1.\" Copyright (c) 1985 Regents of the University of California. 2.\" All rights reserved. 3.\" 4.\" Redistribution and use in source and binary forms, with or without 5.\" modification, are permitted provided that the following conditions 6.\" are met: 7.\" 1. Redistributions of source code must retain the above copyright 8.\" notice, this list of conditions and the following disclaimer. 9.\" 2. Redistributions in binary form must reproduce the above copyright 10.\" notice, this list of conditions and the following disclaimer in the 11.\" documentation and/or other materials provided with the distribution. 12.\" 4. Neither the name of the University nor the names of its contributors 13.\" may be used to endorse or promote products derived from this software 14.\" without specific prior written permission. 15.\" 16.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 17.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 18.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 19.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 20.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 21.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 22.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 23.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 24.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 25.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 26.\" SUCH DAMAGE. 27.\" 28.\" from: @(#)ieee.3 6.4 (Berkeley) 5/6/91 29.\" $FreeBSD$ 30.\" 31.Dd January 26, 2005 32.Dt IEEE 3 33.Os 34.Sh NAME 35.Nm ieee 36.Nd IEEE standard 754 for floating-point arithmetic 37.Sh DESCRIPTION 38The IEEE Standard 754 for Binary Floating-Point Arithmetic 39defines representations of floating-point numbers and abstract 40properties of arithmetic operations relating to precision, 41rounding, and exceptional cases, as described below. 42.Ss IEEE STANDARD 754 Floating-Point Arithmetic 43Radix: Binary. 44.Pp 45Overflow and underflow: 46.Bd -ragged -offset indent -compact 47Overflow goes by default to a signed \*(If. 48Underflow is 49.Em gradual . 50.Ed 51.Pp 52Zero is represented ambiguously as +0 or \-0. 53.Bd -ragged -offset indent -compact 54Its sign transforms correctly through multiplication or 55division, and is preserved by addition of zeros 56with like signs; but x\-x yields +0 for every 57finite x. 58The only operations that reveal zero's 59sign are division by zero and 60.Fn copysign x \(+-0 . 61In particular, comparison (x > y, x \(>= y, etc.)\& 62cannot be affected by the sign of zero; but if 63finite x = y then \*(If = 1/(x\-y) \(!= \-1/(y\-x) = \-\*(If. 64.Ed 65.Pp 66Infinity is signed. 67.Bd -ragged -offset indent -compact 68It persists when added to itself 69or to any finite number. 70Its sign transforms 71correctly through multiplication and division, and 72(finite)/\(+-\*(If\0=\0\(+-0 73(nonzero)/0 = \(+-\*(If. 74But 75\*(If\-\*(If, \*(If\(**0 and \*(If/\*(If 76are, like 0/0 and sqrt(\-3), 77invalid operations that produce \*(Na. ... 78.Ed 79.Pp 80Reserved operands (\*(Nas): 81.Bd -ragged -offset indent -compact 82An \*(Na is 83.Em ( N Ns ot Em a N Ns umber ) . 84Some \*(Nas, called Signaling \*(Nas, trap any floating-point operation 85performed upon them; they are used to mark missing 86or uninitialized values, or nonexistent elements 87of arrays. 88The rest are Quiet \*(Nas; they are 89the default results of Invalid Operations, and 90propagate through subsequent arithmetic operations. 91If x \(!= x then x is \*(Na; every other predicate 92(x > y, x = y, x < y, ...) is FALSE if \*(Na is involved. 93.Ed 94.Pp 95Rounding: 96.Bd -ragged -offset indent -compact 97Every algebraic operation (+, \-, \(**, /, 98\(sr) 99is rounded by default to within half an 100.Em ulp , 101and when the rounding error is exactly half an 102.Em ulp 103then 104the rounded value's least significant bit is zero. 105(An 106.Em ulp 107is one 108.Em U Ns nit 109in the 110.Em L Ns ast 111.Em P Ns lace . ) 112This kind of rounding is usually the best kind, 113sometimes provably so; for instance, for every 114x = 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 ... 116despite that both the quotients and the products 117have been rounded. 118Only rounding like IEEE 754 can do that. 119But no single kind of rounding can be 120proved best for every circumstance, so IEEE 754 121provides rounding towards zero or towards 122+\*(If or towards \-\*(If 123at the programmer's option. 124.Ed 125.Pp 126Exceptions: 127.Bd -ragged -offset indent -compact 128IEEE 754 recognizes five kinds of floating-point exceptions, 129listed below in declining order of probable importance. 130.Bl -column -offset indent "Invalid Operation" "Gradual Underflow" 131.Em "Exception Default Result" 132Invalid Operation \*(Na, or FALSE 133Overflow \(+-\*(If 134Divide by Zero \(+-\*(If 135Underflow Gradual Underflow 136Inexact Rounded value 137.El 138.Pp 139NOTE: An Exception is not an Error unless handled 140badly. 141What makes a class of exceptions exceptional 142is that no single default response can be satisfactory 143in every instance. 144On the other hand, if a default 145response will serve most instances satisfactorily, 146the unsatisfactory instances cannot justify aborting 147computation every time the exception occurs. 148.Ed 149.Ss Data Formats 150Single-precision: 151.Bd -ragged -offset indent -compact 152Type name: 153.Vt float 154.Pp 155Wordsize: 32 bits. 156.Pp 157Precision: 24 significant bits, 158roughly like 7 significant decimals. 159.Bd -ragged -offset indent -compact 160If x and x' are consecutive positive single-precision 161numbers (they differ by 1 162.Em ulp ) , 163then 164.Bd -ragged -compact 1655.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07. 166.Ed 167.Ed 168.Pp 169.Bl -column "XXX" -compact 170Range: Overflow threshold = 2.0**128 = 3.4e38 171 Underflow threshold = 0.5**126 = 1.2e\-38 172.El 173.Bd -ragged -offset indent -compact 174Underflowed results round to the nearest 175integer multiple of 0.5**149 = 1.4e\-45. 176.Ed 177.Ed 178.Pp 179Double-precision: 180.Bd -ragged -offset indent -compact 181Type name: 182.Vt double 183.Bd -ragged -offset indent -compact 184On some architectures, 185.Vt long double 186is the same as 187.Vt double . 188.Ed 189.Pp 190Wordsize: 64 bits. 191.Pp 192Precision: 53 significant bits, 193roughly like 16 significant decimals. 194.Bd -ragged -offset indent -compact 195If x and x' are consecutive positive double-precision 196numbers (they differ by 1 197.Em ulp ) , 198then 199.Bd -ragged -compact 2001.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16. 201.Ed 202.Ed 203.Pp 204.Bl -column "XXX" -compact 205Range: Overflow threshold = 2.0**1024 = 1.8e308 206 Underflow threshold = 0.5**1022 = 2.2e\-308 207.El 208.Bd -ragged -offset indent -compact 209Underflowed results round to the nearest 210integer multiple of 0.5**1074 = 4.9e\-324. 211.Ed 212.Ed 213.Pp 214Extended-precision: 215.Bd -ragged -offset indent -compact 216Type name: 217.Vt long double 218(when supported by the hardware) 219.Pp 220Wordsize: 96 bits. 221.Pp 222Precision: 64 significant bits, 223roughly like 19 significant decimals. 224.Bd -ragged -offset indent -compact 225If x and x' are consecutive positive extended-precision 226numbers (they differ by 1 227.Em ulp ) , 228then 229.Bd -ragged -compact 2301.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19. 231.Ed 232.Ed 233.Pp 234.Bl -column "XXX" -compact 235Range: Overflow threshold = 2.0**16384 = 1.2e4932 236 Underflow threshold = 0.5**16382 = 3.4e\-4932 237.El 238.Bd -ragged -offset indent -compact 239Underflowed results round to the nearest 240integer multiple of 0.5**16445 = 5.7e\-4953. 241.Ed 242.Ed 243.Pp 244Quad-extended-precision: 245.Bd -ragged -offset indent -compact 246Type name: 247.Vt long double 248(when supported by the hardware) 249.Pp 250Wordsize: 128 bits. 251.Pp 252Precision: 113 significant bits, 253roughly like 34 significant decimals. 254.Bd -ragged -offset indent -compact 255If x and x' are consecutive positive quad-extended-precision 256numbers (they differ by 1 257.Em ulp ) , 258then 259.Bd -ragged -compact 2609.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34. 261.Ed 262.Ed 263.Pp 264.Bl -column "XXX" -compact 265Range: Overflow threshold = 2.0**16384 = 1.2e4932 266 Underflow threshold = 0.5**16382 = 3.4e\-4932 267.El 268.Bd -ragged -offset indent -compact 269Underflowed results round to the nearest 270integer multiple of 0.5**16494 = 6.5e\-4966. 271.Ed 272.Ed 273.Ss Additional Information Regarding Exceptions 274For each kind of floating-point exception, IEEE 754 275provides a Flag that is raised each time its exception 276is signaled, and stays raised until the program resets 277it. 278Programs may also test, save and restore a flag. 279Thus, IEEE 754 provides three ways by which programs 280may cope with exceptions for which the default result 281might be unsatisfactory: 282.Bl -enum 283.It 284Test for a condition that might cause an exception 285later, and branch to avoid the exception. 286.It 287Test a flag to see whether an exception has occurred 288since the program last reset its flag. 289.It 290Test a result to see whether it is a value that only 291an exception could have produced. 292.Pp 293CAUTION: The only reliable ways to discover 294whether Underflow has occurred are to test whether 295products or quotients lie closer to zero than the 296underflow threshold, or to test the Underflow 297flag. 298(Sums and differences cannot underflow in 299IEEE 754; if x \(!= y then x\-y is correct to 300full precision and certainly nonzero regardless of 301how tiny it may be.) 302Products and quotients that 303underflow gradually can lose accuracy gradually 304without vanishing, so comparing them with zero 305(as one might on a VAX) will not reveal the loss. 306Fortunately, if a gradually underflowed value is 307destined to be added to something bigger than the 308underflow threshold, as is almost always the case, 309digits lost to gradual underflow will not be missed 310because they would have been rounded off anyway. 311So gradual underflows are usually 312.Em provably 313ignorable. 314The same cannot be said of underflows flushed to 0. 315.El 316.Pp 317At the option of an implementor conforming to IEEE 754, 318other ways to cope with exceptions may be provided: 319.Bl -enum 320.It 321ABORT. 322This mechanism classifies an exception in 323advance as an incident to be handled by means 324traditionally associated with error-handling 325statements like "ON ERROR GO TO ...". 326Different 327languages offer different forms of this statement, 328but most share the following characteristics: 329.Bl -dash 330.It 331No means is provided to substitute a value for 332the offending operation's result and resume 333computation from what may be the middle of an 334expression. 335An exceptional result is abandoned. 336.It 337In a subprogram that lacks an error-handling 338statement, an exception causes the subprogram to 339abort within whatever program called it, and so 340on back up the chain of calling subprograms until 341an error-handling statement is encountered or the 342whole task is aborted and memory is dumped. 343.El 344.It 345STOP. 346This mechanism, requiring an interactive 347debugging environment, is more for the programmer 348than the program. 349It classifies an exception in 350advance as a symptom of a programmer's error; the 351exception suspends execution as near as it can to 352the offending operation so that the programmer can 353look around to see how it happened. 354Quite often 355the first several exceptions turn out to be quite 356unexceptionable, so the programmer ought ideally 357to be able to resume execution after each one as if 358execution had not been stopped. 359.It 360\&... Other ways lie beyond the scope of this document. 361.El 362.Pp 363Ideally, each 364elementary function should act as if it were indivisible, or 365atomic, in the sense that ... 366.Bl -enum 367.It 368No exception should be signaled that is not deserved by 369the data supplied to that function. 370.It 371Any exception signaled should be identified with that 372function rather than with one of its subroutines. 373.It 374The internal behavior of an atomic function should not 375be disrupted when a calling program changes from 376one to another of the five or so ways of handling 377exceptions listed above, although the definition 378of the function may be correlated intentionally 379with exception handling. 380.El 381.Pp 382The functions in 383.Nm libm 384are only approximately atomic. 385They signal no inappropriate exception except possibly ... 386.Bl -tag -width indent -offset indent -compact 387.It Xo 388Over/Underflow 389.Xc 390when a result, if properly computed, might have lain barely within range, and 391.It Xo 392Inexact in 393.Fn cabs , 394.Fn cbrt , 395.Fn hypot , 396.Fn log10 397and 398.Fn pow 399.Xc 400when it happens to be exact, thanks to fortuitous cancellation of errors. 401.El 402Otherwise, ... 403.Bl -tag -width indent -offset indent -compact 404.It Xo 405Invalid Operation is signaled only when 406.Xc 407any result but \*(Na would probably be misleading. 408.It Xo 409Overflow is signaled only when 410.Xc 411the exact result would be finite but beyond the overflow threshold. 412.It Xo 413Divide-by-Zero is signaled only when 414.Xc 415a function takes exactly infinite values at finite operands. 416.It Xo 417Underflow is signaled only when 418.Xc 419the exact result would be nonzero but tinier than the underflow threshold. 420.It Xo 421Inexact is signaled only when 422.Xc 423greater range or precision would be needed to represent the exact result. 424.El 425.Sh SEE ALSO 426.Xr fenv 3 , 427.Xr ieee_test 3 , 428.Xr math 3 429.Pp 430An explanation of IEEE 754 and its proposed extension p854 431was published in the IEEE magazine MICRO in August 1984 under 432the title "A Proposed Radix- and Word-length-independent 433Standard for Floating-point Arithmetic" by 434.An "W. J. Cody" 435et al. 436The manuals for Pascal, C and BASIC on the Apple Macintosh 437document the features of IEEE 754 pretty well. 438Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\& 4391981), and in the ACM SIGNUM Newsletter Special Issue of 440Oct.\& 1979, may be helpful although they pertain to 441superseded drafts of the standard. 442.Sh STANDARDS 443.St -ieee754 444