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28 .\" from: @(#)ieee.3 6.4 (Berkeley) 5/6/91
36 .Nd IEEE standard 754 for floating-point arithmetic
38 The IEEE Standard 754 for Binary Floating-Point Arithmetic
39 defines representations of floating-point numbers and abstract
40 properties of arithmetic operations relating to precision,
41 rounding, and exceptional cases, as described below.
42 .Ss IEEE STANDARD 754 Floating-Point Arithmetic
45 Overflow and underflow:
46 .Bd -ragged -offset indent -compact
47 Overflow goes by default to a signed \*(If.
52 Zero is represented ambiguously as +0 or \-0.
53 .Bd -ragged -offset indent -compact
54 Its sign transforms correctly through multiplication or
55 division, and is preserved by addition of zeros
56 with like signs; but x\-x yields +0 for every
58 The only operations that reveal zero's
59 sign are division by zero and
60 .Fn copysign x \(+-0 .
61 In particular, comparison (x > y, x \(>= y, etc.)\&
62 cannot be affected by the sign of zero; but if
63 finite x = y then \*(If = 1/(x\-y) \(!= \-1/(y\-x) = \-\*(If.
67 .Bd -ragged -offset indent -compact
68 It persists when added to itself
69 or to any finite number.
71 correctly through multiplication and division, and
72 (finite)/\(+-\*(If\0=\0\(+-0
73 (nonzero)/0 = \(+-\*(If.
75 \*(If\-\*(If, \*(If\(**0 and \*(If/\*(If
76 are, like 0/0 and sqrt(\-3),
77 invalid operations that produce \*(Na. ...
80 Reserved operands (\*(Nas):
81 .Bd -ragged -offset indent -compact
83 .Em ( N Ns ot Em a N Ns umber ) .
84 Some \*(Nas, called Signaling \*(Nas, trap any floating-point operation
85 performed upon them; they are used to mark missing
86 or uninitialized values, or nonexistent elements
88 The rest are Quiet \*(Nas; they are
89 the default results of Invalid Operations, and
90 propagate through subsequent arithmetic operations.
91 If x \(!= x then x is \*(Na; every other predicate
92 (x > y, x = y, x < y, ...) is FALSE if \*(Na is involved.
96 .Bd -ragged -offset indent -compact
97 Every algebraic operation (+, \-, \(**, /,
99 is rounded by default to within half an
101 and when the rounding error is exactly half an
104 the rounded value's least significant bit is zero.
112 This kind of rounding is usually the best kind,
113 sometimes provably so; for instance, for every
114 x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
115 (x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ...
116 despite that both the quotients and the products
118 Only rounding like IEEE 754 can do that.
119 But no single kind of rounding can be
120 proved best for every circumstance, so IEEE 754
121 provides rounding towards zero or towards
122 +\*(If or towards \-\*(If
123 at the programmer's option.
127 .Bd -ragged -offset indent -compact
128 IEEE 754 recognizes five kinds of floating-point exceptions,
129 listed below in declining order of probable importance.
130 .Bl -column -offset indent "Invalid Operation" "Gradual Underflow"
131 .Em "Exception Default Result"
132 Invalid Operation \*(Na, or FALSE
134 Divide by Zero \(+-\*(If
135 Underflow Gradual Underflow
136 Inexact Rounded value
139 NOTE: An Exception is not an Error unless handled
141 What makes a class of exceptions exceptional
142 is that no single default response can be satisfactory
144 On the other hand, if a default
145 response will serve most instances satisfactorily,
146 the unsatisfactory instances cannot justify aborting
147 computation every time the exception occurs.
151 .Bd -ragged -offset indent -compact
157 Precision: 24 significant bits,
158 roughly like 7 significant decimals.
160 If x and x' are consecutive positive single-precision
161 numbers (they differ by 1
164 .Bl -column "XXX" -compact
165 5.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07.
168 .Bl -column "XXX" -compact
169 Range: Overflow threshold = 2.0**128 = 3.4e38
170 Underflow threshold = 0.5**126 = 1.2e\-38
173 Underflowed results round to the nearest
175 .Bl -column "XXX" -compact
181 .Bd -ragged -offset indent -compact
184 .Po On some architectures,
192 Precision: 53 significant bits,
193 roughly like 16 significant decimals.
195 If x and x' are consecutive positive double-precision
196 numbers (they differ by 1
199 .Bl -column "XXX" -compact
200 1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
203 .Bl -column "XXX" -compact
204 Range: Overflow threshold = 2.0**1024 = 1.8e308
205 Underflow threshold = 0.5**1022 = 2.2e\-308
208 Underflowed results round to the nearest
210 .Bl -column "XXX" -compact
211 0.5**1074 = 4.9e\-324.
216 .Bd -ragged -offset indent -compact
219 (when supported by the hardware)
223 Precision: 64 significant bits,
224 roughly like 19 significant decimals.
226 If x and x' are consecutive positive extended-precision
227 numbers (they differ by 1
230 .Bl -column "XXX" -compact
231 1.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19.
234 .Bl -column "XXX" -compact
235 Range: Overflow threshold = 2.0**16384 = 1.2e4932
236 Underflow threshold = 0.5**16382 = 3.4e\-4932
239 Underflowed results round to the nearest
241 .Bl -column "XXX" -compact
242 0.5**16445 = 5.7e\-4953.
246 Quad-extended-precision:
247 .Bd -ragged -offset indent -compact
250 (when supported by the hardware)
254 Precision: 113 significant bits,
255 roughly like 34 significant decimals.
257 If x and x' are consecutive positive quad-extended-precision
258 numbers (they differ by 1
261 .Bl -column "XXX" -compact
262 9.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34.
265 .Bl -column "XXX" -compact
266 Range: Overflow threshold = 2.0**16384 = 1.2e4932
267 Underflow threshold = 0.5**16382 = 3.4e\-4932
270 Underflowed results round to the nearest
272 .Bl -column "XXX" -compact
273 0.5**16494 = 6.5e\-4966.
276 .Ss Additional Information Regarding Exceptions
277 For each kind of floating-point exception, IEEE 754
278 provides a Flag that is raised each time its exception
279 is signaled, and stays raised until the program resets
281 Programs may also test, save and restore a flag.
282 Thus, IEEE 754 provides three ways by which programs
283 may cope with exceptions for which the default result
284 might be unsatisfactory:
287 Test for a condition that might cause an exception
288 later, and branch to avoid the exception.
290 Test a flag to see whether an exception has occurred
291 since the program last reset its flag.
293 Test a result to see whether it is a value that only
294 an exception could have produced.
296 CAUTION: The only reliable ways to discover
297 whether Underflow has occurred are to test whether
298 products or quotients lie closer to zero than the
299 underflow threshold, or to test the Underflow
301 (Sums and differences cannot underflow in
302 IEEE 754; if x \(!= y then x\-y is correct to
303 full precision and certainly nonzero regardless of
305 Products and quotients that
306 underflow gradually can lose accuracy gradually
307 without vanishing, so comparing them with zero
308 (as one might on a VAX) will not reveal the loss.
309 Fortunately, if a gradually underflowed value is
310 destined to be added to something bigger than the
311 underflow threshold, as is almost always the case,
312 digits lost to gradual underflow will not be missed
313 because they would have been rounded off anyway.
314 So gradual underflows are usually
317 The same cannot be said of underflows flushed to 0.
320 At the option of an implementor conforming to IEEE 754,
321 other ways to cope with exceptions may be provided:
325 This mechanism classifies an exception in
326 advance as an incident to be handled by means
327 traditionally associated with error-handling
328 statements like "ON ERROR GO TO ...".
330 languages offer different forms of this statement,
331 but most share the following characteristics:
334 No means is provided to substitute a value for
335 the offending operation's result and resume
336 computation from what may be the middle of an
338 An exceptional result is abandoned.
340 In a subprogram that lacks an error-handling
341 statement, an exception causes the subprogram to
342 abort within whatever program called it, and so
343 on back up the chain of calling subprograms until
344 an error-handling statement is encountered or the
345 whole task is aborted and memory is dumped.
349 This mechanism, requiring an interactive
350 debugging environment, is more for the programmer
352 It classifies an exception in
353 advance as a symptom of a programmer's error; the
354 exception suspends execution as near as it can to
355 the offending operation so that the programmer can
356 look around to see how it happened.
358 the first several exceptions turn out to be quite
359 unexceptionable, so the programmer ought ideally
360 to be able to resume execution after each one as if
361 execution had not been stopped.
363 \&... Other ways lie beyond the scope of this document.
367 elementary function should act as if it were indivisible, or
368 atomic, in the sense that ...
371 No exception should be signaled that is not deserved by
372 the data supplied to that function.
374 Any exception signaled should be identified with that
375 function rather than with one of its subroutines.
377 The internal behavior of an atomic function should not
378 be disrupted when a calling program changes from
379 one to another of the five or so ways of handling
380 exceptions listed above, although the definition
381 of the function may be correlated intentionally
382 with exception handling.
387 are only approximately atomic.
388 They signal no inappropriate exception except possibly ...
389 .Bl -tag -width indent -offset indent -compact
393 when a result, if properly computed, might have lain barely within range, and
403 when it happens to be exact, thanks to fortuitous cancellation of errors.
406 .Bl -tag -width indent -offset indent -compact
408 Invalid Operation is signaled only when
410 any result but \*(Na would probably be misleading.
412 Overflow is signaled only when
414 the exact result would be finite but beyond the overflow threshold.
416 Divide-by-Zero is signaled only when
418 a function takes exactly infinite values at finite operands.
420 Underflow is signaled only when
422 the exact result would be nonzero but tinier than the underflow threshold.
424 Inexact is signaled only when
426 greater range or precision would be needed to represent the exact result.
433 An explanation of IEEE 754 and its proposed extension p854
434 was published in the IEEE magazine MICRO in August 1984 under
435 the title "A Proposed Radix- and Word-length-independent
436 Standard for Floating-point Arithmetic" by
439 The manuals for Pascal, C and BASIC on the Apple Macintosh
440 document the features of IEEE 754 pretty well.
441 Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\&
442 1981), and in the ACM SIGNUM Newsletter Special Issue of
443 Oct.\& 1979, may be helpful although they pertain to
444 superseded drafts of the standard.