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  20. <a href="https://luajit.org"><span>Lua<span id="logo">JIT</span></span></a>
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  22. <div id="head">
  23. <h1>FFI Tutorial</h1>
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  59. <div id="main">
  60. <p>
  61. This page is intended to give you an overview of the features of the FFI
  62. library by presenting a few use cases and guidelines.
  63. </p>
  64. <p>
  65. This page makes no attempt to explain all of the FFI library, though.
  66. You'll want to have a look at the <a href="ext_ffi_api.html">ffi.* API
  67. function reference</a> and the <a href="ext_ffi_semantics.html">FFI
  68. semantics</a> to learn more.
  69. </p>
  70. <h2 id="load">Loading the FFI Library</h2>
  71. <p>
  72. The FFI library is built into LuaJIT by default, but it's not loaded
  73. and initialized by default. The suggested way to use the FFI library
  74. is to add the following to the start of every Lua file that needs one
  75. of its functions:
  76. </p>
  77. <pre class="code">
  78. local ffi = require("ffi")
  79. </pre>
  80. <p>
  81. Please note, this doesn't define an <tt>ffi</tt> variable in the table
  82. of globals &mdash; you really need to use the local variable. The
  83. <tt>require</tt> function ensures the library is only loaded once.
  84. </p>
  85. <p style="font-size: 8pt;">
  86. Note: If you want to experiment with the FFI from the interactive prompt
  87. of the command line executable, omit the <tt>local</tt>, as it doesn't
  88. preserve local variables across lines.
  89. </p>
  90. <h2 id="sleep">Accessing Standard System Functions</h2>
  91. <p>
  92. The following code explains how to access standard system functions.
  93. We slowly print two lines of dots by sleeping for 10&nbsp;milliseconds
  94. after each dot:
  95. </p>
  96. <pre class="code mark">
  97. <span class="codemark">&nbsp;
  98. &#9312;
  99. &#9313;
  100. &#9314;
  101. &#9315;
  102. &#9316;
  103. &#9317;</span>local ffi = require("ffi")
  104. ffi.cdef[[
  105. <span style="color:#00a000;">void Sleep(int ms);
  106. int poll(struct pollfd *fds, unsigned long nfds, int timeout);</span>
  107. ]]
  108. local sleep
  109. if ffi.os == "Windows" then
  110. function sleep(s)
  111. ffi.C.Sleep(s*1000)
  112. end
  113. else
  114. function sleep(s)
  115. ffi.C.poll(nil, 0, s*1000)
  116. end
  117. end
  118. for i=1,160 do
  119. io.write("."); io.flush()
  120. sleep(0.01)
  121. end
  122. io.write("\n")
  123. </pre>
  124. <p>
  125. Here's the step-by-step explanation:
  126. </p>
  127. <p>
  128. <span class="mark">&#9312;</span> This defines the
  129. C&nbsp;library functions we're going to use. The part inside the
  130. double-brackets (in green) is just standard C&nbsp;syntax. You can
  131. usually get this info from the C&nbsp;header files or the
  132. documentation provided by each C&nbsp;library or C&nbsp;compiler.
  133. </p>
  134. <p>
  135. <span class="mark">&#9313;</span> The difficulty we're
  136. facing here, is that there are different standards to choose from.
  137. Windows has a simple <tt>Sleep()</tt> function. On other systems there
  138. are a variety of functions available to achieve sub-second sleeps, but
  139. with no clear consensus. Thankfully <tt>poll()</tt> can be used for
  140. this task, too, and it's present on most non-Windows systems. The
  141. check for <tt>ffi.os</tt> makes sure we use the Windows-specific
  142. function only on Windows systems.
  143. </p>
  144. <p>
  145. <span class="mark">&#9314;</span> Here we're wrapping the
  146. call to the C&nbsp;function in a Lua function. This isn't strictly
  147. necessary, but it's helpful to deal with system-specific issues only
  148. in one part of the code. The way we're wrapping it ensures the check
  149. for the OS is only done during initialization and not for every call.
  150. </p>
  151. <p>
  152. <span class="mark">&#9315;</span> A more subtle point is
  153. that we defined our <tt>sleep()</tt> function (for the sake of this
  154. example) as taking the number of seconds, but accepting fractional
  155. seconds. Multiplying this by 1000 gets us milliseconds, but that still
  156. leaves it a Lua number, which is a floating-point value. Alas, the
  157. <tt>Sleep()</tt> function only accepts an integer value. Luckily for
  158. us, the FFI library automatically performs the conversion when calling
  159. the function (truncating the FP value towards zero, like in C).
  160. </p>
  161. <p style="font-size: 8pt;">
  162. Some readers will notice that <tt>Sleep()</tt> is part of
  163. <tt>KERNEL32.DLL</tt> and is also a <tt>stdcall</tt> function. So how
  164. can this possibly work? The FFI library provides the <tt>ffi.C</tt>
  165. default C&nbsp;library namespace, which allows calling functions from
  166. the default set of libraries, like a C&nbsp;compiler would. Also, the
  167. FFI library automatically detects <tt>stdcall</tt> functions, so you
  168. don't need to declare them as such.
  169. </p>
  170. <p>
  171. <span class="mark">&#9316;</span> The <tt>poll()</tt>
  172. function takes a couple more arguments we're not going to use. You can
  173. simply use <tt>nil</tt> to pass a <tt>NULL</tt> pointer and <tt>0</tt>
  174. for the <tt>nfds</tt> parameter. Please note, that the
  175. number&nbsp;<tt>0</tt> <em>does not convert to a pointer value</em>,
  176. unlike in C++. You really have to pass pointers to pointer arguments
  177. and numbers to number arguments.
  178. </p>
  179. <p style="font-size: 8pt;">
  180. The page on <a href="ext_ffi_semantics.html">FFI semantics</a> has all
  181. of the gory details about
  182. <a href="ext_ffi_semantics.html#convert">conversions between Lua
  183. objects and C&nbsp;types</a>. For the most part you don't have to deal
  184. with this, as it's performed automatically and it's carefully designed
  185. to bridge the semantic differences between Lua and C.
  186. </p>
  187. <p>
  188. <span class="mark">&#9317;</span> Now that we have defined
  189. our own <tt>sleep()</tt> function, we can just call it from plain Lua
  190. code. That wasn't so bad, huh? Turning these boring animated dots into
  191. a fascinating best-selling game is left as an exercise for the reader.
  192. :-)
  193. </p>
  194. <h2 id="zlib">Accessing the zlib Compression Library</h2>
  195. <p>
  196. The following code shows how to access the <a
  197. href="https://zlib.net/"><span class="ext">&raquo;</span>&nbsp;zlib</a> compression library from Lua code.
  198. We'll define two convenience wrapper functions that take a string and
  199. compress or uncompress it to another string:
  200. </p>
  201. <pre class="code mark">
  202. <span class="codemark">&nbsp;
  203. &#9312;
  204. &#9313;
  205. &#9314;
  206. &#9315;
  207. &#9316;
  208. &#9317;
  209. &#9318;</span>local ffi = require("ffi")
  210. ffi.cdef[[
  211. <span style="color:#00a000;">unsigned long compressBound(unsigned long sourceLen);
  212. int compress2(uint8_t *dest, unsigned long *destLen,
  213. const uint8_t *source, unsigned long sourceLen, int level);
  214. int uncompress(uint8_t *dest, unsigned long *destLen,
  215. const uint8_t *source, unsigned long sourceLen);</span>
  216. ]]
  217. local zlib = ffi.load(ffi.os == "Windows" and "zlib1" or "z")
  218. local function compress(txt)
  219. local n = zlib.compressBound(#txt)
  220. local buf = ffi.new("uint8_t[?]", n)
  221. local buflen = ffi.new("unsigned long[1]", n)
  222. local res = zlib.compress2(buf, buflen, txt, #txt, 9)
  223. assert(res == 0)
  224. return ffi.string(buf, buflen[0])
  225. end
  226. local function uncompress(comp, n)
  227. local buf = ffi.new("uint8_t[?]", n)
  228. local buflen = ffi.new("unsigned long[1]", n)
  229. local res = zlib.uncompress(buf, buflen, comp, #comp)
  230. assert(res == 0)
  231. return ffi.string(buf, buflen[0])
  232. end
  233. -- Simple test code.
  234. local txt = string.rep("abcd", 1000)
  235. print("Uncompressed size: ", #txt)
  236. local c = compress(txt)
  237. print("Compressed size: ", #c)
  238. local txt2 = uncompress(c, #txt)
  239. assert(txt2 == txt)
  240. </pre>
  241. <p>
  242. Here's the step-by-step explanation:
  243. </p>
  244. <p>
  245. <span class="mark">&#9312;</span> This defines some of the
  246. C&nbsp;functions provided by zlib. For the sake of this example, some
  247. type indirections have been reduced and it uses the predefined
  248. fixed-size integer types, while still adhering to the zlib API/ABI.
  249. </p>
  250. <p>
  251. <span class="mark">&#9313;</span> This loads the zlib shared
  252. library. On POSIX systems, it's named <tt>libz.so</tt> and usually
  253. comes pre-installed. Since <tt>ffi.load()</tt> automatically adds any
  254. missing standard prefixes/suffixes, we can simply load the
  255. <tt>"z"</tt> library. On Windows it's named <tt>zlib1.dll</tt> and
  256. you'll have to download it first from the
  257. <a href="https://zlib.net/"><span class="ext">&raquo;</span>&nbsp;zlib site</a>. The check for
  258. <tt>ffi.os</tt> makes sure we pass the right name to
  259. <tt>ffi.load()</tt>.
  260. </p>
  261. <p>
  262. <span class="mark">&#9314;</span> First, the maximum size of
  263. the compression buffer is obtained by calling the
  264. <tt>zlib.compressBound</tt> function with the length of the
  265. uncompressed string. The next line allocates a byte buffer of this
  266. size. The <tt>[?]</tt> in the type specification indicates a
  267. variable-length array (VLA). The actual number of elements of this
  268. array is given as the 2nd argument to <tt>ffi.new()</tt>.
  269. </p>
  270. <p>
  271. <span class="mark">&#9315;</span> This may look strange at
  272. first, but have a look at the declaration of the <tt>compress2</tt>
  273. function from zlib: the destination length is defined as a pointer!
  274. This is because you pass in the maximum buffer size and get back the
  275. actual length that was used.
  276. </p>
  277. <p>
  278. In C you'd pass in the address of a local variable
  279. (<tt>&amp;buflen</tt>). But since there's no address-of operator in
  280. Lua, we'll just pass in a one-element array. Conveniently, it can be
  281. initialized with the maximum buffer size in one step. Calling the
  282. actual <tt>zlib.compress2</tt> function is then straightforward.
  283. </p>
  284. <p>
  285. <span class="mark">&#9316;</span> We want to return the
  286. compressed data as a Lua string, so we'll use <tt>ffi.string()</tt>.
  287. It needs a pointer to the start of the data and the actual length. The
  288. length has been returned in the <tt>buflen</tt> array, so we'll just
  289. get it from there.
  290. </p>
  291. <p style="font-size: 8pt;">
  292. Note that since the function returns now, the <tt>buf</tt> and
  293. <tt>buflen</tt> variables will eventually be garbage collected. This
  294. is fine, because <tt>ffi.string()</tt> has copied the contents to a
  295. newly created (interned) Lua string. If you plan to call this function
  296. lots of times, consider reusing the buffers and/or handing back the
  297. results in buffers instead of strings. This will reduce the overhead
  298. for garbage collection and string interning.
  299. </p>
  300. <p>
  301. <span class="mark">&#9317;</span> The <tt>uncompress</tt>
  302. functions does the exact opposite of the <tt>compress</tt> function.
  303. The compressed data doesn't include the size of the original string,
  304. so this needs to be passed in. Otherwise, no surprises here.
  305. </p>
  306. <p>
  307. <span class="mark">&#9318;</span> The code, that makes use
  308. of the functions we just defined, is just plain Lua code. It doesn't
  309. need to know anything about the LuaJIT FFI &mdash; the convenience
  310. wrapper functions completely hide it.
  311. </p>
  312. <p>
  313. One major advantage of the LuaJIT FFI is that you are now able to
  314. write those wrappers <em>in Lua</em>. And at a fraction of the time it
  315. would cost you to create an extra C&nbsp;module using the Lua/C API.
  316. Many of the simpler C&nbsp;functions can probably be used directly
  317. from your Lua code, without any wrappers.
  318. </p>
  319. <p style="font-size: 8pt;">
  320. Side note: the zlib API uses the <tt>long</tt> type for passing
  321. lengths and sizes around. But all those zlib functions actually only
  322. deal with 32&nbsp;bit values. This is an unfortunate choice for a
  323. public API, but may be explained by zlib's history &mdash; we'll just
  324. have to deal with it.
  325. </p>
  326. <p style="font-size: 8pt;">
  327. First, you should know that a <tt>long</tt> is a 64&nbsp;bit type e.g.
  328. on POSIX/x64 systems, but a 32&nbsp;bit type on Windows/x64 and on
  329. 32&nbsp;bit systems. Thus a <tt>long</tt> result can be either a plain
  330. Lua number or a boxed 64&nbsp;bit integer cdata object, depending on
  331. the target system.
  332. </p>
  333. <p style="font-size: 8pt;">
  334. Ok, so the <tt>ffi.*</tt> functions generally accept cdata objects
  335. wherever you'd want to use a number. That's why we get a away with
  336. passing <tt>n</tt> to <tt>ffi.string()</tt> above. But other Lua
  337. library functions or modules don't know how to deal with this. So for
  338. maximum portability, one needs to use <tt>tonumber()</tt> on returned
  339. <tt>long</tt> results before passing them on. Otherwise the
  340. application might work on some systems, but would fail in a POSIX/x64
  341. environment.
  342. </p>
  343. <h2 id="metatype">Defining Metamethods for a C&nbsp;Type</h2>
  344. <p>
  345. The following code explains how to define metamethods for a C type.
  346. We define a simple point type and add some operations to it:
  347. </p>
  348. <pre class="code mark">
  349. <span class="codemark">&nbsp;
  350. &#9312;
  351. &#9313;
  352. &#9314;
  353. &#9315;
  354. &#9316;
  355. &#9317;</span>local ffi = require("ffi")
  356. ffi.cdef[[
  357. <span style="color:#00a000;">typedef struct { double x, y; } point_t;</span>
  358. ]]
  359. local point
  360. local mt = {
  361. __add = function(a, b) return point(a.x+b.x, a.y+b.y) end,
  362. __len = function(a) return math.sqrt(a.x*a.x + a.y*a.y) end,
  363. __index = {
  364. area = function(a) return a.x*a.x + a.y*a.y end,
  365. },
  366. }
  367. point = ffi.metatype("point_t", mt)
  368. local a = point(3, 4)
  369. print(a.x, a.y) --> 3 4
  370. print(#a) --> 5
  371. print(a:area()) --> 25
  372. local b = a + point(0.5, 8)
  373. print(#b) --> 12.5
  374. </pre>
  375. <p>
  376. Here's the step-by-step explanation:
  377. </p>
  378. <p>
  379. <span class="mark">&#9312;</span> This defines the C&nbsp;type for a
  380. two-dimensional point object.
  381. </p>
  382. <p>
  383. <span class="mark">&#9313;</span> We have to declare the variable
  384. holding the point constructor first, because it's used inside of a
  385. metamethod.
  386. </p>
  387. <p>
  388. <span class="mark">&#9314;</span> Let's define an <tt>__add</tt>
  389. metamethod which adds the coordinates of two points and creates a new
  390. point object. For simplicity, this function assumes that both arguments
  391. are points. But it could be any mix of objects, if at least one operand
  392. is of the required type (e.g. adding a point plus a number or vice
  393. versa). Our <tt>__len</tt> metamethod returns the distance of a point to
  394. the origin.
  395. </p>
  396. <p>
  397. <span class="mark">&#9315;</span> If we run out of operators, we can
  398. define named methods, too. Here, the <tt>__index</tt> table defines an
  399. <tt>area</tt> function. For custom indexing needs, one might want to
  400. define <tt>__index</tt> and <tt>__newindex</tt> <em>functions</em> instead.
  401. </p>
  402. <p>
  403. <span class="mark">&#9316;</span> This associates the metamethods with
  404. our C&nbsp;type. This only needs to be done once. For convenience, a
  405. constructor is returned by
  406. <a href="ext_ffi_api.html#ffi_metatype"><tt>ffi.metatype()</tt></a>.
  407. We're not required to use it, though. The original C&nbsp;type can still
  408. be used e.g. to create an array of points. The metamethods automatically
  409. apply to any and all uses of this type.
  410. </p>
  411. <p>
  412. Please note, that the association with a metatable is permanent and
  413. <b>the metatable must not be modified afterwards!</b> Ditto for the
  414. <tt>__index</tt> table.
  415. </p>
  416. <p>
  417. <span class="mark">&#9317;</span> Here are some simple usage examples
  418. for the point type and their expected results. The predefined
  419. operations (such as <tt>a.x</tt>) can be freely mixed with the newly
  420. defined metamethods. Note that <tt>area</tt> is a method and must be
  421. called with the Lua syntax for methods: <tt>a:area()</tt>, not
  422. <tt>a.area()</tt>.
  423. </p>
  424. <p>
  425. The C&nbsp;type metamethod mechanism is most useful when used in
  426. conjunction with C&nbsp;libraries that are written in an object-oriented
  427. style. Creators return a pointer to a new instance, and methods take an
  428. instance pointer as the first argument. Sometimes you can just point
  429. <tt>__index</tt> to the library namespace and <tt>__gc</tt> to the
  430. destructor and you're done. But often enough you'll want to add
  431. convenience wrappers, e.g. to return actual Lua strings or when
  432. returning multiple values.
  433. </p>
  434. <p>
  435. Some C libraries only declare instance pointers as an opaque
  436. <tt>void&nbsp;*</tt> type. In this case you can use a fake type for all
  437. declarations, e.g. a pointer to a named (incomplete) struct will do:
  438. <tt>typedef struct foo_type *foo_handle</tt>. The C&nbsp;side doesn't
  439. know what you declare with the LuaJIT FFI, but as long as the underlying
  440. types are compatible, everything still works.
  441. </p>
  442. <h2 id="idioms">Translating C&nbsp;Idioms</h2>
  443. <p>
  444. Here's a list of common C&nbsp;idioms and their translation to the
  445. LuaJIT FFI:
  446. </p>
  447. <table class="idiomtable">
  448. <tr class="idiomhead">
  449. <td class="idiomdesc">Idiom</td>
  450. <td class="idiomc">C&nbsp;code</td>
  451. <td class="idiomlua">Lua code</td>
  452. </tr>
  453. <tr class="odd separate">
  454. <td class="idiomdesc">Pointer dereference<br><tt>int *p;</tt></td><td class="idiomc"><tt>x = *p;<br>*p = y;</tt></td><td class="idiomlua"><tt>x = <b>p[0]</b><br><b>p[0]</b> = y</tt></td></tr>
  455. <tr class="even">
  456. <td class="idiomdesc">Pointer indexing<br><tt>int i, *p;</tt></td><td class="idiomc"><tt>x = p[i];<br>p[i+1] = y;</tt></td><td class="idiomlua"><tt>x = p[i]<br>p[i+1] = y</tt></td></tr>
  457. <tr class="odd">
  458. <td class="idiomdesc">Array indexing<br><tt>int i, a[];</tt></td><td class="idiomc"><tt>x = a[i];<br>a[i+1] = y;</tt></td><td class="idiomlua"><tt>x = a[i]<br>a[i+1] = y</tt></td></tr>
  459. <tr class="even separate">
  460. <td class="idiomdesc"><tt>struct</tt>/<tt>union</tt> dereference<br><tt>struct foo s;</tt></td><td class="idiomc"><tt>x = s.field;<br>s.field = y;</tt></td><td class="idiomlua"><tt>x = s.field<br>s.field = y</tt></td></tr>
  461. <tr class="odd">
  462. <td class="idiomdesc"><tt>struct</tt>/<tt>union</tt> pointer deref.<br><tt>struct foo *sp;</tt></td><td class="idiomc"><tt>x = sp->field;<br>sp->field = y;</tt></td><td class="idiomlua"><tt>x = <b>s.field</b><br><b>s.field</b> = y</tt></td></tr>
  463. <tr class="even separate">
  464. <td class="idiomdesc">Pointer arithmetic<br><tt>int i, *p;</tt></td><td class="idiomc"><tt>x = p + i;<br>y = p - i;</tt></td><td class="idiomlua"><tt>x = p + i<br>y = p - i</tt></td></tr>
  465. <tr class="odd">
  466. <td class="idiomdesc">Pointer difference<br><tt>int *p1, *p2;</tt></td><td class="idiomc"><tt>x = p1 - p2;</tt></td><td class="idiomlua"><tt>x = p1 - p2</tt></td></tr>
  467. <tr class="even">
  468. <td class="idiomdesc">Array element pointer<br><tt>int i, a[];</tt></td><td class="idiomc"><tt>x = &amp;a[i];</tt></td><td class="idiomlua"><tt>x = <b>a+i</b></tt></td></tr>
  469. <tr class="odd">
  470. <td class="idiomdesc">Cast pointer to address<br><tt>int *p;</tt></td><td class="idiomc"><tt>x = (intptr_t)p;</tt></td><td class="idiomlua"><tt>x = <b>tonumber(<br>&nbsp;ffi.cast("intptr_t",<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;p))</b></tt></td></tr>
  471. <tr class="even separate">
  472. <td class="idiomdesc">Functions with outargs<br><tt>void foo(int *inoutlen);</tt></td><td class="idiomc"><tt>int len = x;<br>foo(&amp;len);<br>y = len;</tt></td><td class="idiomlua"><tt><b>local len =<br>&nbsp;&nbsp;ffi.new("int[1]", x)<br>foo(len)<br>y = len[0]</b></tt></td></tr>
  473. <tr class="odd">
  474. <td class="idiomdesc"><a href="ext_ffi_semantics.html#convert_vararg">Vararg conversions</a><br><tt>int printf(char *fmt, ...);</tt></td><td class="idiomc"><tt>printf("%g", 1.0);<br>printf("%d", 1);<br>&nbsp;</tt></td><td class="idiomlua"><tt>printf("%g", 1);<br>printf("%d",<br>&nbsp;&nbsp;<b>ffi.new("int", 1)</b>)</tt></td></tr>
  475. </table>
  476. <h2 id="cache">To Cache or Not to Cache</h2>
  477. <p>
  478. It's a common Lua idiom to cache library functions in local variables
  479. or upvalues, e.g.:
  480. </p>
  481. <pre class="code">
  482. local byte, char = string.byte, string.char
  483. local function foo(x)
  484. return char(byte(x)+1)
  485. end
  486. </pre>
  487. <p>
  488. This replaces several hash-table lookups with a (faster) direct use of
  489. a local or an upvalue. This is less important with LuaJIT, since the
  490. JIT compiler optimizes hash-table lookups a lot and is even able to
  491. hoist most of them out of the inner loops. It can't eliminate
  492. <em>all</em> of them, though, and it saves some typing for often-used
  493. functions. So there's still a place for this, even with LuaJIT.
  494. </p>
  495. <p>
  496. The situation is a bit different with C&nbsp;function calls via the
  497. FFI library. The JIT compiler has special logic to eliminate <em>all
  498. of the lookup overhead</em> for functions resolved from a
  499. <a href="ext_ffi_semantics.html#clib">C&nbsp;library namespace</a>!
  500. Thus it's not helpful and actually counter-productive to cache
  501. individual C&nbsp;functions like this:
  502. </p>
  503. <pre class="code">
  504. local <b>funca</b>, <b>funcb</b> = ffi.C.funca, ffi.C.funcb -- <span style="color:#c00000;">Not helpful!</span>
  505. local function foo(x, n)
  506. for i=1,n do <b>funcb</b>(<b>funca</b>(x, i), 1) end
  507. end
  508. </pre>
  509. <p>
  510. This turns them into indirect calls and generates bigger and slower
  511. machine code. Instead, you'll want to cache the namespace itself and
  512. rely on the JIT compiler to eliminate the lookups:
  513. </p>
  514. <pre class="code">
  515. local <b>C</b> = ffi.C -- <span style="color:#00a000;">Instead use this!</span>
  516. local function foo(x, n)
  517. for i=1,n do <b>C.funcb</b>(<b>C.funca</b>(x, i), 1) end
  518. end
  519. </pre>
  520. <p>
  521. This generates both shorter and faster code. So <b>don't cache
  522. C&nbsp;functions</b>, but <b>do</b> cache namespaces! Most often the
  523. namespace is already in a local variable at an outer scope, e.g. from
  524. <tt>local&nbsp;lib&nbsp;=&nbsp;ffi.load(...)</tt>. Note that copying
  525. it to a local variable in the function scope is unnecessary.
  526. </p>
  527. <br class="flush">
  528. </div>
  529. <div id="foot">
  530. <hr class="hide">
  531. Copyright &copy; 2005-2023
  532. <span class="noprint">
  533. &middot;
  534. <a href="contact.html">Contact</a>
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