Mercurial > hg > xemacs-beta
annotate man/lispref/numbers.texi @ 5917:ffb5abc8dc4e
Fix a bug in the #'equalp compiler macro.
lisp/ChangeLog addition:
2015-06-11 Aidan Kehoe <kehoea@parhasard.net>
* cl-macs.el (equalp):
Fix a bug in this compiler macro when passed constants in a
certain order.
tests/ChangeLog addition:
2015-06-11 Aidan Kehoe <kehoea@parhasard.net>
* automated/lisp-tests.el:
Add a test looking for a bug just fixed in the equalp compiler
macro.
| author | Aidan Kehoe <kehoea@parhasard.net> |
|---|---|
| date | Thu, 11 Jun 2015 16:09:11 +0100 |
| parents | 6174848f3e6c |
| children |
| rev | line source |
|---|---|
| 428 | 1 @c -*-texinfo-*- |
| 2 @c This is part of the XEmacs Lisp Reference Manual. | |
| 444 | 3 @c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc. |
| 428 | 4 @c See the file lispref.texi for copying conditions. |
| 5 @setfilename ../../info/numbers.info | |
| 6 @node Numbers, Strings and Characters, Lisp Data Types, Top | |
| 7 @chapter Numbers | |
| 2090 | 8 @c #### Improve the indexing in this file!!!! |
| 428 | 9 @cindex integers |
| 10 @cindex numbers | |
| 11 | |
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12 XEmacs supports two to five numeric data types. @dfn{Fixnums} and |
| 2028 | 13 @dfn{floating point numbers} are always supported. As a build-time |
| 14 option, @dfn{bignums}, @dfn{ratios}, and @dfn{bigfloats} may be | |
| 15 enabled on some platforms. | |
| 16 | |
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17 Fixnums (called just @dfn{integers} in GNU Emacs and older versions |
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18 of XEmacs) are whole numbers such as @minus{}3, 0, #b0111, #xFEED, |
| 2028 | 19 #o744. Their values are exact, and their range is limited. The |
| 428 | 20 number prefixes `#b', `#o', and `#x' are supported to represent numbers |
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21 in binary, octal, and hexadecimal notation (or radix). The syntax |
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22 `#NNNrNUM' is also supported, allowing a radix to be explicitly |
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23 specified; e.g. `#20r15' denotes the fixnum 25, expressed in base |
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24 twenty. |
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25 |
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26 Floating point numbers are numbers with fractional parts, such as |
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27 @minus{}4.5, 0.0, or 2.71828. They can also be expressed in exponential |
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28 notation: 1.5e2 equals 150; in this example, @samp{e2} stands for ten to |
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29 the second power, and is multiplied by 1.5. Floating point values are |
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30 not exact; they have a fixed, limited amount of precision. |
| 428 | 31 |
| 2028 | 32 Bignums are arbitrary precision integers. When supported, XEmacs can |
| 33 handle any integral calculations you have enough virtual memory to | |
| 34 store. (More precisely, on current architectures the representation | |
| 35 allows integers whose storage would exhaust the address space.) They | |
| 36 are notated in the same way as other integers (fixnums). XEmacs | |
| 37 automatically converts results of computations from fixnum to bignum, | |
| 38 and back, depending on the storage required to represent the number. | |
| 39 Thus use of bignums are entirely transparent to the user, except for a | |
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40 few special applications that expect overflows. |
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41 |
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42 Ratios are rational numbers with arbitrary precision. They are notated |
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43 in the usual way with the solidus, for example 5/3 or @minus{}22/7. The |
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44 syntax for using a particular radix is the same as that for integers, |
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45 so, e.g., @samp{#x8/10} is equivalent to @samp{1/2} and @samp{#30r10/4} |
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46 to @samp{15/2}, respectively. |
| 2090 | 47 |
| 48 Bigfloats are floating point numbers with arbitrary precision, which | |
| 49 may be specified by the user (and may be different for different | |
| 50 bigfloats at the same time). Unlike integers, which are always | |
| 51 infinitely precise if they can be represented, floating point numbers | |
| 52 are inherently imprecise. This means that choice of precision can be a | |
| 53 very delicate issue. XEmacs automatically converts @emph{from float to | |
| 54 bigfloat} when floats and bigfloats are mixed in an expression, but a | |
| 55 bigfloat will never be converted to a float unless the user explicitly | |
| 56 coerces the value. Nor will the result of a float operation be | |
| 57 converted to bigfloat, except for ``contagion'' from another operand | |
| 58 that is already a bigfloat. However, when bigfloats of differing | |
| 59 precision are mixed, the result will always have the larger precision. | |
| 60 The exact rules are more carefully explained elsewhere | |
| 61 (@pxref{Canonicalization and Contagion}). | |
| 2028 | 62 |
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63 Common Lisp terminology and historical Emacs terminology conflict |
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64 here, to an extent. We attempt to use ``fixnum'' and ``integer'' |
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65 consistently, but older XEmacs and GNU Emacs code and documentation use |
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66 the latter to mean the former. ``Float'' is used in Emacs documentation |
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67 to mean ``fixed precision floating point number'', and the Common Lisp |
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68 distinctions among @dfn{short-floats}, @dfn{long-floats}, @emph{etc.}, |
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69 and bigfloats (which are not standardized in Common Lisp) are not |
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70 reflected in XEmacs terminology. We're working on this, but volunteers |
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71 to fix it in the XEmacs manuals would be heartily welcomed. |
| 2028 | 72 |
| 428 | 73 @menu |
| 74 * Integer Basics:: Representation and range of integers. | |
| 2028 | 75 * Rational Basics:: Representation and range of rational numbers. |
| 76 * Float Basics:: Representation and range of floating point. | |
| 77 * The Bignum Extension:: Arbitrary precision integers, ratios, and floats. | |
| 428 | 78 * Predicates on Numbers:: Testing for numbers. |
| 79 * Comparison of Numbers:: Equality and inequality predicates. | |
| 2028 | 80 * Numeric Conversions:: Converting float to integer and vice versa. |
| 428 | 81 * Arithmetic Operations:: How to add, subtract, multiply and divide. |
| 82 * Rounding Operations:: Explicitly rounding floating point numbers. | |
| 83 * Bitwise Operations:: Logical and, or, not, shifting. | |
| 84 * Math Functions:: Trig, exponential and logarithmic functions. | |
| 85 * Random Numbers:: Obtaining random integers, predictable or not. | |
| 86 @end menu | |
| 87 | |
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88 @node Integer Basics, Rational Basics, Numbers, Numbers |
| 428 | 89 @section Integer Basics |
| 90 | |
| 2028 | 91 The range of values for an integer depends on the machine. If a |
| 92 multiple-precision arithmetic library is available on your platform, | |
| 2090 | 93 support for bignums, that is, integers with arbitrary precision, may be |
| 2028 | 94 compiled in to your XEmacs. The rest of this section assumes that the |
| 95 bignum extension is @emph{not} available. The bignum extension and the | |
| 96 user-visible differences in normal integer arithmetic are discussed in a | |
| 97 separate section @ref{The Bignum Extension}. | |
| 98 | |
| 99 The minimum range is @minus{}1073741824 to 1073741823 (31 bits; i.e., | |
| 444 | 100 @ifinfo |
| 2028 | 101 -2**30 |
| 428 | 102 @end ifinfo |
| 444 | 103 @tex |
| 2028 | 104 $-2^{30}$ |
| 428 | 105 @end tex |
| 444 | 106 to |
| 107 @ifinfo | |
| 2028 | 108 2**30 - 1), |
| 428 | 109 @end ifinfo |
| 444 | 110 @tex |
| 2028 | 111 $2^{30}-1$), |
| 428 | 112 @end tex |
| 113 but some machines may provide a wider range. Many examples in this | |
| 2028 | 114 chapter assume an integer has 31 bits. |
| 428 | 115 @cindex overflow |
| 116 | |
| 2028 | 117 The range of fixnums is available to Lisp programs: |
| 118 | |
| 119 @defvar most-positive-fixnum | |
| 120 The fixed-precision integer closest in value to positive infinity. | |
| 121 @end defvar | |
| 122 | |
| 123 @defvar most-negative-fixnum | |
| 124 The fixed-precision integer closest in value to negative infinity. | |
| 125 @end defvar | |
| 126 | |
| 127 Here is a common idiom to temporarily suppress garbage collection: | |
| 128 @example | |
| 129 (garbage-collect) | |
| 130 (let ((gc-cons-threshold most-positive-fixnum)) | |
| 131 ;; allocation-intensive computation | |
| 132 ) | |
| 133 (garbage-collect) | |
| 134 @end example | |
| 135 | |
| 428 | 136 The Lisp reader reads an integer as a sequence of digits with optional |
| 137 initial sign and optional final period. | |
| 138 | |
| 139 @example | |
| 140 1 ; @r{The integer 1.} | |
| 141 1. ; @r{The integer 1.} | |
| 142 +1 ; @r{Also the integer 1.} | |
| 143 -1 ; @r{The integer @minus{}1.} | |
| 2028 | 144 2147483648 ; @r{Read error, due to overflow.} |
| 428 | 145 0 ; @r{The integer 0.} |
| 146 -0 ; @r{The integer 0.} | |
| 147 @end example | |
| 148 | |
| 149 To understand how various functions work on integers, especially the | |
| 150 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
| 151 view the numbers in their binary form. | |
| 152 | |
| 2028 | 153 In 31-bit binary, the decimal integer 5 looks like this: |
| 428 | 154 |
| 155 @example | |
| 2028 | 156 000 0000 0000 0000 0000 0000 0000 0101 |
| 428 | 157 @end example |
| 158 | |
| 159 @noindent | |
| 160 (We have inserted spaces between groups of 4 bits, and two spaces | |
| 161 between groups of 8 bits, to make the binary integer easier to read.) | |
| 162 | |
| 163 The integer @minus{}1 looks like this: | |
| 164 | |
| 165 @example | |
| 2028 | 166 111 1111 1111 1111 1111 1111 1111 1111 |
| 428 | 167 @end example |
| 168 | |
| 169 @noindent | |
| 170 @cindex two's complement | |
| 2028 | 171 @minus{}1 is represented as 31 ones. (This is called @dfn{two's |
| 428 | 172 complement} notation.) |
| 173 | |
| 174 The negative integer, @minus{}5, is creating by subtracting 4 from | |
| 175 @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
| 176 @minus{}5 looks like this: | |
| 177 | |
| 178 @example | |
| 2028 | 179 111 1111 1111 1111 1111 1111 1111 1011 |
| 428 | 180 @end example |
| 181 | |
| 2028 | 182 In this implementation, the largest 31-bit binary integer is the |
| 183 decimal integer 1,073,741,823. In binary, it looks like this: | |
| 428 | 184 |
| 185 @example | |
| 2028 | 186 011 1111 1111 1111 1111 1111 1111 1111 |
| 428 | 187 @end example |
| 188 | |
| 189 Since the arithmetic functions do not check whether integers go | |
| 2028 | 190 outside their range, when you add 1 to 1,073,741,823, the value is the |
| 191 negative integer @minus{}1,073,741,824: | |
| 428 | 192 |
| 193 @example | |
| 2028 | 194 (+ 1 1073741823) |
| 195 @result{} -1073741824 | |
| 196 @result{} 100 0000 0000 0000 0000 0000 0000 0000 | |
| 428 | 197 @end example |
| 198 | |
| 2028 | 199 Many of the arithmetic functions accept markers for arguments as well |
| 428 | 200 as integers. (@xref{Markers}.) More precisely, the actual arguments to |
| 201 such functions may be either integers or markers, which is why we often | |
| 202 give these arguments the name @var{int-or-marker}. When the argument | |
| 203 value is a marker, its position value is used and its buffer is ignored. | |
| 204 | |
| 205 @ignore | |
| 206 In version 19, except where @emph{integer} is specified as an | |
| 207 argument, all of the functions for markers and integers also work for | |
| 208 floating point numbers. | |
| 209 @end ignore | |
| 210 | |
| 2028 | 211 |
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212 @node Rational Basics, Float Basics, Integer Basics, Numbers |
| 2032 | 213 @section Rational Basics |
| 2028 | 214 |
| 215 Ratios (built-in rational numbers) are available only when the bignum | |
| 216 extension is built into your XEmacs. This facility is new and | |
| 217 experimental. It is discussed in a separate section for convenience of | |
| 2090 | 218 updating the documentation @ref{The Bignum Extension}. The following |
| 219 functions are defined regardless of the presence of the extension, but | |
| 220 have trivial results for integers. | |
| 221 | |
| 222 @defun numerator rational | |
| 223 @cindex numbers | |
| 224 Return the numerator of the canonical form of @var{rational}. | |
| 225 If @var{rational} is an integer, @var{rational} is returned. | |
| 226 @var{rational} must be an integer or a ratio. | |
| 227 @end defun | |
| 228 | |
| 229 @defun denominator rational | |
| 230 Return the denominator of the canonical form of @var{rational}. | |
| 231 If @var{rational} is an integer, 1 is returned. @var{rational} must be | |
| 232 an integer or a ratio. | |
| 233 @end defun | |
| 2028 | 234 |
| 235 | |
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236 @node Float Basics, The Bignum Extension, Rational Basics, Numbers |
| 428 | 237 @section Floating Point Basics |
| 238 | |
| 239 XEmacs supports floating point numbers. The precise range of floating | |
| 240 point numbers is machine-specific; it is the same as the range of the C | |
| 2028 | 241 data type @code{double} on the machine in question. If a |
| 242 multiple-precision arithmetic library is available on your platform, | |
| 243 support for bigfloats, that is, floating point numbers with arbitrary | |
| 2090 | 244 precision, may be compiled in to your XEmacs. The rest of this section |
| 2028 | 245 assumes that the bignum extension is @emph{not} available. The bigfloat |
| 246 extension and the user-visible differences in normal float arithmetic | |
| 247 are discussed in a separate section @ref{The Bignum Extension}. | |
| 428 | 248 |
| 249 The printed representation for floating point numbers requires either | |
| 250 a decimal point (with at least one digit following), an exponent, or | |
| 251 both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, | |
| 252 @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point | |
| 253 number whose value is 1500. They are all equivalent. You can also use | |
| 254 a minus sign to write negative floating point numbers, as in | |
| 255 @samp{-1.0}. | |
| 256 | |
| 257 @cindex IEEE floating point | |
| 258 @cindex positive infinity | |
| 259 @cindex negative infinity | |
| 260 @cindex infinity | |
| 261 @cindex NaN | |
| 262 Most modern computers support the IEEE floating point standard, which | |
| 263 provides for positive infinity and negative infinity as floating point | |
| 264 values. It also provides for a class of values called NaN or | |
| 265 ``not-a-number''; numerical functions return such values in cases where | |
| 266 there is no correct answer. For example, @code{(sqrt -1.0)} returns a | |
| 267 NaN. For practical purposes, there's no significant difference between | |
| 268 different NaN values in XEmacs Lisp, and there's no rule for precisely | |
| 269 which NaN value should be used in a particular case, so this manual | |
| 270 doesn't try to distinguish them. XEmacs Lisp has no read syntax for NaNs | |
| 271 or infinities; perhaps we should create a syntax in the future. | |
| 272 | |
| 273 You can use @code{logb} to extract the binary exponent of a floating | |
| 274 point number (or estimate the logarithm of an integer): | |
| 275 | |
| 276 @defun logb number | |
| 277 This function returns the binary exponent of @var{number}. More | |
| 278 precisely, the value is the logarithm of @var{number} base 2, rounded | |
| 279 down to an integer. | |
| 280 @end defun | |
| 281 | |
| 2028 | 282 The range of floats is available to Lisp programs: |
| 283 | |
| 284 @defvar most-positive-float | |
| 285 The fixed-precision floating-point-number closest in value to positive | |
| 286 infinity. | |
| 287 @end defvar | |
| 288 | |
| 289 @defvar most-negative-float | |
| 290 The fixed-precision floating point number closest in value to negative | |
| 291 infinity. | |
| 292 @end defvar | |
| 293 | |
| 294 @defvar least-positive-float | |
| 295 The positive float closest in value to 0. May not be normalized. | |
| 296 @end defvar | |
| 297 | |
| 298 @defvar least-negative-float | |
| 299 The positive float closest in value to 0. Must be normalized. | |
| 300 @end defvar | |
| 301 | |
| 302 @defvar least-positive-normalized-float | |
| 303 The negative float closest in value to 0. May not be normalized. | |
| 304 @end defvar | |
| 305 | |
| 306 @defvar least-negative-normalized-float | |
| 307 The negative float closest in value to 0. Must be normalized. | |
| 308 @end defvar | |
| 309 | |
| 310 Note that for floating point numbers there is an interesting limit on | |
| 311 how small they can get, as well as a limit on how big they can get. In | |
| 312 some representations, a floating point number is @dfn{normalized} if the | |
| 313 leading digit is non-zero. This allows representing numbers smaller | |
| 314 than the most-negative exponent can express, by having fractional | |
| 315 mantissas. This means that the number is less precise than a normalized | |
| 316 floating point number, so Lisp programs can detect loss of precision due | |
| 317 to unnormalized floats by checking whether the number is between | |
| 318 @code{least-positive-float} and @code{least-positive-normalized-float}. | |
| 319 | |
| 320 | |
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321 @node The Bignum Extension, Predicates on Numbers, Float Basics, Numbers |
| 2028 | 322 @section The Bignum Extension |
| 323 | |
| 324 In XEmacs 21.5.18, an extension was added by @email{james@@xemacs.org, | |
| 325 Jerry James} to allow linking with arbitrary-precision arithmetic | |
| 326 libraries if they are available on your platform. ``Arbitrary'' | |
| 327 precision means precisely what it says. Your ability to work with large | |
| 328 numbers is limited only by the amount of virtual memory (and time) you | |
| 329 can throw at them. | |
| 330 | |
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331 XEmacs supports the GNU Multiple Precision arithmetic library (GMP), |
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332 the Multiple Precision Integers and Rationals library (MPIR), and the |
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333 BSD Multiple Precision arithmetic library (MP). To enable bignum |
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334 support using GMP, MPIR, or MP, invoke configure with your usual options |
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335 and add @samp{--use-number-lib=gmp}, @samp{--use-number-lib=mpir}, or |
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336 @samp{--use-number-lib=mp}, respectively. The default is to disable |
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337 bignum support, but if you are using a script to automate the build |
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338 process, it may be convenient to explicitly disable support by |
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339 @emph{appending} @samp{--use-number-lib=no} to your invocation of |
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340 configure. GMP has an MP compatibility mode, but it is not recommended, |
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341 as there remain poorly understood bugs (even more so than for other |
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342 vendors' versions of MP). |
| 2028 | 343 |
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344 With GMP and MPIR, exact arithmetic with integers and ratios of |
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345 arbitrary precision and approximate (``floating point'') arithmetic of |
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346 arbitrary precision are implemented efficiently in the library. (Note |
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347 that numerical implementations are quite delicate and sensitive to |
| 2028 | 348 optimization. If the library was poorly optimized for your hardware, as |
| 349 is often the case with Linux distributions for 80x86, you may achieve | |
| 350 gains of @emph{several orders of magnitude} by rebuilding the MP | |
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351 library. See @uref{http://gmplib.org/gmpbench.html}.) The MP |
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352 implementation provides arbitrary precision integers. Ratios and |
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353 arbitrary precision floats are not available with MP. |
| 2028 | 354 |
| 2033 | 355 If your code needs to run correctly whether or not the feature is |
| 356 provided, you may test for the features @code{bignum}, @code{ratio}, and | |
| 357 @code{bigfloat}. | |
| 358 | |
| 2090 | 359 The XEmacs bignum facility implements the Common Lisp notions of |
| 360 @dfn{canonicalization} and @dfn{contagion}. Canonicalization means that | |
| 361 in exact (integer and ratio) arithmetic, a result of an operation is | |
| 362 always converted to the ``smallest'' type that can represent it | |
| 363 exactly. For exact numbers, the user only cares if efficiency is | |
| 364 extremely important; Lisp does not try to determine an order of | |
| 365 computation that avoids conversion to bignum (or ratio) even if one is | |
| 366 available. (Note that integers are never silently converted to | |
| 367 ratios: the result of @code{(/ 1 2)} is the integer @code{0}. You can | |
| 368 @emph{request} that a ratio be used if needed with @code{(div 1 2)}.) | |
| 369 | |
| 370 Since floating point arithmetic is inherently imprecise, numbers are | |
| 371 implicitly coerced to bigfloats only if other operands in the expression | |
| 372 are bigfloat, and bigfloats are only coerced to other numerical types by | |
| 373 explicit calls to the function @code{coerce}. | |
| 2028 | 374 |
| 375 @menu | |
| 376 * Bignum Basics:: Representation and range of integers. | |
| 377 * Ratio Basics:: Representation and range of rational numbers. | |
| 378 * Bigfloat Basics:: Representation and range of floating point. | |
| 2090 | 379 * Canonicalization and Contagion:: Automatic coercion to other types. |
| 2028 | 380 * Compatibility Issues:: Changes in fixed-precision arithmetic. |
| 381 @end menu | |
| 382 | |
| 383 | |
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384 @node Bignum Basics, Ratio Basics, The Bignum Extension, The Bignum Extension |
| 2028 | 385 @subsection Bignum Basics |
| 386 | |
| 387 In most cases, bignum support should be transparent to users and Lisp | |
| 388 programmers. A bignum-enabled XEmacs will automatically convert from | |
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389 fixnums to bignums and back in pure integer arithmetic, and for GMP and |
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390 MPIR, from floats to bigfloats. (Bigfloats must be explicitly coerced |
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391 to other types, even if they are exactly representable by less precise |
| 2028 | 392 types.) The Lisp reader and printer have been enhanced to handle |
| 393 bignums, as have the mathematical functions. Rationals (fixnums, | |
| 394 bignums, and ratios) are printed using the @samp{%d}, @samp{%o}, | |
| 395 @samp{%x}, and @samp{%u} format conversions. | |
| 396 | |
| 397 | |
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398 @node Ratio Basics, Bigfloat Basics, Bignum Basics, The Bignum Extension |
| 2028 | 399 @subsection Ratio Basics |
| 400 | |
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401 Ratios, when available, have the read syntax and print representation |
| 2028 | 402 @samp{3/5}. Like other rationals (fixnums and bignums), they are |
| 403 printed using the @samp{%d}, @samp{%o}, @samp{%x}, and @samp{%u} format | |
| 404 conversions. | |
| 405 | |
| 406 | |
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407 @node Bigfloat Basics, Canonicalization and Contagion, Ratio Basics, The Bignum Extension |
| 2028 | 408 @subsection Bigfloat Basics |
| 409 | |
| 410 Bigfloats, when available, have the same read syntax and print | |
| 411 representations as fixed-precision floats. | |
| 412 | |
| 2182 | 413 It is possible to make bigfloat the default floating point format by |
| 414 setting @code{default-float-precision} to a non-zero value. Precision | |
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415 is given in bits, with a maximum precision of |
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416 @code{bigfloat-maximum-precision}. |
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417 |
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418 @example |
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419 (let* ((float1 (string-to-number "9999999999999999.99999999999999999999")) |
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420 (default-float-precision 256) |
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421 (float2 (string-to-number "9999999999999999.99999999999999999999"))) |
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422 (+ (if (bigfloatp float1) 1 0) |
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423 (if (bigfloatp float2) 2 0))) |
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424 @result{} 2 |
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425 @end example |
| 2182 | 426 |
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427 @example |
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428 (let* ((float1 (float 999999999999999999999999999999999999999999999999999999)) |
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429 (default-float-precision 256) |
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430 (float2 (float 999999999999999999999999999999999999999999999999999999))) |
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431 (princ float1) |
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432 (terpri) |
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433 (princ float2) |
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434 (terpri)) |
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435 @result{} 9.999999999999999e+53 |
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436 9.99999999999999999999999999999999999999999999999999999E53 |
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437 t |
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438 @end example |
| 2182 | 439 |
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440 Explicit type coercion is also available, although then the precision of |
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441 the bigfloat is no greater than the source type. |
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442 |
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443 @example |
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444 (coerce 999999999999999 'bigfloat) |
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445 @result{} 9.99999999999999E14 |
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446 @end example |
| 2028 | 447 |
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448 @node Canonicalization and Contagion, Compatibility Issues, Bigfloat Basics, The Bignum Extension |
| 2090 | 449 @subsection Canonicalization and Contagion |
| 450 | |
| 451 @dfn{Canonicalization} is a rule intended to enhance the time and space | |
| 452 efficiency of exact arithmetic. Because bignums and ratios are | |
| 453 implemented as record objects, they take up much more space than | |
| 454 fixnums, which are implemented as an immediate object. Conversions and | |
| 455 calls to the MP library also take time. So the implementation always | |
| 456 converts the result of exact arithmetic to the smallest representation | |
| 457 that can exactly represent the quantity. | |
| 458 | |
| 459 @example | |
| 460 (+ 3/4 5) | |
| 461 @result{} 23/4 | |
| 462 | |
| 463 (+ 3/4 1/4 2) | |
| 464 @result{} 3 | |
| 465 @end example | |
| 466 | |
| 467 Conversely, if an integer (read or computed) cannot be represented as a | |
| 468 fixnum, a bignum will be used. Integer division is a somewhat | |
| 469 exceptional case. Because it is useful and is the historical meaning of | |
| 470 the function @code{/}, a separate function @code{div} is provided. | |
| 471 @code{div} is identical to @code{/} except that when the rational result | |
| 472 is not an integer, it is represented exactly as a ratio. In both cases | |
| 473 if a rational result is an integer, it is automatically converted to the | |
| 474 appropriate integral representation. | |
| 475 | |
| 476 Note that the efficiency gain from canonicalization is likely to be | |
| 477 less than you might think. Experience with numerical analysis shows that | |
| 478 in very precise calculations, the required precision tends to increase. | |
| 479 Thus it is typically wasted effort to attempt to convert to smaller | |
| 480 representations, as the number is often reused and requires a larger | |
| 481 representation. However, XEmacs Lisp presumes that calculations using | |
| 482 bignums are the exception, so it applies canonicalization. | |
| 2028 | 483 |
| 484 @dfn{Contagion} is one way to address the requirement that an arithmetic | |
| 2090 | 485 operation should not fail because of differing types of the operands. |
| 486 Contagion is the idea that less precise operands are converted to the | |
| 487 more precise type, and then the operation is performed. While changing | |
| 488 precision is a delicate issue, contagion is so useful that XEmacs | |
| 489 performs it automatically. | |
| 2028 | 490 |
| 491 In XEmacs, the following rules of contagion are used: | |
| 492 | |
| 493 @c #### this probably wants names for each rule | |
| 494 @enumerate | |
| 495 @item | |
| 2090 | 496 If an expression mixes an integral type with a ratio, then the usual |
| 497 rules of rational arithmetic apply. (If the result of the expression | |
| 498 happens to be an integer, it will be canonicalized to integer.) | |
| 2028 | 499 |
| 500 @item | |
| 501 If an expression mixes a rational type (fixnum, bignum, or ratio) with a | |
| 502 float, the rational operand is converted to a float and the operation | |
| 503 performed if the result would fit in a float, otherwise both operands | |
| 504 are promoted to bigfloat, and the operation performed. | |
| 505 | |
| 506 @item | |
| 507 If an expression mixes any other type with a bigfloat, the other operand | |
| 508 is converted to bigfloat and the operation performed. | |
| 509 | |
| 510 @item | |
| 2090 | 511 If bigfloats of different precision are mixed, all are converted to the |
| 512 @emph{highest} precision, and the operation performed. | |
| 2028 | 513 @end enumerate |
| 514 | |
| 515 Note that there are no rules to canonicalize floats or bigfloats. This | |
| 516 might seem surprising, but in both cases information will be lost. Any | |
| 517 floating point representation is implicitly approximate. A conversion | |
| 518 to a rational type, even if it seems exact, loses this information. | |
| 519 More subtly, demoting a bigfloat to a smaller bigfloat or to a float | |
| 520 would lose information about the precision of the result, and thus some | |
| 521 information about the accuracy. Thus floating point numbers are always | |
| 522 already in canonical form. | |
| 523 | |
| 524 Of course the programmer can explicitly request canonicalization, or | |
| 525 more coercion to another type. Coercion uses the Common Lisp | |
| 526 compatibility function @code{coerce} from the @file{cl-extra.el} | |
| 527 library. A number can be explicitly converted to canonical form | |
| 528 according to the above rules using | |
| 529 | |
| 530 @defun canonicalize-number number | |
| 531 Return the canonical form of @var{number}. | |
| 532 @end defun | |
| 533 | |
| 2090 | 534 However, if we've done our job properly, this is always a no-op. That |
| 535 is, if you find a number in un-canonicalized form, please report it as a | |
| 536 bug. | |
| 537 | |
| 2028 | 538 |
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539 @node Compatibility Issues, , Canonicalization and Contagion, The Bignum Extension |
| 2028 | 540 @subsection Compatibility Issues |
| 541 | |
| 542 @emph{Surgeon General's Warning}: The automatic conversions cannot be | |
| 543 disabled at runtime. Old functions will not produce ratios unless there | |
| 544 is a ratio operand, so there should be few surprises with type | |
| 545 conflicts (the contagion rules are quite natural for Lisp programmers | |
| 546 used to the behavior of integers and floats in pre-21.5.18 XEmacsen), | |
| 547 but they can't be ruled out. Also, if you work with extremely large | |
| 548 numbers, your machine may arbitrarily decide to hand you an unpleasant | |
| 549 surprise rather than a bignum. | |
| 550 | |
| 551 User-visible changes in behavior include (in probable order of annoyance) | |
| 552 | |
| 553 @itemize | |
| 554 @item | |
| 555 Arithmetic can cause a segfault, depending on your MP library. | |
| 556 | |
| 557 GMP by default allocates temporaries on the stack. If you run out of | |
| 558 stack space, you're dead; there is no way that we know of to reliably | |
| 559 detect this condition, because @samp{alloca} is typically implemented to | |
| 560 be @emph{fast} rather than robust. If you just need a little more | |
| 561 oomph, use a bigger stack (@emph{e.g.}, the @file{ulimit -s} command in | |
| 562 bash(1)). If you want robustness at the cost of speed, configure GMP | |
| 563 with @samp{--disable-alloca} and rebuild the GMP library. | |
| 564 | |
| 565 We do not know whether BSD MP uses @samp{alloca} or not. Please send | |
| 566 any information you have as a bug report (@kbd{M-x report-xemacs-bug | |
| 567 @key{RET}}), which will give us platform information. (We do know that | |
| 568 BSD MP implementations vary across vendors, but how much, we do not know | |
| 569 yet.) | |
| 570 | |
| 571 @item | |
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572 Our documentation's terminology, and our API terminology, is not always |
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573 Common-Lisp-conforming. Many places use ``integer'' where ``fixnum'' |
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574 better reflects what the code accepts or produces; there are similar |
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575 issues for the varying types of floating point numbers. Since Emacs |
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576 Lisp has not had a ratio type before, there are no problems there. |
| 2028 | 577 |
| 578 @item | |
| 579 An atom with ratio read syntax now returns a number, not a symbol. | |
| 580 | |
| 581 @item | |
| 582 Many operations that used to cause a range error now succeed, with | |
| 583 intermediate results and return values coerced to bignums as needed. | |
| 584 | |
| 585 @item | |
| 586 The @samp{%u} format conversion will now give an error if its argument | |
| 587 is negative. (Without MP, it prints a number which Lisp can't read.) | |
| 588 @end itemize | |
| 589 | |
| 590 This is not a compatibility issue in the sense of specification, but | |
| 591 careless programmers who have taken advantage of the immediate | |
| 592 representation for numbers and written @code{(eq x y)} are in for a | |
| 593 surprise. This doesn't work with bignums, even if both arguments are | |
| 594 bignums! Arbitrary precision obviously requires consing new objects | |
| 595 because the objects are ``large'' and of variable size, and the | |
| 596 definition of @samp{eq} does not permit different objects to compare as | |
| 597 equal. Instead of @code{eq}, use @code{eql}, in which numbers of the | |
| 598 same type which have equal values compare equal, or @code{=}, which does | |
| 599 any necessary type coercions before comparing for equality | |
| 600 @ref{Comparison of Numbers}. | |
| 601 | |
| 602 | |
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603 @node Predicates on Numbers, Comparison of Numbers, The Bignum Extension, Numbers |
| 428 | 604 @section Type Predicates for Numbers |
| 605 | |
| 606 The functions in this section test whether the argument is a number or | |
| 2090 | 607 whether it is a certain sort of number. The functions which test for |
| 608 type can take any type of Lisp object as argument (the more general | |
| 609 predicates would not be of much use otherwise). However, the | |
| 610 @code{zerop} predicate requires a number as its argument, and the | |
| 611 @code{evenp}, and @code{oddp} predicates require integers as their | |
| 612 arguments. See also @code{integer-or-marker-p}, | |
| 613 @code{integer-char-or-marker-p}, @code{number-or-marker-p} and | |
| 614 @code{number-char-or-marker-p}, in @ref{Predicates on Markers}. | |
| 428 | 615 |
| 2090 | 616 @defun numberp object |
| 617 This predicate tests whether its argument is a number (either integer or | |
| 618 floating point), and returns @code{t} if so, @code{nil} otherwise. | |
| 619 @end defun | |
| 428 | 620 |
| 2090 | 621 @defun realp object |
| 622 @cindex numbers | |
| 623 The @code{realp} predicate tests to see whether @var{object} is a | |
| 624 rational or floating point number, and returns @code{t} if so, | |
| 625 @code{nil} otherwise. Currently equivalent to @code{numberp}. | |
| 626 @end defun | |
| 627 | |
| 628 @defun zerop number | |
| 629 This predicate tests whether its argument is zero, and returns @code{t} | |
| 630 if so, @code{nil} otherwise. The argument must be a number. | |
| 631 | |
| 632 These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. | |
| 428 | 633 @end defun |
| 634 | |
| 635 @defun integerp object | |
| 636 This predicate tests whether its argument is an integer, and returns | |
| 637 @code{t} if so, @code{nil} otherwise. | |
| 638 @end defun | |
| 639 | |
| 2090 | 640 @defun oddp integer |
| 641 @cindex integers | |
| 642 The @code{oddp} predicate tests to see whether @var{integer} is odd, and | |
| 643 returns @code{t} if so, @code{nil} otherwise. @var{integer} must be an | |
| 644 integer. | |
| 645 @end defun | |
| 646 | |
| 647 @defun evenp integer | |
| 648 @cindex integers | |
| 649 The @code{evenp} predicate tests to see whether @var{integer} is even, | |
| 650 and returns @code{t} if so, @code{nil} otherwise. @var{integer} must be | |
| 651 an integer. | |
| 428 | 652 @end defun |
| 653 | |
| 654 @defun natnump object | |
| 655 @cindex natural numbers | |
| 656 The @code{natnump} predicate (whose name comes from the phrase | |
| 657 ``natural-number-p'') tests to see whether its argument is a nonnegative | |
| 658 integer, and returns @code{t} if so, @code{nil} otherwise. 0 is | |
| 659 considered non-negative. | |
| 660 @end defun | |
| 661 | |
| 2090 | 662 @defun fixnump object |
| 663 @cindex integers | |
| 664 The @code{} predicate tests to see whether its argument is an integer | |
| 665 represented as a fixnum, and returns @code{t} if so, @code{nil} | |
| 666 otherwise. | |
| 667 @end defun | |
| 668 | |
| 669 @defun bignump object | |
| 670 @cindex integers | |
| 671 The @code{bignump} predicate tests to see whether @var{object} is an | |
| 672 integer represented as a bignum, and returns @code{t} if so, @code{nil} | |
| 673 otherwise. | |
| 674 @end defun | |
| 675 | |
| 676 @defun rationalp object | |
| 677 @cindex numbers | |
| 678 The @code{rationalp} predicate tests to see whether @var{object} is a | |
| 679 rational number, and returns @code{t} if so, @code{nil} otherwise. | |
| 680 @end defun | |
| 428 | 681 |
| 2090 | 682 @defun ratiop object |
| 683 @cindex ratios | |
| 684 The @code{ratiop} predicate tests to see whether @var{object} is a | |
| 685 number represented as a ratio, and returns @code{t} if so, @code{nil} | |
| 686 otherwise. | |
| 687 @end defun | |
| 688 | |
| 689 @defun floatingp object | |
| 690 @cindex floats | |
| 691 The @code{floatingp} predicate tests to see whether @var{object} is a | |
| 692 floating point number represented as a float or a bigfloat, and returns | |
| 693 @code{t} if so, @code{nil} otherwise. | |
| 428 | 694 @end defun |
| 695 | |
| 2090 | 696 @defun floatp object |
| 697 @cindex floats | |
| 698 This predicate tests whether its argument is a floating point | |
| 699 number and returns @code{t} if so, @code{nil} otherwise. | |
| 700 | |
| 701 @code{floatp} does not exist in Emacs versions 18 and earlier. If the | |
| 702 bignum extension is present, it returns @code{nil} for a bigfloat. | |
| 703 @end defun | |
| 704 | |
| 705 @defun bigfloatp object | |
| 706 @cindex floats | |
| 707 The @code{bigfloatp} predicate tests to see whether @var{object} is an | |
| 2091 | 708 floating point number represented as a bigfloat, and returns @code{t} if |
| 709 so, @code{nil} otherwise. | |
| 2090 | 710 @end defun |
| 711 | |
| 712 | |
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713 @node Comparison of Numbers, Numeric Conversions, Predicates on Numbers, Numbers |
| 428 | 714 @section Comparison of Numbers |
| 715 @cindex number equality | |
| 716 | |
| 717 To test numbers for numerical equality, you should normally use | |
| 2090 | 718 @code{=}, not @code{eq}. There can be many distinct floating point, |
| 719 bignum, and ratio number objects with the same numeric value. If you | |
| 720 use @code{eq} to compare them, then you test whether two values are the | |
| 721 same @emph{object}. By contrast, @code{=} compares only the numeric | |
| 722 values of the objects. | |
| 428 | 723 |
| 2028 | 724 In versions before 21.5.18, each integer value had a unique Lisp |
| 725 object in XEmacs Lisp. Therefore, @code{eq} was equivalent to @code{=} | |
| 726 where integers are concerned. Even with the introduction of bignums, it | |
| 727 is sometimes convenient to use @code{eq} for comparing an unknown value | |
| 728 with an integer, because @code{eq} does not report an error if the | |
| 729 unknown value is not a number---it accepts arguments of any type. By | |
| 730 contrast, @code{=} signals an error if the arguments are not numbers or | |
| 731 markers. However, it is a good idea to use @code{=} if you can, even | |
| 732 for comparing exact values, because two bignums or ratios with the same | |
| 733 value will often not be the same object. | |
| 428 | 734 |
| 2090 | 735 On the other hand, some functions, such as the string- and |
| 736 buffer-searching functions, will return an integer on success, but | |
| 737 something else (usually @code{nil}) on failure. If it is known what the | |
| 738 numerical subtype (float, bigfloat, or exact) of the returned object | |
| 739 will be if it is a number, then the predicate @code{eql} can be used for | |
| 740 comparison without signaling an error on some expected return values. | |
| 741 Because of canonicalization, @code{eql} can be used to compare a fixnum | |
| 742 value to something that might be a ratio; if the potential ratio value | |
| 743 is representable as a fixnum, it will be canonicalized to fixnum before | |
| 2091 | 744 comparing. However, although floats and bigfloats are of different |
| 745 types for the purpose of comparisons via @code{eql}, two bigfloats of | |
| 746 different @emph{precision} that are @code{=} will always be @code{eql}. | |
| 2090 | 747 |
| 748 @example | |
| 749 (eql 2 (string-match "ere" "there")) | |
| 750 @result{} t | |
| 751 | |
| 752 (eql 2 (string-match "ere" "three")) | |
| 753 @result{} nil | |
| 754 | |
| 755 (eql 2 2.0) | |
| 756 @result{} nil | |
| 757 | |
| 758 (= 2 (string-match "ere" "there")) | |
| 759 @result{} t | |
| 760 | |
| 761 (= 2 (string-match "ere" "three")) | |
| 762 @error{} Wrong type argument: number-char-or-marker-p, nil | |
| 763 | |
| 764 (= 2 2.0) | |
| 765 @result{} t | |
| 766 @end example | |
| 767 | |
| 768 | |
| 769 | |
| 428 | 770 There is another wrinkle: because floating point arithmetic is not |
| 771 exact, it is often a bad idea to check for equality of two floating | |
| 772 point values. Usually it is better to test for approximate equality. | |
| 773 Here's a function to do this: | |
| 774 | |
| 775 @example | |
| 776 (defconst fuzz-factor 1.0e-6) | |
| 777 (defun approx-equal (x y) | |
| 778 (or (and (= x 0) (= y 0)) | |
| 779 (< (/ (abs (- x y)) | |
| 780 (max (abs x) (abs y))) | |
| 781 fuzz-factor))) | |
| 782 @end example | |
| 783 | |
| 784 @cindex CL note---integers vrs @code{eq} | |
| 785 @quotation | |
| 786 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
| 787 @code{=} because Common Lisp implements multi-word integers, and two | |
| 788 distinct integer objects can have the same numeric value. XEmacs Lisp | |
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789 can have just one fixnum object for any given value because it has a |
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790 limited range of fixnum values. |
| 428 | 791 @end quotation |
| 792 | |
| 793 In addition to numbers, all of the following functions also accept | |
| 794 characters and markers as arguments, and treat them as their number | |
| 795 equivalents. | |
| 796 | |
| 797 @defun = number &rest more-numbers | |
| 798 This function returns @code{t} if all of its arguments are numerically | |
| 799 equal, @code{nil} otherwise. | |
| 800 | |
| 801 @example | |
| 802 (= 5) | |
| 803 @result{} t | |
| 804 (= 5 6) | |
| 805 @result{} nil | |
| 806 (= 5 5.0) | |
| 807 @result{} t | |
| 808 (= 5 5 6) | |
| 809 @result{} nil | |
| 810 @end example | |
| 811 @end defun | |
| 812 | |
| 813 @defun /= number &rest more-numbers | |
| 814 This function returns @code{t} if no two arguments are numerically | |
| 815 equal, @code{nil} otherwise. | |
| 816 | |
| 817 @example | |
| 818 (/= 5 6) | |
| 819 @result{} t | |
| 820 (/= 5 5 6) | |
| 821 @result{} nil | |
| 822 (/= 5 6 1) | |
| 823 @result{} t | |
| 824 @end example | |
| 825 @end defun | |
| 826 | |
| 827 @defun < number &rest more-numbers | |
| 828 This function returns @code{t} if the sequence of its arguments is | |
| 829 monotonically increasing, @code{nil} otherwise. | |
| 830 | |
| 831 @example | |
| 832 (< 5 6) | |
| 833 @result{} t | |
| 834 (< 5 6 6) | |
| 835 @result{} nil | |
| 836 (< 5 6 7) | |
| 837 @result{} t | |
| 838 @end example | |
| 839 @end defun | |
| 840 | |
| 841 @defun <= number &rest more-numbers | |
| 842 This function returns @code{t} if the sequence of its arguments is | |
| 843 monotonically nondecreasing, @code{nil} otherwise. | |
| 844 | |
| 845 @example | |
| 846 (<= 5 6) | |
| 847 @result{} t | |
| 848 (<= 5 6 6) | |
| 849 @result{} t | |
| 850 (<= 5 6 5) | |
| 851 @result{} nil | |
| 852 @end example | |
| 853 @end defun | |
| 854 | |
| 855 @defun > number &rest more-numbers | |
| 856 This function returns @code{t} if the sequence of its arguments is | |
| 857 monotonically decreasing, @code{nil} otherwise. | |
| 858 @end defun | |
| 859 | |
| 860 @defun >= number &rest more-numbers | |
| 861 This function returns @code{t} if the sequence of its arguments is | |
| 862 monotonically nonincreasing, @code{nil} otherwise. | |
| 863 @end defun | |
| 864 | |
| 865 @defun max number &rest more-numbers | |
| 866 This function returns the largest of its arguments. | |
| 867 | |
| 868 @example | |
| 869 (max 20) | |
| 870 @result{} 20 | |
| 871 (max 1 2.5) | |
| 872 @result{} 2.5 | |
| 873 (max 1 3 2.5) | |
| 874 @result{} 3 | |
| 875 @end example | |
| 876 @end defun | |
| 877 | |
| 878 @defun min number &rest more-numbers | |
| 879 This function returns the smallest of its arguments. | |
| 880 | |
| 881 @example | |
| 882 (min -4 1) | |
| 883 @result{} -4 | |
| 884 @end example | |
| 885 @end defun | |
| 886 | |
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887 @node Numeric Conversions, Arithmetic Operations, Comparison of Numbers, Numbers |
| 428 | 888 @section Numeric Conversions |
| 889 @cindex rounding in conversions | |
| 890 | |
| 891 To convert an integer to floating point, use the function @code{float}. | |
| 892 | |
| 893 @defun float number | |
| 894 This returns @var{number} converted to floating point. | |
| 895 If @var{number} is already a floating point number, @code{float} returns | |
| 896 it unchanged. | |
| 897 @end defun | |
| 898 | |
| 899 There are four functions to convert floating point numbers to integers; | |
| 900 they differ in how they round. These functions accept integer arguments | |
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901 also, and return such arguments unchanged. They return multiple values, |
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902 @pxref{(cl.info)Multiple values}. |
| 428 | 903 |
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904 All these functions take optional @var{divisor} arguments, and if this |
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905 argument is specified, the @var{number} argument is divided by |
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906 @var{divisor} before the calculation is made. An @code{arith-error} |
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907 results if @var{divisor} is 0. |
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908 |
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909 @defun truncate number &optional divisor |
| 428 | 910 This returns @var{number}, converted to an integer by rounding towards |
| 911 zero. | |
| 912 @end defun | |
| 913 | |
| 914 @defun floor number &optional divisor | |
| 915 This returns @var{number}, converted to an integer by rounding downward | |
| 916 (towards negative infinity). | |
| 917 @end defun | |
| 918 | |
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919 @defun ceiling number &optional divisor |
| 428 | 920 This returns @var{number}, converted to an integer by rounding upward |
| 921 (towards positive infinity). | |
| 922 @end defun | |
| 923 | |
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924 @defun round number &optional divisor |
| 428 | 925 This returns @var{number}, converted to an integer by rounding towards the |
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926 nearest integer. |
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927 |
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928 Rounding a value equidistant between two integers chooses the even |
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929 integer. GNU Emacs and older XEmacs did not guarantee this, and the |
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930 direction of rounding depended on the underlying machine and the C |
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931 implementation. |
| 428 | 932 @end defun |
| 933 | |
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934 @node Arithmetic Operations, Rounding Operations, Numeric Conversions, Numbers |
| 428 | 935 @section Arithmetic Operations |
| 936 | |
| 937 XEmacs Lisp provides the traditional four arithmetic operations: | |
| 938 addition, subtraction, multiplication, and division. Remainder and modulus | |
| 939 functions supplement the division functions. The functions to | |
| 940 add or subtract 1 are provided because they are traditional in Lisp and | |
| 941 commonly used. | |
| 942 | |
| 943 All of these functions except @code{%} return a floating point value | |
| 944 if any argument is floating. | |
| 945 | |
| 946 It is important to note that in XEmacs Lisp, arithmetic functions | |
| 947 do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to | |
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948 @minus{}134217728, depending on your hardware and whether your XEmacs |
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949 supports bignums. |
| 428 | 950 |
| 444 | 951 @defun 1+ number |
| 952 This function returns @var{number} plus one. @var{number} may be a | |
| 953 number, character or marker. Markers and characters are converted to | |
| 954 integers. | |
| 955 | |
| 428 | 956 For example, |
| 957 | |
| 958 @example | |
| 959 (setq foo 4) | |
| 960 @result{} 4 | |
| 961 (1+ foo) | |
| 962 @result{} 5 | |
| 963 @end example | |
| 964 | |
| 965 This function is not analogous to the C operator @code{++}---it does not | |
| 966 increment a variable. It just computes a sum. Thus, if we continue, | |
| 967 | |
| 968 @example | |
| 969 foo | |
| 970 @result{} 4 | |
| 971 @end example | |
| 972 | |
| 973 If you want to increment the variable, you must use @code{setq}, | |
| 974 like this: | |
| 975 | |
| 976 @example | |
| 977 (setq foo (1+ foo)) | |
| 978 @result{} 5 | |
| 979 @end example | |
| 980 | |
| 981 Now that the @code{cl} package is always available from lisp code, a | |
| 982 more convenient and natural way to increment a variable is | |
| 983 @w{@code{(incf foo)}}. | |
| 984 @end defun | |
| 985 | |
| 444 | 986 @defun 1- number |
| 987 This function returns @var{number} minus one. @var{number} may be a | |
| 988 number, character or marker. Markers and characters are converted to | |
| 989 integers. | |
| 428 | 990 @end defun |
| 991 | |
| 992 @defun abs number | |
| 993 This returns the absolute value of @var{number}. | |
| 994 @end defun | |
| 995 | |
| 444 | 996 @defun + &rest numbers |
| 428 | 997 This function adds its arguments together. When given no arguments, |
| 998 @code{+} returns 0. | |
| 999 | |
| 444 | 1000 If any of the arguments are characters or markers, they are first |
| 1001 converted to integers. | |
| 1002 | |
| 428 | 1003 @example |
| 1004 (+) | |
| 1005 @result{} 0 | |
| 1006 (+ 1) | |
| 1007 @result{} 1 | |
| 1008 (+ 1 2 3 4) | |
| 1009 @result{} 10 | |
| 1010 @end example | |
| 1011 @end defun | |
| 1012 | |
| 444 | 1013 @defun - &optional number &rest other-numbers |
| 428 | 1014 The @code{-} function serves two purposes: negation and subtraction. |
| 1015 When @code{-} has a single argument, the value is the negative of the | |
| 1016 argument. When there are multiple arguments, @code{-} subtracts each of | |
| 444 | 1017 the @var{other-numbers} from @var{number}, cumulatively. If there are |
| 1018 no arguments, an error is signaled. | |
| 1019 | |
| 1020 If any of the arguments are characters or markers, they are first | |
| 1021 converted to integers. | |
| 428 | 1022 |
| 1023 @example | |
| 1024 (- 10 1 2 3 4) | |
| 1025 @result{} 0 | |
| 1026 (- 10) | |
| 1027 @result{} -10 | |
| 1028 (-) | |
| 1029 @result{} 0 | |
| 1030 @end example | |
| 1031 @end defun | |
| 1032 | |
| 444 | 1033 @defun * &rest numbers |
| 428 | 1034 This function multiplies its arguments together, and returns the |
| 1035 product. When given no arguments, @code{*} returns 1. | |
| 1036 | |
| 444 | 1037 If any of the arguments are characters or markers, they are first |
| 1038 converted to integers. | |
| 1039 | |
| 428 | 1040 @example |
| 1041 (*) | |
| 1042 @result{} 1 | |
| 1043 (* 1) | |
| 1044 @result{} 1 | |
| 1045 (* 1 2 3 4) | |
| 1046 @result{} 24 | |
| 1047 @end example | |
| 1048 @end defun | |
| 1049 | |
| 444 | 1050 @defun / dividend &rest divisors |
| 1051 The @code{/} function serves two purposes: inversion and division. When | |
| 1052 @code{/} has a single argument, the value is the inverse of the | |
| 1053 argument. When there are multiple arguments, @code{/} divides | |
| 1054 @var{dividend} by each of the @var{divisors}, cumulatively, returning | |
| 1055 the quotient. If there are no arguments, an error is signaled. | |
| 428 | 1056 |
| 444 | 1057 If none of the arguments are floats, then the result is an integer. |
| 428 | 1058 This means the result has to be rounded. On most machines, the result |
| 1059 is rounded towards zero after each division, but some machines may round | |
| 1060 differently with negative arguments. This is because the Lisp function | |
| 1061 @code{/} is implemented using the C division operator, which also | |
| 1062 permits machine-dependent rounding. As a practical matter, all known | |
| 1063 machines round in the standard fashion. | |
| 1064 | |
| 444 | 1065 If any of the arguments are characters or markers, they are first |
| 1066 converted to integers. | |
| 1067 | |
| 428 | 1068 @cindex @code{arith-error} in division |
| 1069 If you divide by 0, an @code{arith-error} error is signaled. | |
| 1070 (@xref{Errors}.) | |
| 1071 | |
| 1072 @example | |
| 1073 @group | |
| 1074 (/ 6 2) | |
| 1075 @result{} 3 | |
| 1076 @end group | |
| 1077 (/ 5 2) | |
| 1078 @result{} 2 | |
| 1079 (/ 25 3 2) | |
| 1080 @result{} 4 | |
| 444 | 1081 (/ 3.0) |
| 1082 @result{} 0.3333333333333333 | |
| 428 | 1083 (/ -17 6) |
| 1084 @result{} -2 | |
| 1085 @end example | |
| 1086 | |
| 1087 The result of @code{(/ -17 6)} could in principle be -3 on some | |
| 1088 machines. | |
| 1089 @end defun | |
| 1090 | |
| 1091 @defun % dividend divisor | |
| 1092 @cindex remainder | |
| 1093 This function returns the integer remainder after division of @var{dividend} | |
| 1094 by @var{divisor}. The arguments must be integers or markers. | |
| 1095 | |
| 1096 For negative arguments, the remainder is in principle machine-dependent | |
| 1097 since the quotient is; but in practice, all known machines behave alike. | |
| 1098 | |
| 1099 An @code{arith-error} results if @var{divisor} is 0. | |
| 1100 | |
| 1101 @example | |
| 1102 (% 9 4) | |
| 1103 @result{} 1 | |
| 1104 (% -9 4) | |
| 1105 @result{} -1 | |
| 1106 (% 9 -4) | |
| 1107 @result{} 1 | |
| 1108 (% -9 -4) | |
| 1109 @result{} -1 | |
| 1110 @end example | |
| 1111 | |
| 1112 For any two integers @var{dividend} and @var{divisor}, | |
| 1113 | |
| 1114 @example | |
| 1115 @group | |
| 1116 (+ (% @var{dividend} @var{divisor}) | |
| 1117 (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
| 1118 @end group | |
| 1119 @end example | |
| 1120 | |
| 1121 @noindent | |
| 1122 always equals @var{dividend}. | |
| 1123 @end defun | |
| 1124 | |
| 1125 @defun mod dividend divisor | |
| 1126 @cindex modulus | |
| 1127 This function returns the value of @var{dividend} modulo @var{divisor}; | |
| 1128 in other words, the remainder after division of @var{dividend} | |
| 1129 by @var{divisor}, but with the same sign as @var{divisor}. | |
| 1130 The arguments must be numbers or markers. | |
| 1131 | |
| 1132 Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
| 1133 arguments. It also permits floating point arguments; it rounds the | |
| 1134 quotient downward (towards minus infinity) to an integer, and uses that | |
| 1135 quotient to compute the remainder. | |
| 1136 | |
| 1137 An @code{arith-error} results if @var{divisor} is 0. | |
| 1138 | |
| 1139 @example | |
| 1140 @group | |
| 1141 (mod 9 4) | |
| 1142 @result{} 1 | |
| 1143 @end group | |
| 1144 @group | |
| 1145 (mod -9 4) | |
| 1146 @result{} 3 | |
| 1147 @end group | |
| 1148 @group | |
| 1149 (mod 9 -4) | |
| 1150 @result{} -3 | |
| 1151 @end group | |
| 1152 @group | |
| 1153 (mod -9 -4) | |
| 1154 @result{} -1 | |
| 1155 @end group | |
| 1156 @group | |
| 1157 (mod 5.5 2.5) | |
| 1158 @result{} .5 | |
| 1159 @end group | |
| 1160 @end example | |
| 1161 | |
| 1162 For any two numbers @var{dividend} and @var{divisor}, | |
| 1163 | |
| 1164 @example | |
| 1165 @group | |
| 1166 (+ (mod @var{dividend} @var{divisor}) | |
| 1167 (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
| 1168 @end group | |
| 1169 @end example | |
| 1170 | |
| 1171 @noindent | |
| 1172 always equals @var{dividend}, subject to rounding error if either | |
| 1173 argument is floating point. For @code{floor}, see @ref{Numeric | |
| 1174 Conversions}. | |
| 1175 @end defun | |
| 1176 | |
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1177 @node Rounding Operations, Bitwise Operations, Arithmetic Operations, Numbers |
| 428 | 1178 @section Rounding Operations |
| 1179 @cindex rounding without conversion | |
| 1180 | |
| 1181 The functions @code{ffloor}, @code{fceiling}, @code{fround} and | |
| 1182 @code{ftruncate} take a floating point argument and return a floating | |
| 1183 point result whose value is a nearby integer. @code{ffloor} returns the | |
| 1184 nearest integer below; @code{fceiling}, the nearest integer above; | |
| 1185 @code{ftruncate}, the nearest integer in the direction towards zero; | |
| 1186 @code{fround}, the nearest integer. | |
| 1187 | |
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1188 All these functions take optional @var{divisor} arguments, and if this |
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1189 argument is specified, the @var{number} argument is divided by |
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1190 @var{divisor} before the calculation is made. An @code{arith-error} |
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1191 results if @var{divisor} is 0. Also, they return multiple values, |
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1192 @pxref{(cl.info)Multiple values}; the second value is the remainder. |
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1193 |
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1194 @defun ffloor number &optional divisor |
| 444 | 1195 This function rounds @var{number} to the next lower integral value, and |
| 428 | 1196 returns that value as a floating point number. |
| 1197 @end defun | |
| 1198 | |
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1199 @defun fceiling number &optional divisor |
| 444 | 1200 This function rounds @var{number} to the next higher integral value, and |
| 428 | 1201 returns that value as a floating point number. |
| 1202 @end defun | |
| 1203 | |
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1204 @defun ftruncate number &optional divisor |
| 444 | 1205 This function rounds @var{number} towards zero to an integral value, and |
| 428 | 1206 returns that value as a floating point number. |
| 1207 @end defun | |
| 1208 | |
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1209 @defun fround number &optional divisor |
| 444 | 1210 This function rounds @var{number} to the nearest integral value, |
| 428 | 1211 and returns that value as a floating point number. |
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1212 |
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1213 Rounding a value equidistant between two integral values chooses the |
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1214 even value. While this is specified by Common Lisp, GNU Emacs and older |
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1215 XEmacs did not make this guarantee, and the direction of rounding |
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1216 depended on the underlying machine and the C implementation. |
| 428 | 1217 @end defun |
| 1218 | |
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1219 @node Bitwise Operations, Math Functions, Rounding Operations, Numbers |
| 428 | 1220 @section Bitwise Operations on Integers |
| 1221 | |
| 1222 In a computer, an integer is represented as a binary number, a | |
| 1223 sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
| 1224 operation acts on the individual bits of such a sequence. For example, | |
| 1225 @dfn{shifting} moves the whole sequence left or right one or more places, | |
| 1226 reproducing the same pattern ``moved over''. | |
| 1227 | |
| 1228 The bitwise operations in XEmacs Lisp apply only to integers. | |
| 1229 | |
| 1230 @defun lsh integer1 count | |
| 1231 @cindex logical shift | |
| 1232 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
| 1233 bits in @var{integer1} to the left @var{count} places, or to the right | |
| 1234 if @var{count} is negative, bringing zeros into the vacated bits. If | |
| 1235 @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
| 1236 (most-significant) bit, producing a positive result even if | |
| 1237 @var{integer1} is negative. Contrast this with @code{ash}, below. | |
| 1238 | |
| 1239 Here are two examples of @code{lsh}, shifting a pattern of bits one | |
| 1240 place to the left. We show only the low-order eight bits of the binary | |
| 1241 pattern; the rest are all zero. | |
| 1242 | |
| 1243 @example | |
| 1244 @group | |
| 1245 (lsh 5 1) | |
| 1246 @result{} 10 | |
| 1247 ;; @r{Decimal 5 becomes decimal 10.} | |
| 1248 00000101 @result{} 00001010 | |
| 1249 | |
| 1250 (lsh 7 1) | |
| 1251 @result{} 14 | |
| 1252 ;; @r{Decimal 7 becomes decimal 14.} | |
| 1253 00000111 @result{} 00001110 | |
| 1254 @end group | |
| 1255 @end example | |
| 1256 | |
| 1257 @noindent | |
| 1258 As the examples illustrate, shifting the pattern of bits one place to | |
| 1259 the left produces a number that is twice the value of the previous | |
| 1260 number. | |
| 1261 | |
| 1262 Shifting a pattern of bits two places to the left produces results | |
| 1263 like this (with 8-bit binary numbers): | |
| 1264 | |
| 1265 @example | |
| 1266 @group | |
| 1267 (lsh 3 2) | |
| 1268 @result{} 12 | |
| 1269 ;; @r{Decimal 3 becomes decimal 12.} | |
| 444 | 1270 00000011 @result{} 00001100 |
| 428 | 1271 @end group |
| 1272 @end example | |
| 1273 | |
| 1274 On the other hand, shifting one place to the right looks like this: | |
| 1275 | |
| 1276 @example | |
| 1277 @group | |
| 1278 (lsh 6 -1) | |
| 1279 @result{} 3 | |
| 1280 ;; @r{Decimal 6 becomes decimal 3.} | |
| 444 | 1281 00000110 @result{} 00000011 |
| 428 | 1282 @end group |
| 1283 | |
| 1284 @group | |
| 1285 (lsh 5 -1) | |
| 1286 @result{} 2 | |
| 1287 ;; @r{Decimal 5 becomes decimal 2.} | |
| 444 | 1288 00000101 @result{} 00000010 |
| 428 | 1289 @end group |
| 1290 @end example | |
| 1291 | |
| 1292 @noindent | |
| 1293 As the example illustrates, shifting one place to the right divides the | |
| 1294 value of a positive integer by two, rounding downward. | |
| 1295 | |
| 1296 The function @code{lsh}, like all XEmacs Lisp arithmetic functions, does | |
| 1297 not check for overflow, so shifting left can discard significant bits | |
| 1298 and change the sign of the number. For example, left shifting | |
| 1299 134,217,727 produces @minus{}2 on a 28-bit machine: | |
| 1300 | |
| 1301 @example | |
| 1302 (lsh 134217727 1) ; @r{left shift} | |
| 1303 @result{} -2 | |
| 1304 @end example | |
| 1305 | |
| 1306 In binary, in the 28-bit implementation, the argument looks like this: | |
| 1307 | |
| 1308 @example | |
| 1309 @group | |
| 1310 ;; @r{Decimal 134,217,727} | |
| 444 | 1311 0111 1111 1111 1111 1111 1111 1111 |
| 428 | 1312 @end group |
| 1313 @end example | |
| 1314 | |
| 1315 @noindent | |
| 1316 which becomes the following when left shifted: | |
| 1317 | |
| 1318 @example | |
| 1319 @group | |
| 1320 ;; @r{Decimal @minus{}2} | |
| 444 | 1321 1111 1111 1111 1111 1111 1111 1110 |
| 428 | 1322 @end group |
| 1323 @end example | |
| 1324 @end defun | |
| 1325 | |
| 1326 @defun ash integer1 count | |
| 1327 @cindex arithmetic shift | |
| 1328 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
| 1329 to the left @var{count} places, or to the right if @var{count} | |
| 1330 is negative. | |
| 1331 | |
| 1332 @code{ash} gives the same results as @code{lsh} except when | |
| 1333 @var{integer1} and @var{count} are both negative. In that case, | |
| 1334 @code{ash} puts ones in the empty bit positions on the left, while | |
| 1335 @code{lsh} puts zeros in those bit positions. | |
| 1336 | |
| 1337 Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
| 1338 looks like this: | |
| 1339 | |
| 1340 @example | |
| 1341 @group | |
| 444 | 1342 (ash -6 -1) @result{} -3 |
| 428 | 1343 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} |
| 1344 1111 1111 1111 1111 1111 1111 1010 | |
| 444 | 1345 @result{} |
| 428 | 1346 1111 1111 1111 1111 1111 1111 1101 |
| 1347 @end group | |
| 1348 @end example | |
| 1349 | |
| 1350 In contrast, shifting the pattern of bits one place to the right with | |
| 1351 @code{lsh} looks like this: | |
| 1352 | |
| 1353 @example | |
| 1354 @group | |
| 1355 (lsh -6 -1) @result{} 134217725 | |
| 1356 ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.} | |
| 1357 1111 1111 1111 1111 1111 1111 1010 | |
| 444 | 1358 @result{} |
| 428 | 1359 0111 1111 1111 1111 1111 1111 1101 |
| 1360 @end group | |
| 1361 @end example | |
| 1362 | |
| 1363 Here are other examples: | |
| 1364 | |
| 1365 @c !!! Check if lined up in smallbook format! XDVI shows problem | |
| 1366 @c with smallbook but not with regular book! --rjc 16mar92 | |
| 1367 @smallexample | |
| 1368 @group | |
| 1369 ; @r{ 28-bit binary values} | |
| 1370 | |
| 1371 (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1372 @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100} | |
| 1373 @end group | |
| 1374 @group | |
| 1375 (ash 5 2) | |
| 1376 @result{} 20 | |
| 1377 (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} | |
| 1378 @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100} | |
| 1379 (ash -5 2) | |
| 1380 @result{} -20 | |
| 1381 @end group | |
| 1382 @group | |
| 1383 (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1384 @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001} | |
| 1385 @end group | |
| 1386 @group | |
| 1387 (ash 5 -2) | |
| 1388 @result{} 1 | |
| 1389 @end group | |
| 1390 @group | |
| 1391 (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} | |
| 1392 @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110} | |
| 1393 @end group | |
| 1394 @group | |
| 1395 (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} | |
| 1396 @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110} | |
| 1397 @end group | |
| 1398 @end smallexample | |
| 1399 @end defun | |
| 1400 | |
| 1401 @defun logand &rest ints-or-markers | |
| 1402 @cindex logical and | |
| 1403 @cindex bitwise and | |
| 1404 This function returns the ``logical and'' of the arguments: the | |
| 1405 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
| 1406 set in all the arguments. (``Set'' means that the value of the bit is 1 | |
| 1407 rather than 0.) | |
| 1408 | |
| 1409 For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
| 1410 12 is 12: 1101 combined with 1100 produces 1100. | |
| 1411 In both the binary numbers, the leftmost two bits are set (i.e., they | |
| 1412 are 1's), so the leftmost two bits of the returned value are set. | |
| 1413 However, for the rightmost two bits, each is zero in at least one of | |
| 1414 the arguments, so the rightmost two bits of the returned value are 0's. | |
| 1415 | |
| 1416 @noindent | |
| 1417 Therefore, | |
| 1418 | |
| 1419 @example | |
| 1420 @group | |
| 1421 (logand 13 12) | |
| 1422 @result{} 12 | |
| 1423 @end group | |
| 1424 @end example | |
| 1425 | |
| 1426 If @code{logand} is not passed any argument, it returns a value of | |
| 1427 @minus{}1. This number is an identity element for @code{logand} | |
| 1428 because its binary representation consists entirely of ones. If | |
| 1429 @code{logand} is passed just one argument, it returns that argument. | |
| 1430 | |
| 1431 @smallexample | |
| 1432 @group | |
| 1433 ; @r{ 28-bit binary values} | |
| 1434 | |
| 1435 (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} | |
| 1436 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
| 1437 @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
| 1438 @end group | |
| 1439 | |
| 1440 @group | |
| 1441 (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} | |
| 1442 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
| 1443 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
| 1444 @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
| 1445 @end group | |
| 1446 | |
| 1447 @group | |
| 1448 (logand) | |
| 1449 @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111} | |
| 1450 @end group | |
| 1451 @end smallexample | |
| 1452 @end defun | |
| 1453 | |
| 1454 @defun logior &rest ints-or-markers | |
| 1455 @cindex logical inclusive or | |
| 1456 @cindex bitwise or | |
| 1457 This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
| 1458 is set in the result if, and only if, the @var{n}th bit is set in at least | |
| 1459 one of the arguments. If there are no arguments, the result is zero, | |
| 1460 which is an identity element for this operation. If @code{logior} is | |
| 1461 passed just one argument, it returns that argument. | |
| 1462 | |
| 1463 @smallexample | |
| 1464 @group | |
| 1465 ; @r{ 28-bit binary values} | |
| 1466 | |
| 1467 (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
| 1468 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1469 @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
| 1470 @end group | |
| 1471 | |
| 1472 @group | |
| 1473 (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
| 1474 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1475 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
| 1476 @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111} | |
| 1477 @end group | |
| 1478 @end smallexample | |
| 1479 @end defun | |
| 1480 | |
| 1481 @defun logxor &rest ints-or-markers | |
| 1482 @cindex bitwise exclusive or | |
| 1483 @cindex logical exclusive or | |
| 1484 This function returns the ``exclusive or'' of its arguments: the | |
| 1485 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
| 1486 set in an odd number of the arguments. If there are no arguments, the | |
| 1487 result is 0, which is an identity element for this operation. If | |
| 1488 @code{logxor} is passed just one argument, it returns that argument. | |
| 1489 | |
| 1490 @smallexample | |
| 1491 @group | |
| 1492 ; @r{ 28-bit binary values} | |
| 1493 | |
| 1494 (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
| 1495 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1496 @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001} | |
| 1497 @end group | |
| 1498 | |
| 1499 @group | |
| 1500 (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
| 1501 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1502 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
| 1503 @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} | |
| 1504 @end group | |
| 1505 @end smallexample | |
| 1506 @end defun | |
| 1507 | |
| 1508 @defun lognot integer | |
| 1509 @cindex logical not | |
| 1510 @cindex bitwise not | |
| 1511 This function returns the logical complement of its argument: the @var{n}th | |
| 1512 bit is one in the result if, and only if, the @var{n}th bit is zero in | |
| 1513 @var{integer}, and vice-versa. | |
| 1514 | |
| 1515 @example | |
| 444 | 1516 (lognot 5) |
| 428 | 1517 @result{} -6 |
| 1518 ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
| 1519 ;; @r{becomes} | |
| 1520 ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010} | |
| 1521 @end example | |
| 1522 @end defun | |
| 1523 | |
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1524 @node Math Functions, Random Numbers, Bitwise Operations, Numbers |
| 428 | 1525 @section Standard Mathematical Functions |
| 1526 @cindex transcendental functions | |
| 1527 @cindex mathematical functions | |
| 1528 | |
| 1529 These mathematical functions are available if floating point is | |
| 1530 supported (which is the normal state of affairs). They allow integers | |
| 1531 as well as floating point numbers as arguments. | |
| 1532 | |
| 444 | 1533 @defun sin number |
| 1534 @defunx cos number | |
| 1535 @defunx tan number | |
| 428 | 1536 These are the ordinary trigonometric functions, with argument measured |
| 1537 in radians. | |
| 1538 @end defun | |
| 1539 | |
| 444 | 1540 @defun asin number |
| 1541 The value of @code{(asin @var{number})} is a number between @minus{}pi/2 | |
| 1542 and pi/2 (inclusive) whose sine is @var{number}; if, however, @var{number} | |
| 428 | 1543 is out of range (outside [-1, 1]), then the result is a NaN. |
| 1544 @end defun | |
| 1545 | |
| 444 | 1546 @defun acos number |
| 1547 The value of @code{(acos @var{number})} is a number between 0 and pi | |
| 1548 (inclusive) whose cosine is @var{number}; if, however, @var{number} | |
| 428 | 1549 is out of range (outside [-1, 1]), then the result is a NaN. |
| 1550 @end defun | |
| 1551 | |
| 444 | 1552 @defun atan number &optional number2 |
| 1553 The value of @code{(atan @var{number})} is a number between @minus{}pi/2 | |
| 1554 and pi/2 (exclusive) whose tangent is @var{number}. | |
| 1555 | |
| 1556 If optional argument @var{number2} is supplied, the function returns | |
| 1557 @code{atan2(@var{number},@var{number2})}. | |
| 428 | 1558 @end defun |
| 1559 | |
| 444 | 1560 @defun sinh number |
| 1561 @defunx cosh number | |
| 1562 @defunx tanh number | |
| 428 | 1563 These are the ordinary hyperbolic trigonometric functions. |
| 1564 @end defun | |
| 1565 | |
| 444 | 1566 @defun asinh number |
| 1567 @defunx acosh number | |
| 1568 @defunx atanh number | |
| 428 | 1569 These are the inverse hyperbolic trigonometric functions. |
| 1570 @end defun | |
| 1571 | |
| 444 | 1572 @defun exp number |
| 428 | 1573 This is the exponential function; it returns @i{e} to the power |
| 444 | 1574 @var{number}. @i{e} is a fundamental mathematical constant also called the |
| 428 | 1575 base of natural logarithms. |
| 1576 @end defun | |
| 1577 | |
| 444 | 1578 @defun log number &optional base |
| 1579 This function returns the logarithm of @var{number}, with base @var{base}. | |
| 1738 | 1580 If you don't specify @var{base}, the base @code{e} is used. If @var{number} |
| 428 | 1581 is negative, the result is a NaN. |
| 1582 @end defun | |
| 1583 | |
| 444 | 1584 @defun log10 number |
| 1585 This function returns the logarithm of @var{number}, with base 10. If | |
| 1586 @var{number} is negative, the result is a NaN. @code{(log10 @var{x})} | |
| 428 | 1587 @equiv{} @code{(log @var{x} 10)}, at least approximately. |
| 1588 @end defun | |
| 1589 | |
| 1590 @defun expt x y | |
| 1591 This function returns @var{x} raised to power @var{y}. If both | |
| 1592 arguments are integers and @var{y} is positive, the result is an | |
| 1593 integer; in this case, it is truncated to fit the range of possible | |
| 1594 integer values. | |
| 1595 @end defun | |
| 1596 | |
| 444 | 1597 @defun sqrt number |
| 1598 This returns the square root of @var{number}. If @var{number} is negative, | |
| 428 | 1599 the value is a NaN. |
| 1600 @end defun | |
| 1601 | |
| 444 | 1602 @defun cube-root number |
| 1603 This returns the cube root of @var{number}. | |
| 428 | 1604 @end defun |
| 1605 | |
|
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Silence texinfo 5.2 warnings, primarily by adding next, prev, and up
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|
1606 @node Random Numbers, , Math Functions, Numbers |
| 428 | 1607 @section Random Numbers |
| 1608 @cindex random numbers | |
| 1609 | |
| 1610 A deterministic computer program cannot generate true random numbers. | |
| 1611 For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
| 1612 pseudo-random numbers is generated in a deterministic fashion. The | |
| 1613 numbers are not truly random, but they have certain properties that | |
| 1614 mimic a random series. For example, all possible values occur equally | |
| 1615 often in a pseudo-random series. | |
| 1616 | |
| 1617 In XEmacs, pseudo-random numbers are generated from a ``seed'' number. | |
| 1618 Starting from any given seed, the @code{random} function always | |
| 1619 generates the same sequence of numbers. XEmacs always starts with the | |
| 1620 same seed value, so the sequence of values of @code{random} is actually | |
| 1621 the same in each XEmacs run! For example, in one operating system, the | |
| 1622 first call to @code{(random)} after you start XEmacs always returns | |
| 1623 -1457731, and the second one always returns -7692030. This | |
| 1624 repeatability is helpful for debugging. | |
| 1625 | |
| 2090 | 1626 If you want reasonably unpredictable random numbers, execute |
| 1627 @code{(random t)}. This chooses a new seed based on the current time of | |
| 1628 day and on XEmacs's process @sc{id} number. (This is not | |
| 1629 cryptographically strong, it's just hard for a @emph{human} to | |
| 1630 anticipate.) | |
| 428 | 1631 |
| 1632 @defun random &optional limit | |
| 1633 This function returns a pseudo-random integer. Repeated calls return a | |
| 1634 series of pseudo-random integers. | |
| 1635 | |
| 1636 If @var{limit} is a positive integer, the value is chosen to be | |
| 1637 nonnegative and less than @var{limit}. | |
| 1638 | |
| 1639 If @var{limit} is @code{t}, it means to choose a new seed based on the | |
| 1640 current time of day and on XEmacs's process @sc{id} number. | |
| 1641 @c "XEmacs'" is incorrect usage! | |
| 2090 | 1642 @end defun |
| 428 | 1643 |
| 2090 | 1644 The range of random is implementation-dependent. On any machine, the |
| 1645 result of @code{(random)} is an arbitrary fixnum, so on 32-bit | |
| 1646 architectures it is normally in the range -2^30 (inclusive) to +2^30 | |
| 1647 (exclusive). With the optional integer argument @var{limit}, the result | |
| 1648 is in the range 0 (inclusive) to @var{limit} (exclusive). Note this is | |
| 1649 regardless of the presence of the bignum extension. | |
| 1650 |