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.\" 3. 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.Pp 160If x and x' are consecutive positive single-precision 161numbers (they differ by 1 162.Em ulp ) , 163then 164.Bl -column "XXX" -compact 1655.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07. 166.El 167.Pp 168.Bl -column "XXX" -compact 169Range: Overflow threshold = 2.0**128 = 3.4e38 170 Underflow threshold = 0.5**126 = 1.2e\-38 171.El 172.Pp 173Underflowed results round to the nearest 174integer multiple of 175.Bl -column "XXX" -compact 1760.5**149 = 1.4e\-45. 177.El 178.Ed 179.Pp 180Double-precision: 181.Bd -ragged -offset indent -compact 182Type name: 183.Vt double 184.Po On some architectures, 185.Vt long double 186is the same as 187.Vt double 188.Pc 189.Pp 190Wordsize: 64 bits. 191.Pp 192Precision: 53 significant bits, 193roughly like 16 significant decimals. 194.Pp 195If x and x' are consecutive positive double-precision 196numbers (they differ by 1 197.Em ulp ) , 198then 199.Bl -column "XXX" -compact 2001.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16. 201.El 202.Pp 203.Bl -column "XXX" -compact 204Range: Overflow threshold = 2.0**1024 = 1.8e308 205 Underflow threshold = 0.5**1022 = 2.2e\-308 206.El 207.Pp 208Underflowed results round to the nearest 209integer multiple of 210.Bl -column "XXX" -compact 2110.5**1074 = 4.9e\-324. 212.El 213.Ed 214.Pp 215Extended-precision: 216.Bd -ragged -offset indent -compact 217Type name: 218.Vt long double 219(when supported by the hardware) 220.Pp 221Wordsize: 96 bits. 222.Pp 223Precision: 64 significant bits, 224roughly like 19 significant decimals. 225.Pp 226If x and x' are consecutive positive extended-precision 227numbers (they differ by 1 228.Em ulp ) , 229then 230.Bl -column "XXX" -compact 2311.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19. 232.El 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.Pp 239Underflowed results round to the nearest 240integer multiple of 241.Bl -column "XXX" -compact 2420.5**16445 = 5.7e\-4953. 243.El 244.Ed 245.Pp 246Quad-extended-precision: 247.Bd -ragged -offset indent -compact 248Type name: 249.Vt long double 250(when supported by the hardware) 251.Pp 252Wordsize: 128 bits. 253.Pp 254Precision: 113 significant bits, 255roughly like 34 significant decimals. 256.Pp 257If x and x' are consecutive positive quad-extended-precision 258numbers (they differ by 1 259.Em ulp ) , 260then 261.Bl -column "XXX" -compact 2629.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34. 263.El 264.Pp 265.Bl -column "XXX" -compact 266Range: Overflow threshold = 2.0**16384 = 1.2e4932 267 Underflow threshold = 0.5**16382 = 3.4e\-4932 268.El 269.Pp 270Underflowed results round to the nearest 271integer multiple of 272.Bl -column "XXX" -compact 2730.5**16494 = 6.5e\-4966. 274.El 275.Ed 276.Ss Additional Information Regarding Exceptions 277For each kind of floating-point exception, IEEE 754 278provides a Flag that is raised each time its exception 279is signaled, and stays raised until the program resets 280it. 281Programs may also test, save and restore a flag. 282Thus, IEEE 754 provides three ways by which programs 283may cope with exceptions for which the default result 284might be unsatisfactory: 285.Bl -enum 286.It 287Test for a condition that might cause an exception 288later, and branch to avoid the exception. 289.It 290Test a flag to see whether an exception has occurred 291since the program last reset its flag. 292.It 293Test a result to see whether it is a value that only 294an exception could have produced. 295.Pp 296CAUTION: The only reliable ways to discover 297whether Underflow has occurred are to test whether 298products or quotients lie closer to zero than the 299underflow threshold, or to test the Underflow 300flag. 301(Sums and differences cannot underflow in 302IEEE 754; if x \(!= y then x\-y is correct to 303full precision and certainly nonzero regardless of 304how tiny it may be.) 305Products and quotients that 306underflow gradually can lose accuracy gradually 307without vanishing, so comparing them with zero 308(as one might on a VAX) will not reveal the loss. 309Fortunately, if a gradually underflowed value is 310destined to be added to something bigger than the 311underflow threshold, as is almost always the case, 312digits lost to gradual underflow will not be missed 313because they would have been rounded off anyway. 314So gradual underflows are usually 315.Em provably 316ignorable. 317The same cannot be said of underflows flushed to 0. 318.El 319.Pp 320At the option of an implementor conforming to IEEE 754, 321other ways to cope with exceptions may be provided: 322.Bl -enum 323.It 324ABORT. 325This mechanism classifies an exception in 326advance as an incident to be handled by means 327traditionally associated with error-handling 328statements like "ON ERROR GO TO ...". 329Different 330languages offer different forms of this statement, 331but most share the following characteristics: 332.Bl -dash 333.It 334No means is provided to substitute a value for 335the offending operation's result and resume 336computation from what may be the middle of an 337expression. 338An exceptional result is abandoned. 339.It 340In a subprogram that lacks an error-handling 341statement, an exception causes the subprogram to 342abort within whatever program called it, and so 343on back up the chain of calling subprograms until 344an error-handling statement is encountered or the 345whole task is aborted and memory is dumped. 346.El 347.It 348STOP. 349This mechanism, requiring an interactive 350debugging environment, is more for the programmer 351than the program. 352It classifies an exception in 353advance as a symptom of a programmer's error; the 354exception suspends execution as near as it can to 355the offending operation so that the programmer can 356look around to see how it happened. 357Quite often 358the first several exceptions turn out to be quite 359unexceptionable, so the programmer ought ideally 360to be able to resume execution after each one as if 361execution had not been stopped. 362.It 363\&... Other ways lie beyond the scope of this document. 364.El 365.Pp 366Ideally, each 367elementary function should act as if it were indivisible, or 368atomic, in the sense that ... 369.Bl -enum 370.It 371No exception should be signaled that is not deserved by 372the data supplied to that function. 373.It 374Any exception signaled should be identified with that 375function rather than with one of its subroutines. 376.It 377The internal behavior of an atomic function should not 378be disrupted when a calling program changes from 379one to another of the five or so ways of handling 380exceptions listed above, although the definition 381of the function may be correlated intentionally 382with exception handling. 383.El 384.Pp 385The functions in 386.Nm libm 387are only approximately atomic. 388They signal no inappropriate exception except possibly ... 389.Bl -tag -width indent -offset indent -compact 390.It Xo 391Over/Underflow 392.Xc 393when a result, if properly computed, might have lain barely within range, and 394.It Xo 395Inexact in 396.Fn cabs , 397.Fn cbrt , 398.Fn hypot , 399.Fn log10 400and 401.Fn pow 402.Xc 403when it happens to be exact, thanks to fortuitous cancellation of errors. 404.El 405Otherwise, ... 406.Bl -tag -width indent -offset indent -compact 407.It Xo 408Invalid Operation is signaled only when 409.Xc 410any result but \*(Na would probably be misleading. 411.It Xo 412Overflow is signaled only when 413.Xc 414the exact result would be finite but beyond the overflow threshold. 415.It Xo 416Divide-by-Zero is signaled only when 417.Xc 418a function takes exactly infinite values at finite operands. 419.It Xo 420Underflow is signaled only when 421.Xc 422the exact result would be nonzero but tinier than the underflow threshold. 423.It Xo 424Inexact is signaled only when 425.Xc 426greater range or precision would be needed to represent the exact result. 427.El 428.Sh SEE ALSO 429.Xr fenv 3 , 430.Xr ieee_test 3 , 431.Xr math 3 432.Pp 433An explanation of IEEE 754 and its proposed extension p854 434was published in the IEEE magazine MICRO in August 1984 under 435the title "A Proposed Radix- and Word-length-independent 436Standard for Floating-point Arithmetic" by 437.An "W. J. Cody" 438et al. 439The manuals for Pascal, C and BASIC on the Apple Macintosh 440document the features of IEEE 754 pretty well. 441Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\& 4421981), and in the ACM SIGNUM Newsletter Special Issue of 443Oct.\& 1979, may be helpful although they pertain to 444superseded drafts of the standard. 445.Sh STANDARDS 446.St -ieee754 447