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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.
159 .Bd -ragged -offset indent -compact
160 If x and x' are consecutive positive single-precision
161 numbers (they differ by 1
165 5.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07.
169 .Bl -column "XXX" -compact
170 Range: Overflow threshold = 2.0**128 = 3.4e38
171 Underflow threshold = 0.5**126 = 1.2e\-38
173 .Bd -ragged -offset indent -compact
174 Underflowed results round to the nearest
175 integer multiple of 0.5**149 = 1.4e\-45.
180 .Bd -ragged -offset indent -compact
183 .Bd -ragged -offset indent -compact
184 On some architectures,
192 Precision: 53 significant bits,
193 roughly like 16 significant decimals.
194 .Bd -ragged -offset indent -compact
195 If x and x' are consecutive positive double-precision
196 numbers (they differ by 1
200 1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
204 .Bl -column "XXX" -compact
205 Range: Overflow threshold = 2.0**1024 = 1.8e308
206 Underflow threshold = 0.5**1022 = 2.2e\-308
208 .Bd -ragged -offset indent -compact
209 Underflowed results round to the nearest
210 integer multiple of 0.5**1074 = 4.9e\-324.
215 .Bd -ragged -offset indent -compact
218 (when supported by the hardware)
222 Precision: 64 significant bits,
223 roughly like 19 significant decimals.
224 .Bd -ragged -offset indent -compact
225 If x and x' are consecutive positive extended-precision
226 numbers (they differ by 1
230 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
238 .Bd -ragged -offset indent -compact
239 Underflowed results round to the nearest
240 integer multiple of 0.5**16445 = 5.7e\-4953.
244 Quad-extended-precision:
245 .Bd -ragged -offset indent -compact
248 (when supported by the hardware)
252 Precision: 113 significant bits,
253 roughly like 34 significant decimals.
254 .Bd -ragged -offset indent -compact
255 If x and x' are consecutive positive quad-extended-precision
256 numbers (they differ by 1
260 9.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34.
264 .Bl -column "XXX" -compact
265 Range: Overflow threshold = 2.0**16384 = 1.2e4932
266 Underflow threshold = 0.5**16382 = 3.4e\-4932
268 .Bd -ragged -offset indent -compact
269 Underflowed results round to the nearest
270 integer multiple of 0.5**16494 = 6.5e\-4966.
273 .Ss Additional Information Regarding Exceptions
275 For each kind of floating-point exception, IEEE 754
276 provides a Flag that is raised each time its exception
277 is signaled, and stays raised until the program resets
279 Programs may also test, save and restore a flag.
280 Thus, IEEE 754 provides three ways by which programs
281 may cope with exceptions for which the default result
282 might be unsatisfactory:
285 Test for a condition that might cause an exception
286 later, and branch to avoid the exception.
288 Test a flag to see whether an exception has occurred
289 since the program last reset its flag.
291 Test a result to see whether it is a value that only
292 an exception could have produced.
294 CAUTION: The only reliable ways to discover
295 whether Underflow has occurred are to test whether
296 products or quotients lie closer to zero than the
297 underflow threshold, or to test the Underflow
299 (Sums and differences cannot underflow in
300 IEEE 754; if x \(!= y then x\-y is correct to
301 full precision and certainly nonzero regardless of
303 Products and quotients that
304 underflow gradually can lose accuracy gradually
305 without vanishing, so comparing them with zero
306 (as one might on a VAX) will not reveal the loss.
307 Fortunately, if a gradually underflowed value is
308 destined to be added to something bigger than the
309 underflow threshold, as is almost always the case,
310 digits lost to gradual underflow will not be missed
311 because they would have been rounded off anyway.
312 So gradual underflows are usually
315 The same cannot be said of underflows flushed to 0.
318 At the option of an implementor conforming to IEEE 754,
319 other ways to cope with exceptions may be provided:
323 This mechanism classifies an exception in
324 advance as an incident to be handled by means
325 traditionally associated with error-handling
326 statements like "ON ERROR GO TO ...".
328 languages offer different forms of this statement,
329 but most share the following characteristics:
332 No means is provided to substitute a value for
333 the offending operation's result and resume
334 computation from what may be the middle of an
336 An exceptional result is abandoned.
338 In a subprogram that lacks an error-handling
339 statement, an exception causes the subprogram to
340 abort within whatever program called it, and so
341 on back up the chain of calling subprograms until
342 an error-handling statement is encountered or the
343 whole task is aborted and memory is dumped.
347 This mechanism, requiring an interactive
348 debugging environment, is more for the programmer
350 It classifies an exception in
351 advance as a symptom of a programmer's error; the
352 exception suspends execution as near as it can to
353 the offending operation so that the programmer can
354 look around to see how it happened.
356 the first several exceptions turn out to be quite
357 unexceptionable, so the programmer ought ideally
358 to be able to resume execution after each one as if
359 execution had not been stopped.
361 \&... Other ways lie beyond the scope of this document.
365 elementary function should act as if it were indivisible, or
366 atomic, in the sense that ...
369 No exception should be signaled that is not deserved by
370 the data supplied to that function.
372 Any exception signaled should be identified with that
373 function rather than with one of its subroutines.
375 The internal behavior of an atomic function should not
376 be disrupted when a calling program changes from
377 one to another of the five or so ways of handling
378 exceptions listed above, although the definition
379 of the function may be correlated intentionally
380 with exception handling.
385 are only approximately atomic.
386 They signal no inappropriate exception except possibly ...
387 .Bl -tag -width indent -offset indent -compact
391 when a result, if properly computed, might have lain barely within range, and
401 when it happens to be exact, thanks to fortuitous cancellation of errors.
404 .Bl -tag -width indent -offset indent -compact
406 Invalid Operation is signaled only when
408 any result but \*(Na would probably be misleading.
410 Overflow is signaled only when
412 the exact result would be finite but beyond the overflow threshold.
414 Divide-by-Zero is signaled only when
416 a function takes exactly infinite values at finite operands.
418 Underflow is signaled only when
420 the exact result would be nonzero but tinier than the underflow threshold.
422 Inexact is signaled only when
424 greater range or precision would be needed to represent the exact result.
431 An explanation of IEEE 754 and its proposed extension p854
432 was published in the IEEE magazine MICRO in August 1984 under
433 the title "A Proposed Radix- and Word-length-independent
434 Standard for Floating-point Arithmetic" by
437 The manuals for Pascal, C and BASIC on the Apple Macintosh
438 document the features of IEEE 754 pretty well.
439 Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\&
440 1981), and in the ACM SIGNUM Newsletter Special Issue of
441 Oct.\& 1979, may be helpful although they pertain to
442 superseded drafts of the standard.