1.\" $NetBSD: math.3,v 1.26 2012/11/10 15:59:58 njoly Exp $ 2.\" 3.\" Copyright (c) 1985 Regents of the University of California. 4.\" All rights reserved. 5.\" 6.\" Redistribution and use in source and binary forms, with or without 7.\" modification, are permitted provided that the following conditions 8.\" are met: 9.\" 1. Redistributions of source code must retain the above copyright 10.\" notice, this list of conditions and the following disclaimer. 11.\" 2. Redistributions in binary form must reproduce the above copyright 12.\" notice, this list of conditions and the following disclaimer in the 13.\" documentation and/or other materials provided with the distribution. 14.\" 3. Neither the name of the University nor the names of its contributors 15.\" may be used to endorse or promote products derived from this software 16.\" without specific prior written permission. 17.\" 18.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28.\" SUCH DAMAGE. 29.\" 30.\" from: @(#)math.3 6.10 (Berkeley) 5/6/91 31.\" 32.Dd February 23, 2007 33.Dt MATH 3 34.Os 35.Sh NAME 36.Nm math 37.Nd introduction to mathematical library functions 38.Sh LIBRARY 39.Lb libm 40.Sh SYNOPSIS 41.In math.h 42.Sh DESCRIPTION 43These functions constitute the C 44.Lb libm . 45Declarations for these functions may be obtained from the include file 46.In math.h . 47.\" The Fortran math library is described in ``man 3f intro''. 48.Ss List of Functions 49.Bl -column "copysignX" "gammaX3XX" "inverse trigonometric funcX" 50.It Sy Name Ta Sy Man page Ta Sy Description Ta Sy Error Bound Dv ( ULP Ns No s) 51.It acos Ta Xr acos 3 Ta inverse trigonometric function Ta 3 52.It acosh Ta Xr acosh 3 Ta inverse hyperbolic function Ta 3 53.It asin Ta Xr asin 3 Ta inverse trigonometric function Ta 3 54.It asinh Ta Xr asinh 3 Ta inverse hyperbolic function Ta 3 55.It atan Ta Xr atan 3 Ta inverse trigonometric function Ta 1 56.It atanh Ta Xr atanh 3 Ta inverse hyperbolic function Ta 3 57.It atan2 Ta Xr atan2 3 Ta inverse trigonometric function Ta 2 58.It cbrt Ta Xr sqrt 3 Ta cube root Ta 1 59.It ceil Ta Xr ceil 3 Ta integer no less than Ta 0 60.It copysign Ta Xr copysign 3 Ta copy sign bit Ta 0 61.It cos Ta Xr cos 3 Ta trigonometric function Ta 1 62.It cosh Ta Xr cosh 3 Ta hyperbolic function Ta 3 63.It erf Ta Xr erf 3 Ta error function Ta ??? 64.It erfc Ta Xr erf 3 Ta complementary error function Ta ??? 65.It exp Ta Xr exp 3 Ta exponential Ta 1 66.It expm1 Ta Xr exp 3 Ta exp(x)\-1 Ta 1 67.It fabs Ta Xr fabs 3 Ta absolute value Ta 0 68.It finite Ta Xr finite 3 Ta test for finity Ta 0 69.It floor Ta Xr floor 3 Ta integer no greater than Ta 0 70.It fmod Ta Xr fmod 3 Ta remainder Ta ??? 71.It hypot Ta Xr hypot 3 Ta Euclidean distance Ta 1 72.It ilogb Ta Xr ilogb 3 Ta exponent extraction Ta 0 73.It isinf Ta Xr isinf 3 Ta test for infinity Ta 0 74.It isnan Ta Xr isnan 3 Ta test for not-a-number Ta 0 75.It j0 Ta Xr j0 3 Ta Bessel function Ta ??? 76.It j1 Ta Xr j0 3 Ta Bessel function Ta ??? 77.It jn Ta Xr j0 3 Ta Bessel function Ta ??? 78.It lgamma Ta Xr lgamma 3 Ta log gamma function Ta ??? 79.It log Ta Xr log 3 Ta natural logarithm Ta 1 80.It log10 Ta Xr log 3 Ta logarithm to base 10 Ta 3 81.It log1p Ta Xr log 3 Ta log(1+x) Ta 1 82.It nan Ta Xr nan 3 Ta return quiet \*(Na Ta 0 83.It nextafter Ta Xr nextafter 3 Ta next representable number Ta 0 84.It pow Ta Xr pow 3 Ta exponential x**y Ta 60\-500 85.It remainder Ta Xr remainder 3 Ta remainder Ta 0 86.It rint Ta Xr rint 3 Ta round to nearest integer Ta 0 87.It scalbn Ta Xr scalbn 3 Ta exponent adjustment Ta 0 88.It sin Ta Xr sin 3 Ta trigonometric function Ta 1 89.It sinh Ta Xr sinh 3 Ta hyperbolic function Ta 3 90.It sqrt Ta Xr sqrt 3 Ta square root Ta 1 91.It tan Ta Xr tan 3 Ta trigonometric function Ta 3 92.It tanh Ta Xr tanh 3 Ta hyperbolic function Ta 3 93.It trunc Ta Xr trunc 3 Ta nearest integral value Ta 3 94.It y0 Ta Xr j0 3 Ta Bessel function Ta ??? 95.It y1 Ta Xr j0 3 Ta Bessel function Ta ??? 96.It yn Ta Xr j0 3 Ta Bessel function Ta ??? 97.El 98.Ss List of Defined Values 99.Bl -column "M_2_SQRTPIXX" "1.12837916709551257390XX" "2/sqrt(pi)XXX" 100.It Sy Name Ta Sy Value Ta Sy Description 101.It M_E 2.7182818284590452354 e 102.It M_LOG2E 1.4426950408889634074 log 2e 103.It M_LOG10E 0.43429448190325182765 log 10e 104.It M_LN2 0.69314718055994530942 log e2 105.It M_LN10 2.30258509299404568402 log e10 106.It M_PI 3.14159265358979323846 pi 107.It M_PI_2 1.57079632679489661923 pi/2 108.It M_PI_4 0.78539816339744830962 pi/4 109.It M_1_PI 0.31830988618379067154 1/pi 110.It M_2_PI 0.63661977236758134308 2/pi 111.It M_2_SQRTPI 1.12837916709551257390 2/sqrt(pi) 112.It M_SQRT2 1.41421356237309504880 sqrt(2) 113.It M_SQRT1_2 0.70710678118654752440 1/sqrt(2) 114.El 115.Sh NOTES 116In 4.3 BSD, distributed from the University of California 117in late 1985, most of the foregoing functions come in two 118versions, one for the double\-precision "D" format in the 119DEC VAX\-11 family of computers, another for double\-precision 120arithmetic conforming to the IEEE Standard 754 for Binary 121Floating\-Point Arithmetic. 122The two versions behave very 123similarly, as should be expected from programs more accurate 124and robust than was the norm when UNIX was born. 125For instance, the programs are accurate to within the numbers 126of 127.Dv ULPs 128tabulated above; an 129.Dv ULP 130is one Unit in the Last Place. 131And the programs have been cured of anomalies that 132afflicted the older math library 133in which incidents like 134the following had been reported: 135.Bd -literal -offset indent 136sqrt(\-1.0) = 0.0 and log(\-1.0) = \-1.7e38. 137cos(1.0e\-11) \*[Gt] cos(0.0) \*[Gt] 1.0. 138pow(x,1.0) \(!= x when x = 2.0, 3.0, 4.0, ..., 9.0. 139pow(\-1.0,1.0e10) trapped on Integer Overflow. 140sqrt(1.0e30) and sqrt(1.0e\-30) were very slow. 141.Ed 142However the two versions do differ in ways that have to be 143explained, to which end the following notes are provided. 144.Ss DEC VAX\-11 D_floating\-point 145This is the format for which the original math library 146was developed, and to which this manual is still principally dedicated. 147It is 148.Em the 149double\-precision format for the PDP\-11 150and the earlier VAX\-11 machines; VAX\-11s after 1983 were 151provided with an optional "G" format closer to the IEEE 152double\-precision format. 153The earlier DEC MicroVAXs have no D format, only G double\-precision. 154(Why? 155Why not?) 156.Pp 157Properties of D_floating\-point: 158.Bl -hang -offset indent 159.It Wordsize : 16064 bits, 8 bytes. 161.It Radix : 162Binary. 163.It Precision : 16456 significant bits, roughly like 17 significant decimals. 165If x and x' are consecutive positive D_floating\-point 166numbers (they differ by 1 167.Dv ULP ) , 168then 169.Dl 1.3e\-17 \*[Lt] 0.5**56 \*[Lt] (x'\-x)/x \*[Le] 0.5**55 \*[Lt] 2.8e\-17. 170.It Range : 171.Bl -column "Underflow thresholdX" "2.0**127X" 172.It Overflow threshold = 2.0**127 = 1.7e38. 173.It Underflow threshold = 0.5**128 = 2.9e\-39. 174.El 175.Em NOTE: THIS RANGE IS COMPARATIVELY NARROW. 176.Pp 177Overflow customarily stops computation. 178Underflow is customarily flushed quietly to zero. 179.Em CAUTION : 180It is possible to have x 181\(!= 182y and yet x\-y = 0 because of underflow. 183Similarly x \*[Gt] y \*[Gt] 0 cannot prevent either x\(**y = 0 184or y/x = 0 from happening without warning. 185.It Zero is represented ambiguously : 186Although 2**55 different representations of zero are accepted by 187the hardware, only the obvious representation is ever produced. 188There is no \-0 on a VAX. 189.It \*(If is not part of the VAX architecture . 190.It Reserved operands : 191of the 2**55 that the hardware 192recognizes, only one of them is ever produced. 193Any floating\-point operation upon a reserved 194operand, even a MOVF or MOVD, customarily stops 195computation, so they are not much used. 196.It Exceptions : 197Divisions by zero and operations that 198overflow are invalid operations that customarily 199stop computation or, in earlier machines, produce 200reserved operands that will stop computation. 201.It Rounding : 202Every rational operation (+, \-, \(**, /) on a 203VAX (but not necessarily on a PDP\-11), if not an 204over/underflow nor division by zero, is rounded to 205within half an 206.Dv ULP , 207and when the rounding error is 208exactly half an 209.Dv ULP 210then rounding is away from 0. 211.El 212.Pp 213Except for its narrow range, D_floating\-point is one of the 214better computer arithmetics designed in the 1960's. 215Its properties are reflected fairly faithfully in the elementary 216functions for a VAX distributed in 4.3 BSD. 217They over/underflow only if their results have to lie out of range 218or very nearly so, and then they behave much as any rational 219arithmetic operation that over/underflowed would behave. 220Similarly, expressions like log(0) and atanh(1) behave 221like 1/0; and sqrt(\-3) and acos(3) behave like 0/0; 222they all produce reserved operands and/or stop computation! 223The situation is described in more detail in manual pages. 224.Pp 225.Em This response seems excessively punitive, so it is destined 226.Em to be replaced at some time in the foreseeable future by a 227.Em more flexible but still uniform scheme being developed to 228.Em handle all floating\-point arithmetic exceptions neatly. 229.Pp 230How do the functions in 4.3 BSD's new math library for UNIX 231compare with their counterparts in DEC's VAX/VMS library? 232Some of the VMS functions are a little faster, some are 233a little more accurate, some are more puritanical about 234exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)), 235and most occupy much more memory than their counterparts in 236libm. 237The VMS codes interpolate in large table to achieve 238speed and accuracy; the libm codes use tricky formulas 239compact enough that all of them may some day fit into a ROM. 240.Pp 241More important, DEC regards the VMS codes as proprietary 242and guards them zealously against unauthorized use. 243But the libm codes in 4.3 BSD are intended for the public domain; 244they may be copied freely provided their provenance is always 245acknowledged, and provided users assist the authors in their 246researches by reporting experience with the codes. 247Therefore no user of UNIX on a machine whose arithmetic resembles 248VAX D_floating\-point need use anything worse than the new libm. 249.Ss IEEE STANDARD 754 Floating\-Point Arithmetic 250This standard is on its way to becoming more widely adopted 251than any other design for computer arithmetic. 252VLSI chips that conform to some version of that standard have been 253produced by a host of manufacturers, among them ... 254.Bl -column "Intel i8070, i80287XX" 255.It Intel i8087, i80287 National Semiconductor 32081 256.It 68881 Weitek WTL-1032, ... , -1165 257.It Zilog Z8070 Western Electric (AT\*[Am]T) WE32106. 258.El 259Other implementations range from software, done thoroughly 260in the Apple Macintosh, through VLSI in the Hewlett\-Packard 2619000 series, to the ELXSI 6400 running ECL at 3 Megaflops. 262Several other companies have adopted the formats 263of IEEE 754 without, alas, adhering to the standard's way 264of handling rounding and exceptions like over/underflow. 265The DEC VAX G_floating\-point format is very similar to the IEEE 266754 Double format, so similar that the C programs for the 267IEEE versions of most of the elementary functions listed 268above could easily be converted to run on a MicroVAX, though 269nobody has volunteered to do that yet. 270.Pp 271The codes in 4.3 BSD's libm for machines that conform to 272IEEE 754 are intended primarily for the National Semiconductor 32081 273and WTL 1164/65. 274To use these codes with the Intel or Zilog 275chips, or with the Apple Macintosh or ELXSI 6400, is to 276forego the use of better codes provided (perhaps freely) by 277those companies and designed by some of the authors of the 278codes above. 279Except for 280.Fn atan , 281.Fn cbrt , 282.Fn erf , 283.Fn erfc , 284.Fn hypot , 285.Fn j0-jn , 286.Fn lgamma , 287.Fn pow , 288and 289.Fn y0\-yn , 290the Motorola 68881 has all the functions in libm on chip, 291and faster and more accurate; 292it, Apple, the i8087, Z8070 and WE32106 all use 64 significant bits. 293The main virtue of 4.3 BSD's 294libm codes is that they are intended for the public domain; 295they may be copied freely provided their provenance is always 296acknowledged, and provided users assist the authors in their 297researches by reporting experience with the codes. 298Therefore no user of UNIX on a machine that conforms to 299IEEE 754 need use anything worse than the new libm. 300.Pp 301Properties of IEEE 754 Double\-Precision: 302.Bl -hang -offset indent 303.It Wordsize : 30464 bits, 8 bytes. 305.It Radix : 306Binary. 307.It Precision : 30853 significant bits, roughly like 16 significant decimals. 309If x and x' are consecutive positive Double\-Precision 310numbers (they differ by 1 311.Dv ULP ) , 312then 313.Dl 1.1e\-16 \*[Lt] 0.5**53 \*[Lt] (x'\-x)/x \*[Le] 0.5**52 \*[Lt] 2.3e\-16. 314.It Range : 315.Bl -column "Underflow thresholdX" "2.0**1024X" 316.It Overflow threshold = 2.0**1024 = 1.8e308 317.It Underflow threshold = 0.5**1022 = 2.2e\-308 318.El 319Overflow goes by default to a signed \*(If. 320Underflow is 321.Sy Gradual , 322rounding to the nearest 323integer multiple of 0.5**1074 = 4.9e\-324. 324.It Zero is represented ambiguously as +0 or \-0: 325Its sign transforms correctly through multiplication or 326division, and is preserved by addition of zeros 327with like signs; but x\-x yields +0 for every 328finite x. 329The only operations that reveal zero's 330sign are division by zero and copysign(x,\(+-0). 331In particular, comparison (x \*[Gt] y, x \*[Ge] y, etc.) 332cannot be affected by the sign of zero; but if 333finite x = y then \*(If 334\&= 1/(x\-y) 335\(!= 336\-1/(y\-x) = 337\- \*(If . 338.It \*(If is signed : 339it persists when added to itself 340or to any finite number. 341Its sign transforms 342correctly through multiplication and division, and 343\*(If (finite)/\(+- \0=\0\(+-0 344(nonzero)/0 = 345\(+- \*(If. 346But 347\(if\-\(if, \(if\(**0 and \(if/\(if 348are, like 0/0 and sqrt(\-3), 349invalid operations that produce \*(Na. 350.It Reserved operands : 351there are 2**53\-2 of them, all 352called \*(Na (Not A Number). 353Some, called Signaling \*[Na]s, trap any floating\-point operation 354performed upon them; they are used to mark missing 355or uninitialized values, or nonexistent elements of arrays. 356The rest are Quiet \*[Na]s; they are 357the default results of Invalid Operations, and 358propagate through subsequent arithmetic operations. 359If x 360\(!= 361x then x is \*(Na; every other predicate 362(x \*[Gt] y, x = y, x \*[Lt] y, ...) is FALSE if \*(Na is involved. 363.Pp 364.Em NOTE : 365Trichotomy is violated by \*(Na. 366Besides being FALSE, predicates that entail ordered 367comparison, rather than mere (in)equality, 368signal Invalid Operation when \*(Na is involved. 369.It Rounding : 370Every algebraic operation (+, \-, \(**, /, 371\(sr) 372is rounded by default to within half an 373.Dv ULP , 374and when the rounding error is exactly half an 375.Dv ULP 376then the rounded value's least significant bit is zero. 377This kind of rounding is usually the best kind, 378sometimes provably so; for instance, for every 379x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find 380(x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ... 381despite that both the quotients and the products 382have been rounded. 383Only rounding like IEEE 754 can do that. 384But no single kind of rounding can be 385proved best for every circumstance, so IEEE 754 386provides rounding towards zero or towards 387+\*(If 388or towards 389\-\*(If 390at the programmer's option. 391And the same kinds of rounding are specified for 392Binary\-Decimal Conversions, at least for magnitudes 393between roughly 1.0e\-10 and 1.0e37. 394.It Exceptions : 395IEEE 754 recognizes five kinds of floating\-point exceptions, 396listed below in declining order of probable importance. 397.Bl -column "Invalid OperationX" "Gradual OverflowX" 398.It Sy Exception Ta Sy Default Result 399.It Invalid Operation \*(Na, or FALSE 400.It Overflow \(+-\(if 401.It Divide by Zero \(+-\(if \} 402.It Underflow Gradual Underflow 403.It Inexact Rounded value 404.El 405.Pp 406.Em NOTE : 407An Exception is not an Error unless handled badly. 408What makes a class of exceptions exceptional 409is that no single default response can be satisfactory 410in every instance. 411On the other hand, if a default 412response will serve most instances satisfactorily, 413the unsatisfactory instances cannot justify aborting 414computation every time the exception occurs. 415.El 416.Pp 417For each kind of floating\-point exception, IEEE 754 418provides a Flag that is raised each time its exception 419is signaled, and stays raised until the program resets it. 420Programs may also test, save and restore a flag. 421Thus, IEEE 754 provides three ways by which programs 422may cope with exceptions for which the default result 423might be unsatisfactory: 424.Bl -enum 425.It 426Test for a condition that might cause an exception 427later, and branch to avoid the exception. 428.It 429Test a flag to see whether an exception has occurred 430since the program last reset its flag. 431.It 432Test a result to see whether it is a value that only 433an exception could have produced. 434.Em CAUTION : 435The only reliable ways to discover 436whether Underflow has occurred are to test whether 437products or quotients lie closer to zero than the 438underflow threshold, or to test the Underflow flag. 439(Sums and differences cannot underflow in 440IEEE 754; if x 441\(!= 442y then x\-y is correct to 443full precision and certainly nonzero regardless of 444how tiny it may be.) 445Products and quotients that 446underflow gradually can lose accuracy gradually 447without vanishing, so comparing them with zero 448(as one might on a VAX) will not reveal the loss. 449Fortunately, if a gradually underflowed value is 450destined to be added to something bigger than the 451underflow threshold, as is almost always the case, 452digits lost to gradual underflow will not be missed 453because they would have been rounded off anyway. 454So gradual underflows are usually 455.Em provably 456ignorable. 457The same cannot be said of underflows flushed to 0. 458.Pp 459At the option of an implementor conforming to IEEE 754, 460other ways to cope with exceptions may be provided: 461.It 462ABORT. 463This mechanism classifies an exception in 464advance as an incident to be handled by means 465traditionally associated with error\-handling 466statements like "ON ERROR GO TO ...". 467Different languages offer different forms of this statement, 468but most share the following characteristics: 469.Bl -dash 470.It 471No means is provided to substitute a value for 472the offending operation's result and resume 473computation from what may be the middle of an expression. 474An exceptional result is abandoned. 475.It 476In a subprogram that lacks an error\-handling 477statement, an exception causes the subprogram to 478abort within whatever program called it, and so 479on back up the chain of calling subprograms until 480an error\-handling statement is encountered or the 481whole task is aborted and memory is dumped. 482.El 483.It 484STOP. 485This mechanism, requiring an interactive 486debugging environment, is more for the programmer 487than the program. 488It classifies an exception in 489advance as a symptom of a programmer's error; the 490exception suspends execution as near as it can to 491the offending operation so that the programmer can 492look around to see how it happened. 493Quite often 494the first several exceptions turn out to be quite 495unexceptionable, so the programmer ought ideally 496to be able to resume execution after each one as if 497execution had not been stopped. 498.It 499\&... Other ways lie beyond the scope of this document. 500.El 501.Pp 502The crucial problem for exception handling is the problem of 503Scope, and the problem's solution is understood, but not 504enough manpower was available to implement it fully in time 505to be distributed in 4.3 BSD's libm. 506Ideally, each elementary function should act 507as if it were indivisible, or atomic, in the sense that ... 508.Bl -enum 509.It 510No exception should be signaled that is not deserved by 511the data supplied to that function. 512.It 513Any exception signaled should be identified with that 514function rather than with one of its subroutines. 515.It 516The internal behavior of an atomic function should not 517be disrupted when a calling program changes from 518one to another of the five or so ways of handling 519exceptions listed above, although the definition 520of the function may be correlated intentionally 521with exception handling. 522.El 523.Pp 524Ideally, every programmer should be able 525.Em conveniently 526to turn a debugged subprogram into one that appears atomic to 527its users. 528But simulating all three characteristics of an 529atomic function is still a tedious affair, entailing hosts 530of tests and saves\-restores; work is under way to ameliorate 531the inconvenience. 532.Pp 533Meanwhile, the functions in libm are only approximately atomic. 534They signal no inappropriate exception except possibly ... 535.Bl -ohang -offset indent 536.It Over/Underflow 537when a result, if properly computed, might have lain barely within range, and 538.It Inexact in Fn cbrt , Fn hypot , Fn log10 No and Fn pow 539when it happens to be exact, thanks to fortuitous cancellation of errors. 540.El 541Otherwise, ... 542.Bl -ohang -offset indent 543.It Invalid Operation is signaled only when 544any result but \*(Na would probably be misleading. 545.It Overflow is signaled only when 546the exact result would be finite but beyond the overflow threshold. 547.It Divide\-by\-Zero is signaled only when 548a function takes exactly infinite values at finite operands. 549.It Underflow is signaled only when 550the exact result would be nonzero but tinier than the underflow threshold. 551.It Inexact is signaled only when 552greater range or precision would be needed to represent the exact result. 553.El 554.\" .Sh FILES 555.\" .Bl -tag -width /usr/lib/libm_p.a -compact 556.\" .It Pa /usr/lib/libm.a 557.\" the static math library 558.\" .It Pa /usr/lib/libm.so 559.\" the dynamic math library 560.\" .It Pa /usr/lib/libm_p.a 561.\" the static math library compiled for profiling 562.\" .El 563.Sh SEE ALSO 564An explanation of IEEE 754 and its proposed extension p854 565was published in the IEEE magazine MICRO in August 1984 under 566the title "A Proposed Radix\- and Word\-length\-independent 567Standard for Floating\-point Arithmetic" by W. J. Cody et al. 568The manuals for Pascal, C and BASIC on the Apple Macintosh 569document the features of IEEE 754 pretty well. 570Articles in the IEEE magazine COMPUTER vol. 14 no. 3 (Mar. 1981), 571and in the ACM SIGNUM Newsletter Special Issue of 572Oct. 1979, may be helpful although they pertain to 573superseded drafts of the standard. 574.Sh BUGS 575When signals are appropriate, they are emitted by certain 576operations within the codes, so a subroutine\-trace may be 577needed to identify the function with its signal in case 578method 5) above is in use. 579And the codes all take the 580IEEE 754 defaults for granted; this means that a decision to 581trap all divisions by zero could disrupt a code that would 582otherwise get correct results despite division by zero. 583