Skip site navigation (1)Skip section navigation (2)

FreeBSD Man Pages

Man Page or Keyword Search:
Man Architecture
Apropos Keyword Search (all sections) Output format
home | help
IEEE(3)                FreeBSD Library Functions Manual                IEEE(3)

     ieee - IEEE standard 754 for floating-point arithmetic

     The IEEE Standard 754 for Binary Floating-Point Arithmetic defines
     representations of floating-point numbers and abstract properties of
     arithmetic operations relating to precision, rounding, and exceptional
     cases, as described below.

   IEEE STANDARD 754 Floating-Point Arithmetic
     Radix: Binary.

     Overflow and underflow:
           Overflow goes by default to a signed infinity.  Underflow is

     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 (NaNs):
           An NaN is (Not a Number).  Some NaNs, 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.

           Every algebraic operation (+, -, *, /, \/) 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.  (An
           ulp is one Unit in the Last Place.) 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.

           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             +-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.

   Data Formats
           Type name: float

           Wordsize: 32 bits.

           Precision: 24 significant bits, roughly like 7 significant
                 If x and x' are consecutive positive single-precision numbers
                 (they differ by 1 ulp), then
                 5.9e-08 < 0.5**24 < (x'-x)/x <= 0.5**23 < 1.2e-07.

           Range: Overflow threshold = 2.0**128 = 3.4e38
                  Underflow threshold = 0.5**126 = 1.2e-38
                 Underflowed results round to the nearest integer multiple of
                 0.5**149 = 1.4e-45.

           Type name: double
                 On some architectures, long double is the the same as double.

           Wordsize: 64 bits.

           Precision: 53 significant bits, roughly like 16 significant
                 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
                 Underflowed results round to the nearest integer multiple of
                 0.5**1074 = 4.9e-324.

           Type name: long double (when supported by the hardware)

           Wordsize: 96 bits.

           Precision: 64 significant bits, roughly like 19 significant
                 If x and x' are consecutive positive double-precision numbers
                 (they differ by 1 ulp), then
                 1.0e-19 < 0.5**63 < (x'-x)/x <= 0.5**62 < 2.2e-19.

           Range: Overflow threshold = 2.0**16384 = 1.2e4932
                  Underflow threshold = 0.5**16382 = 3.4e-4932
                 Underflowed results round to the nearest integer multiple of
                 0.5**16445 = 5.7e-4953.

           Type name: long double (when supported by the hardware)

           Wordsize: 128 bits.

           Precision: 113 significant bits, roughly like 34 significant
                 If x and x' are consecutive positive double-precision numbers
                 (they differ by 1 ulp), then
                 9.6e-35 < 0.5**113 < (x'-x)/x <= 0.5**112 < 2.0e-34.

           Range: Overflow threshold = 2.0**16384 = 1.2e4932
                  Underflow threshold = 0.5**16382 = 3.4e-4932
                 Underflowed results round to the nearest integer multiple of
                 0.5**16494 = 6.5e-4966.

   Additional Information Regarding Exceptions
     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:

     1.   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.

     2.   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

     3.   ... Other ways lie beyond the scope of this document.

     Ideally, each elementary function should act as if it were indivisible,
     or atomic, in the sense that ...

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

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

     3.   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

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

     fenv(3), ieee_test(3), math(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.

     IEEE Std 754-1985

FreeBSD 11.0-PRERELEASE        January 26, 2005        FreeBSD 11.0-PRERELEASE


Want to link to this manual page? Use this URL:

home | help