x)-1
     fabs         floor(3)    absolute value
     floor        floor(3)    integer no greater than
     hypot        hypot(3)    Euclidean distance
     ilogb        ieee(3)     exponent extraction
     j0           j0(3)       bessel function
     j1           j0(3)       bessel function
     jn           j0(3)       bessel function
     log          exp(3)      natural logarithm
     logb         ieee(3)     exponent extraction
     log10        exp(3)      logarithm to base 10
     log1p        exp(3)      log(1+x)
     pow          exp(3)      exponential x**y
     rint         floor(3)    round to nearest integer
     scalb        ieee(3)     exponent adjustment
     scalbn       ieee(3)     exponent adjustment
     sin          sin(3)      trigonometric function
     sinh         sinh(3)     hyperbolic function
     sqrt         sqrt(3)     square root
     tan          sin(3)      trigonometric function
     tanh         sinh(3)     hyperbolic function
     y0           j0(3)       bessel function
     y1           j0(3)       bessel function
     yn           j0(3)       bessel function

     From the FDLIBM README

     FDLIBM (double precision version) assumes:
      a.  IEEE 754 style (if not precise compliance) arithmetic;
      b.  32 bit 2's complement integer arithmetic;
      c.  Each double precision floating-point number must be in IEEE 754
          double format, and that each number can be retrieved as two 32-bit

          Example: let y = 2.0
             double fp number y:     2.0
             IEEE double format:     0x4000000000000000

             Referencing y as two integers:
             *(int*)&y,*(1+(int*)&y) =       {0x40000000,0x0} (on sparc)
                                             {0x0,0x40000000} (on 386)

             Note: FDLIBM will detect, at run time, the correct ordering of
                   the high and low part of a floating-point number.

       d. IEEE exceptions may trigger "signals" as is common in Unix

     All exception cases in the FDLIBM functions will be mapped
     to one of the following four exceptions:

         +huge*huge, +tiny*tiny,     +1.0/0.0,       +0.0/0.0
         (overflow)  (underflow)  (divided-by-zero)  (invalid)

     For example, log(0) is a singularity and is thus mapped to
             -1.0/0.0 = -infinity.
     That is, FDLIBM's log will compute -one/zero and return the
     computed value.  On an IEEE machine, this will trigger the
     divided-by-zero exception and a negative infinity is returned by

     Similarly, exp(-huge) will be mapped to tiny*tiny to generate
     an underflow signal.

     IEEE STANDARD 754 Floating-Point Arithmetic:

     Properties of IEEE 754 Double-Precision:
          Wordsize: 64 bits, 8 bytes.  Radix: Binary.
          Precision: 53 significant bits, roughly 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 Inf.
               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 Inf = 1/(x-y) != -1/(y-x) = -Inf.
          Inf is signed.
               it persists when added to itself or to any finite number.   Its
               sign  transforms correctly through multiplication and division,
               and (finite)/+Inf = +0 (nonzero)/0 = +Inf.  But Inf-Inf,  Inf*0
               and Inf/Inf are, like 0/0 and sqrt(-3), invalid operations that
               produce NaN. ...
          Reserved operands:
               there are 2**53-2 of them,  all  called  NaN  (Not  a  Number).
               Some,  called Signaling NaNs, trap any floating-point operation
               performed  upon  them;  they  are  used  to  mark  missing   or
               uninitialized  values,  or nonexistent elements of arrays.  The
               rest are Quiet NaNs; they are the default  results  of  Invalid
               Operations,   and   propagate   through  subsequent  arithmetic
               operations.  If x != x then x is NaN; every other predicate  (x
               > y, x = y, x < y, ...) is FALSE if NaN is involved.
               NOTE: Trichotomy is violated by NaN.
                    Besides  being  FALSE,  predicates  that  entail   ordered
                    comparison,  rather than mere (in)equality, signal Invalid
                    Operation when NaN is involved.
               Every algebraic operation (+, -, *,  /,  sqrt)  is  rounded  by
               default  to  within half an ulp, and when the rounding error is
               exactly half an ulp then the rounded value's least  significant
               bit  is  zero.  This kind of rounding is usually the best kind,
               sometimes provably so; for instance, for every x  =  1.0,  2.0,
               3.0,   4.0,   ...,  2.0**52,  we  find  (x/3.0)*3.0  ==  x  and
               (x/10.0)*10.0 == x and ...  despite that both the quotients and
               the  products  have  been rounded.  Only rounding like IEEE 754
               can do that.  But no single kind of rounding can be proved best
               for every circumstance, so IEEE 754 provides  rounding  towards
               zero  or  towards  +Inf  or  towards  -Inf  at the programmer's
               option.  And the same  kinds  of  rounding  are  specified  for
               Binary-Decimal  Conversions,  at  least  for magnitudes between
               roughly 1.0e-10 and 1.0e37.
               IEEE 754 recognizes five kinds  of  floating-point  exceptions,
               listed below in declining order of probable importance.
                    Exception              Default Result

                    Invalid Operation      NaN, or FALSE
                    Overflow               +Inf
                    Divide by Zero         +Inf
                    Underflow              Gradual Underflow
                    Inexact                Rounded value
               NOTE:  An Exception is not an Error unless handled badly.  What
               makes  a  class  of  exceptions  exceptional  is that no single
               default response can be satisfactory in every instance.  On the
               other  hand,  if  a  default response will serve most instances
               satisfactorily, the  unsatisfactory  instances  cannot  justify
               aborting computation every time the exception occurs.

     For each kind of floating-point exception, IEEE 754 provides a Flag  that
     is raised each time its exception is signaled, and stays raised until the
     program resets it.  Programs may also test,  save  and  restore  a  flag.
     Thus,  IEEE  754  provides  three  ways  by  which programs may cope with
     exceptions for which the default result might be unsatisfactory:

     1)      Test for a condition that might cause  an  exception  later,  and
             branch to avoid the exception.

     2)      Test a flag to see whether an exception has  occurred  since  the
             program last reset its flag.

     3)      Test a result to see whether it is a value that only an exception
             could have produced.
                  CAUTION:  The  only  reliable  ways  to   discover   whether
                  Underflow  has  occurred  are  to  test  whether products or
                  quotients lie closer to zero than the  underflow  threshold,
                  or to test the Underflow flag.  (Sums and differences cannot
                  underflow in IEEE 754; if x != y then x-y is correct to full
                  precision  and  certainly  nonzero regardless of how tiny it
                  may be.)  Products and quotients  that  underflow  gradually
                  can  lose accuracy gradually without vanishing, so comparing
                  them with zero (as one might on a VAX) will not  reveal  the
                  loss.   Fortunately,  if  a  gradually  underflowed value is
                  destined to be added to something bigger than the  underflow
                  threshold,  as  is  almost  always  the case, digits lost to
                  gradual underflow will not be missed because they would have
                  been  rounded off anyway.  So gradual underflows are usually
                  provably ignorable.  The same cannot be said  of  underflows
                  flushed to 0.

     At the option of an implementor conforming to IEEE  754,  other  ways  to
     cope with exceptions may be provided:

     4)      ABORT.  This mechanism classifies an exception in advance  as  an
             incident  to  be  handled  by means traditionally associated with
             error-handling statements like "ON ERROR GO TO  ...".   Different
             languages offer different forms of this statement, but most share
             the following characteristics:

     --      No means is provided to substitute  a  value  for  the  offending
             operation's  result  and  resume computation from what may be the
             middle of an expression.  An exceptional result is abandoned.

     --      In a  subprogram  that  lacks  an  error-handling  statement,  an
             exception  causes the subprogram to abort within whatever program
             called it, and so on back up the  chain  of  calling  subprograms
             until  an  error-handling  statement  is encountered or the whole
             task is aborted and memory is dumped.

     5)      STOP.   This  mechanism,  requiring  an   interactive   debugging
             environment,  is  more  for  the programmer than the program.  It
             classifies an exception in advance as a symptom of a programmer's
             error;  the exception suspends execution as near as it can to the
             offending operation so that the programmer can look around to see
             how  it  happened.  Quite often the first several exceptions turn
             out to be quite unexceptionable, so the programmer ought  ideally
             to be able to resume execution after each one as if execution had
             not been stopped.

     6)      ... Other ways lie beyond the scope of this document.

     The crucial problem for exception handling is the problem of  Scope,  and
     the  problem's  solution  is  understood,  but  not  enough  manpower was
     available to implement it fully in time to be distributed  in  4.3  BSD's
     libm.   Ideally,  each  elementary  function  should  act  as  if it were
     indivisible, or atomic, in the sense that ...

     i)    No exception should be signaled that is not deserved  by  the  data
           supplied to that function.

     ii)   Any exception signaled should  be  identified  with  that  function
           rather than with one of its subroutines.

     iii)  The internal behavior of an atomic function should not be disrupted
           when  a  calling program changes from one to another of the five or
           so  ways  of  handling  exceptions  listed  above,   although   the
           definition  of  the  function  may be correlated intentionally with
           exception handling.

     Ideally, every programmer should be able conveniently to turn a  debugged
     subprogram into one that appears atomic to its users.  But simulating all
     three characteristics of an atomic function is still  a  tedious  affair,
     entailing  hosts  of  tests  and  saves-restores;  work  is  under way to
     ameliorate the inconvenience.

     Meanwhile, the functions in libm are  only  approximately  atomic.   They
     signal no inappropriate exception except possibly ...
               when a result, if properly computed,  might  have  lain  barely
               within range, and
          Inexact in cbrt, hypot, log10 and pow
               when it happens to be exact, thanks to fortuitous  cancellation
               of errors.
     Otherwise, ...
          Invalid Operation is signaled only when
               any result but NaN would probably be misleading.
          Overflow is signaled only when
               the exact result  would  be  finite  but  beyond  the  overflow
          Divide-by-Zero is signaled only when
               a function takes exactly infinite values at finite operands.
          Underflow is signaled only when
               the exact result would be nonzero but tinier than the underflow
          Inexact is signaled only when
               greater range or precision would be  needed  to  represent  the
               exact result.

     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.

     The math manual pages have been adapted from the 4.3BSD 3M  manual  pages
     for  FDLIBM  by  Kees  J. Bot <> who normally avoids floating
     point like the plague.  Some text may not apply to FDLIBM, but KJB didn't
     know  whether  to  remove  it  or  not.  Don't blame the original authors
     mentioned on these pages for inaccuracies introduced.

     asinh(3), erf(3), exp(3), floor(3),  hypot(3),  ieee(3),  j0(3),  sin(3),
     sinh(3), sqrt(3).

     An explanation of IEEE 754 and its proposed extension p854 was  published
     in  the  IEEE  magazine  MICRO in August 1984 under the title "A Proposed
     Radix-   and   Word-length-independent   Standard   for    Floating-point
     Arithmetic"  by W. J. Cody et al.  The manuals for Pascal, C and BASIC on
     the Apple Macintosh document  the  features  of  IEEE  754  pretty  well.
     Articles in the IEEE magazine COMPUTER vol. 14 no. 3 (Mar.  1981), and in
     the ACM SIGNUM Newsletter Special Issue of  Oct.  1979,  may  be  helpful
     although they pertain to superseded drafts of the standard.

     W. Kahan, with the help  of  Z-S.  Alex  Liu,  Stuart  I.  McDonald,  Dr.
     Kwok-Choi Ng, Peter Tang.