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diff --git a/libm/man/ieee.3 b/libm/man/ieee.3 new file mode 100644 index 0000000..3b7087d --- /dev/null +++ b/libm/man/ieee.3 @@ -0,0 +1,448 @@ +.\" Copyright (c) 1985 Regents of the University of California. +.\" All rights reserved. +.\" +.\" Redistribution and use in source and binary forms, with or without +.\" modification, are permitted provided that the following conditions +.\" are met: +.\" 1. Redistributions of source code must retain the above copyright +.\" notice, this list of conditions and the following disclaimer. +.\" 2. Redistributions in binary form must reproduce the above copyright +.\" notice, this list of conditions and the following disclaimer in the +.\" documentation and/or other materials provided with the distribution. +.\" 3. All advertising materials mentioning features or use of this software +.\" must display the following acknowledgement: +.\" This product includes software developed by the University of +.\" California, Berkeley and its contributors. +.\" 4. Neither the name of the University nor the names of its contributors +.\" may be used to endorse or promote products derived from this software +.\" without specific prior written permission. +.\" +.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND +.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE +.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS +.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) +.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT +.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY +.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF +.\" SUCH DAMAGE. +.\" +.\" from: @(#)ieee.3 6.4 (Berkeley) 5/6/91 +.\" $FreeBSD: src/lib/msun/man/ieee.3,v 1.22 2005/06/16 21:55:45 ru Exp $ +.\" +.Dd January 26, 2005 +.Dt IEEE 3 +.Os +.Sh NAME +.Nm ieee +.Nd IEEE standard 754 for floating-point arithmetic +.Sh DESCRIPTION +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. +.Ss IEEE STANDARD 754 Floating-Point Arithmetic +Radix: Binary. +.Pp +Overflow and underflow: +.Bd -ragged -offset indent -compact +Overflow goes by default to a signed \*(If. +Underflow is +.Em gradual . +.Ed +.Pp +Zero is represented ambiguously as +0 or \-0. +.Bd -ragged -offset indent -compact +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 +.Fn 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 \*(If = 1/(x\-y) \(!= \-1/(y\-x) = \-\*(If. +.Ed +.Pp +Infinity is signed. +.Bd -ragged -offset indent -compact +It persists when added to itself +or to any finite number. +Its sign transforms +correctly through multiplication and division, and +(finite)/\(+-\*(If\0=\0\(+-0 +(nonzero)/0 = \(+-\*(If. +But +\*(If\-\*(If, \*(If\(**0 and \*(If/\*(If +are, like 0/0 and sqrt(\-3), +invalid operations that produce \*(Na. ... +.Ed +.Pp +Reserved operands (\*(Nas): +.Bd -ragged -offset indent -compact +An \*(Na is +.Em ( N Ns ot Em a N Ns umber ) . +Some \*(Nas, called Signaling \*(Nas, 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 \*(Nas; they are +the default results of Invalid Operations, and +propagate through subsequent arithmetic operations. +If x \(!= x then x is \*(Na; every other predicate +(x > y, x = y, x < y, ...) is FALSE if \*(Na is involved. +.Ed +.Pp +Rounding: +.Bd -ragged -offset indent -compact +Every algebraic operation (+, \-, \(**, /, +\(sr) +is rounded by default to within half an +.Em ulp , +and when the rounding error is exactly half an +.Em ulp +then +the rounded value's least significant bit is zero. +(An +.Em ulp +is one +.Em U Ns nit +in the +.Em L Ns ast +.Em P Ns lace . ) +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 ++\*(If or towards \-\*(If +at the programmer's option. +.Ed +.Pp +Exceptions: +.Bd -ragged -offset indent -compact +IEEE 754 recognizes five kinds of floating-point exceptions, +listed below in declining order of probable importance. +.Bl -column -offset indent "Invalid Operation" "Gradual Underflow" +.Em "Exception Default Result" +Invalid Operation \*(Na, or FALSE +Overflow \(+-\*(If +Divide by Zero \(+-\*(If +Underflow Gradual Underflow +Inexact Rounded value +.El +.Pp +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. +.Ed +.Ss Data Formats +Single-precision: +.Bd -ragged -offset indent -compact +Type name: +.Vt float +.Pp +Wordsize: 32 bits. +.Pp +Precision: 24 significant bits, +roughly like 7 significant decimals. +.Bd -ragged -offset indent -compact +If x and x' are consecutive positive single-precision +numbers (they differ by 1 +.Em ulp ) , +then +.Bd -ragged -compact +5.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07. +.Ed +.Ed +.Pp +.Bl -column "XXX" -compact +Range: Overflow threshold = 2.0**128 = 3.4e38 + Underflow threshold = 0.5**126 = 1.2e\-38 +.El +.Bd -ragged -offset indent -compact +Underflowed results round to the nearest +integer multiple of 0.5**149 = 1.4e\-45. +.Ed +.Ed +.Pp +Double-precision: +.Bd -ragged -offset indent -compact +Type name: +.Vt double +.Bd -ragged -offset indent -compact +On some architectures, +.Vt long double +is the the same as +.Vt double . +.Ed +.Pp +Wordsize: 64 bits. +.Pp +Precision: 53 significant bits, +roughly like 16 significant decimals. +.Bd -ragged -offset indent -compact +If x and x' are consecutive positive double-precision +numbers (they differ by 1 +.Em ulp ) , +then +.Bd -ragged -compact +1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16. +.Ed +.Ed +.Pp +.Bl -column "XXX" -compact +Range: Overflow threshold = 2.0**1024 = 1.8e308 + Underflow threshold = 0.5**1022 = 2.2e\-308 +.El +.Bd -ragged -offset indent -compact +Underflowed results round to the nearest +integer multiple of 0.5**1074 = 4.9e\-324. +.Ed +.Ed +.Pp +Extended-precision: +.Bd -ragged -offset indent -compact +Type name: +.Vt long double +(when supported by the hardware) +.Pp +Wordsize: 96 bits. +.Pp +Precision: 64 significant bits, +roughly like 19 significant decimals. +.Bd -ragged -offset indent -compact +If x and x' are consecutive positive double-precision +numbers (they differ by 1 +.Em ulp ) , +then +.Bd -ragged -compact +1.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19. +.Ed +.Ed +.Pp +.Bl -column "XXX" -compact +Range: Overflow threshold = 2.0**16384 = 1.2e4932 + Underflow threshold = 0.5**16382 = 3.4e\-4932 +.El +.Bd -ragged -offset indent -compact +Underflowed results round to the nearest +integer multiple of 0.5**16445 = 5.7e\-4953. +.Ed +.Ed +.Pp +Quad-extended-precision: +.Bd -ragged -offset indent -compact +Type name: +.Vt long double +(when supported by the hardware) +.Pp +Wordsize: 128 bits. +.Pp +Precision: 113 significant bits, +roughly like 34 significant decimals. +.Bd -ragged -offset indent -compact +If x and x' are consecutive positive double-precision +numbers (they differ by 1 +.Em ulp ) , +then +.Bd -ragged -compact +9.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34. +.Ed +.Ed +.Pp +.Bl -column "XXX" -compact +Range: Overflow threshold = 2.0**16384 = 1.2e4932 + Underflow threshold = 0.5**16382 = 3.4e\-4932 +.El +.Bd -ragged -offset indent -compact +Underflowed results round to the nearest +integer multiple of 0.5**16494 = 6.5e\-4966. +.Ed +.Ed +.Ss Additional Information Regarding Exceptions +.Pp +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: +.Bl -enum +.It +Test for a condition that might cause an exception +later, and branch to avoid the exception. +.It +Test a flag to see whether an exception has occurred +since the program last reset its flag. +.It +Test a result to see whether it is a value that only +an exception could have produced. +.Pp +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 +.Em provably +ignorable. +The same cannot be said of underflows flushed to 0. +.El +.Pp +At the option of an implementor conforming to IEEE 754, +other ways to cope with exceptions may be provided: +.Bl -enum +.It +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: +.Bl -dash +.It +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. +.It +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. +.El +.It +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. +.It +\&... Other ways lie beyond the scope of this document. +.El +.Pp +Ideally, each +elementary function should act as if it were indivisible, or +atomic, in the sense that ... +.Bl -enum +.It +No exception should be signaled that is not deserved by +the data supplied to that function. +.It +Any exception signaled should be identified with that +function rather than with one of its subroutines. +.It +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. +.El +.Pp +The functions in +.Nm libm +are only approximately atomic. +They signal no inappropriate exception except possibly ... +.Bl -tag -width indent -offset indent -compact +.It Xo +Over/Underflow +.Xc +when a result, if properly computed, might have lain barely within range, and +.It Xo +Inexact in +.Fn cabs , +.Fn cbrt , +.Fn hypot , +.Fn log10 +and +.Fn pow +.Xc +when it happens to be exact, thanks to fortuitous cancellation of errors. +.El +Otherwise, ... +.Bl -tag -width indent -offset indent -compact +.It Xo +Invalid Operation is signaled only when +.Xc +any result but \*(Na would probably be misleading. +.It Xo +Overflow is signaled only when +.Xc +the exact result would be finite but beyond the overflow threshold. +.It Xo +Divide-by-Zero is signaled only when +.Xc +a function takes exactly infinite values at finite operands. +.It Xo +Underflow is signaled only when +.Xc +the exact result would be nonzero but tinier than the underflow threshold. +.It Xo +Inexact is signaled only when +.Xc +greater range or precision would be needed to represent the exact result. +.El +.Sh SEE ALSO +.Xr fenv 3 , +.Xr ieee_test 3 , +.Xr math 3 +.Pp +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 +.An "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. +.Sh STANDARDS +.St -ieee754 |