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55c14926 | 1 | @c We need some definitions here. |
7a68c94a | 2 | @ifclear mult |
55c14926 | 3 | @ifhtml |
7a68c94a UD |
4 | @set mult · |
5 | @set infty ∞ | |
6 | @set pie π | |
55c14926 | 7 | @end ifhtml |
ca34d7a7 | 8 | @iftex |
838e5ffe | 9 | @set mult @cdot |
7a68c94a | 10 | @set infty @infty |
ca34d7a7 | 11 | @end iftex |
838e5ffe | 12 | @ifclear mult |
7a68c94a UD |
13 | @set mult * |
14 | @set infty oo | |
15 | @set pie pi | |
838e5ffe | 16 | @end ifclear |
fe0ec73e | 17 | @macro mul |
838e5ffe | 18 | @value{mult} |
fe0ec73e | 19 | @end macro |
ca34d7a7 UD |
20 | @macro infinity |
21 | @value{infty} | |
22 | @end macro | |
7a68c94a UD |
23 | @ifnottex |
24 | @macro pi | |
25 | @value{pie} | |
26 | @end macro | |
27 | @end ifnottex | |
28 | @end ifclear | |
55c14926 | 29 | |
d52b6462 | 30 | @node Mathematics, Arithmetic, Syslog, Top |
7a68c94a | 31 | @c %MENU% Math functions, useful constants, random numbers |
28f540f4 RM |
32 | @chapter Mathematics |
33 | ||
34 | This chapter contains information about functions for performing | |
35 | mathematical computations, such as trigonometric functions. Most of | |
36 | these functions have prototypes declared in the header file | |
7a68c94a UD |
37 | @file{math.h}. The complex-valued functions are defined in |
38 | @file{complex.h}. | |
28f540f4 | 39 | @pindex math.h |
7a68c94a UD |
40 | @pindex complex.h |
41 | ||
42 | All mathematical functions which take a floating-point argument | |
43 | have three variants, one each for @code{double}, @code{float}, and | |
44 | @code{long double} arguments. The @code{double} versions are mostly | |
ec751a23 UD |
45 | defined in @w{ISO C89}. The @code{float} and @code{long double} |
46 | versions are from the numeric extensions to C included in @w{ISO C99}. | |
7a68c94a UD |
47 | |
48 | Which of the three versions of a function should be used depends on the | |
49 | situation. For most calculations, the @code{float} functions are the | |
50 | fastest. On the other hand, the @code{long double} functions have the | |
51 | highest precision. @code{double} is somewhere in between. It is | |
04b9968b | 52 | usually wise to pick the narrowest type that can accommodate your data. |
7a68c94a UD |
53 | Not all machines have a distinct @code{long double} type; it may be the |
54 | same as @code{double}. | |
28f540f4 RM |
55 | |
56 | @menu | |
7a68c94a UD |
57 | * Mathematical Constants:: Precise numeric values for often-used |
58 | constants. | |
59 | * Trig Functions:: Sine, cosine, tangent, and friends. | |
60 | * Inverse Trig Functions:: Arcsine, arccosine, etc. | |
61 | * Exponents and Logarithms:: Also pow and sqrt. | |
62 | * Hyperbolic Functions:: sinh, cosh, tanh, etc. | |
63 | * Special Functions:: Bessel, gamma, erf. | |
aaa1276e | 64 | * Errors in Math Functions:: Known Maximum Errors in Math Functions. |
7a68c94a UD |
65 | * Pseudo-Random Numbers:: Functions for generating pseudo-random |
66 | numbers. | |
67 | * FP Function Optimizations:: Fast code or small code. | |
28f540f4 RM |
68 | @end menu |
69 | ||
55c14926 UD |
70 | @node Mathematical Constants |
71 | @section Predefined Mathematical Constants | |
72 | @cindex constants | |
73 | @cindex mathematical constants | |
74 | ||
7a68c94a UD |
75 | The header @file{math.h} defines several useful mathematical constants. |
76 | All values are defined as preprocessor macros starting with @code{M_}. | |
77 | The values provided are: | |
55c14926 UD |
78 | |
79 | @vtable @code | |
80 | @item M_E | |
7a68c94a | 81 | The base of natural logarithms. |
55c14926 | 82 | @item M_LOG2E |
7a68c94a | 83 | The logarithm to base @code{2} of @code{M_E}. |
55c14926 | 84 | @item M_LOG10E |
7a68c94a | 85 | The logarithm to base @code{10} of @code{M_E}. |
55c14926 | 86 | @item M_LN2 |
7a68c94a | 87 | The natural logarithm of @code{2}. |
55c14926 | 88 | @item M_LN10 |
7a68c94a | 89 | The natural logarithm of @code{10}. |
55c14926 | 90 | @item M_PI |
04b9968b | 91 | Pi, the ratio of a circle's circumference to its diameter. |
55c14926 | 92 | @item M_PI_2 |
7a68c94a | 93 | Pi divided by two. |
55c14926 | 94 | @item M_PI_4 |
7a68c94a | 95 | Pi divided by four. |
55c14926 | 96 | @item M_1_PI |
7a68c94a | 97 | The reciprocal of pi (1/pi) |
55c14926 | 98 | @item M_2_PI |
7a68c94a | 99 | Two times the reciprocal of pi. |
55c14926 | 100 | @item M_2_SQRTPI |
7a68c94a | 101 | Two times the reciprocal of the square root of pi. |
55c14926 | 102 | @item M_SQRT2 |
7a68c94a | 103 | The square root of two. |
55c14926 | 104 | @item M_SQRT1_2 |
7a68c94a | 105 | The reciprocal of the square root of two (also the square root of 1/2). |
55c14926 UD |
106 | @end vtable |
107 | ||
7a68c94a | 108 | These constants come from the Unix98 standard and were also available in |
04b9968b | 109 | 4.4BSD; therefore they are only defined if @code{_BSD_SOURCE} or |
7a68c94a UD |
110 | @code{_XOPEN_SOURCE=500}, or a more general feature select macro, is |
111 | defined. The default set of features includes these constants. | |
112 | @xref{Feature Test Macros}. | |
113 | ||
114 | All values are of type @code{double}. As an extension, the GNU C | |
115 | library also defines these constants with type @code{long double}. The | |
116 | @code{long double} macros have a lowercase @samp{l} appended to their | |
117 | names: @code{M_El}, @code{M_PIl}, and so forth. These are only | |
118 | available if @code{_GNU_SOURCE} is defined. | |
55c14926 UD |
119 | |
120 | @vindex PI | |
121 | @emph{Note:} Some programs use a constant named @code{PI} which has the | |
7a68c94a UD |
122 | same value as @code{M_PI}. This constant is not standard; it may have |
123 | appeared in some old AT&T headers, and is mentioned in Stroustrup's book | |
124 | on C++. It infringes on the user's name space, so the GNU C library | |
125 | does not define it. Fixing programs written to expect it is simple: | |
126 | replace @code{PI} with @code{M_PI} throughout, or put @samp{-DPI=M_PI} | |
127 | on the compiler command line. | |
61eb22d3 | 128 | |
28f540f4 RM |
129 | @node Trig Functions |
130 | @section Trigonometric Functions | |
131 | @cindex trigonometric functions | |
132 | ||
133 | These are the familiar @code{sin}, @code{cos}, and @code{tan} functions. | |
134 | The arguments to all of these functions are in units of radians; recall | |
135 | that pi radians equals 180 degrees. | |
136 | ||
137 | @cindex pi (trigonometric constant) | |
7a68c94a UD |
138 | The math library normally defines @code{M_PI} to a @code{double} |
139 | approximation of pi. If strict ISO and/or POSIX compliance | |
140 | are requested this constant is not defined, but you can easily define it | |
141 | yourself: | |
28f540f4 RM |
142 | |
143 | @smallexample | |
b4012b75 | 144 | #define M_PI 3.14159265358979323846264338327 |
28f540f4 RM |
145 | @end smallexample |
146 | ||
147 | @noindent | |
148 | You can also compute the value of pi with the expression @code{acos | |
149 | (-1.0)}. | |
150 | ||
28f540f4 | 151 | @comment math.h |
f65fd747 | 152 | @comment ISO |
28f540f4 | 153 | @deftypefun double sin (double @var{x}) |
4260bc74 UD |
154 | @comment math.h |
155 | @comment ISO | |
779ae82e | 156 | @deftypefunx float sinf (float @var{x}) |
4260bc74 UD |
157 | @comment math.h |
158 | @comment ISO | |
779ae82e | 159 | @deftypefunx {long double} sinl (long double @var{x}) |
b4012b75 | 160 | These functions return the sine of @var{x}, where @var{x} is given in |
28f540f4 RM |
161 | radians. The return value is in the range @code{-1} to @code{1}. |
162 | @end deftypefun | |
163 | ||
164 | @comment math.h | |
f65fd747 | 165 | @comment ISO |
28f540f4 | 166 | @deftypefun double cos (double @var{x}) |
4260bc74 UD |
167 | @comment math.h |
168 | @comment ISO | |
779ae82e | 169 | @deftypefunx float cosf (float @var{x}) |
4260bc74 UD |
170 | @comment math.h |
171 | @comment ISO | |
779ae82e | 172 | @deftypefunx {long double} cosl (long double @var{x}) |
b4012b75 | 173 | These functions return the cosine of @var{x}, where @var{x} is given in |
28f540f4 RM |
174 | radians. The return value is in the range @code{-1} to @code{1}. |
175 | @end deftypefun | |
176 | ||
177 | @comment math.h | |
f65fd747 | 178 | @comment ISO |
28f540f4 | 179 | @deftypefun double tan (double @var{x}) |
4260bc74 UD |
180 | @comment math.h |
181 | @comment ISO | |
779ae82e | 182 | @deftypefunx float tanf (float @var{x}) |
4260bc74 UD |
183 | @comment math.h |
184 | @comment ISO | |
779ae82e | 185 | @deftypefunx {long double} tanl (long double @var{x}) |
b4012b75 | 186 | These functions return the tangent of @var{x}, where @var{x} is given in |
28f540f4 RM |
187 | radians. |
188 | ||
28f540f4 RM |
189 | Mathematically, the tangent function has singularities at odd multiples |
190 | of pi/2. If the argument @var{x} is too close to one of these | |
7a68c94a | 191 | singularities, @code{tan} will signal overflow. |
28f540f4 RM |
192 | @end deftypefun |
193 | ||
7a68c94a UD |
194 | In many applications where @code{sin} and @code{cos} are used, the sine |
195 | and cosine of the same angle are needed at the same time. It is more | |
196 | efficient to compute them simultaneously, so the library provides a | |
197 | function to do that. | |
b4012b75 UD |
198 | |
199 | @comment math.h | |
200 | @comment GNU | |
201 | @deftypefun void sincos (double @var{x}, double *@var{sinx}, double *@var{cosx}) | |
4260bc74 UD |
202 | @comment math.h |
203 | @comment GNU | |
779ae82e | 204 | @deftypefunx void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx}) |
4260bc74 UD |
205 | @comment math.h |
206 | @comment GNU | |
779ae82e | 207 | @deftypefunx void sincosl (long double @var{x}, long double *@var{sinx}, long double *@var{cosx}) |
b4012b75 UD |
208 | These functions return the sine of @var{x} in @code{*@var{sinx}} and the |
209 | cosine of @var{x} in @code{*@var{cos}}, where @var{x} is given in | |
210 | radians. Both values, @code{*@var{sinx}} and @code{*@var{cosx}}, are in | |
211 | the range of @code{-1} to @code{1}. | |
ca34d7a7 | 212 | |
7a68c94a UD |
213 | This function is a GNU extension. Portable programs should be prepared |
214 | to cope with its absence. | |
b4012b75 UD |
215 | @end deftypefun |
216 | ||
217 | @cindex complex trigonometric functions | |
218 | ||
ec751a23 | 219 | @w{ISO C99} defines variants of the trig functions which work on |
7a68c94a UD |
220 | complex numbers. The GNU C library provides these functions, but they |
221 | are only useful if your compiler supports the new complex types defined | |
222 | by the standard. | |
ec751a23 | 223 | @c XXX Change this when gcc is fixed. -zw |
7a68c94a UD |
224 | (As of this writing GCC supports complex numbers, but there are bugs in |
225 | the implementation.) | |
b4012b75 UD |
226 | |
227 | @comment complex.h | |
228 | @comment ISO | |
229 | @deftypefun {complex double} csin (complex double @var{z}) | |
4260bc74 UD |
230 | @comment complex.h |
231 | @comment ISO | |
779ae82e | 232 | @deftypefunx {complex float} csinf (complex float @var{z}) |
4260bc74 UD |
233 | @comment complex.h |
234 | @comment ISO | |
779ae82e | 235 | @deftypefunx {complex long double} csinl (complex long double @var{z}) |
7a68c94a | 236 | These functions return the complex sine of @var{z}. |
b4012b75 UD |
237 | The mathematical definition of the complex sine is |
238 | ||
4c78249d | 239 | @ifnottex |
779ae82e | 240 | @math{sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i))}. |
4c78249d | 241 | @end ifnottex |
779ae82e UD |
242 | @tex |
243 | $$\sin(z) = {1\over 2i} (e^{zi} - e^{-zi})$$ | |
244 | @end tex | |
b4012b75 UD |
245 | @end deftypefun |
246 | ||
247 | @comment complex.h | |
248 | @comment ISO | |
249 | @deftypefun {complex double} ccos (complex double @var{z}) | |
4260bc74 UD |
250 | @comment complex.h |
251 | @comment ISO | |
779ae82e | 252 | @deftypefunx {complex float} ccosf (complex float @var{z}) |
4260bc74 UD |
253 | @comment complex.h |
254 | @comment ISO | |
779ae82e | 255 | @deftypefunx {complex long double} ccosl (complex long double @var{z}) |
7a68c94a | 256 | These functions return the complex cosine of @var{z}. |
b4012b75 UD |
257 | The mathematical definition of the complex cosine is |
258 | ||
4c78249d | 259 | @ifnottex |
779ae82e | 260 | @math{cos (z) = 1/2 * (exp (z*i) + exp (-z*i))} |
4c78249d | 261 | @end ifnottex |
779ae82e UD |
262 | @tex |
263 | $$\cos(z) = {1\over 2} (e^{zi} + e^{-zi})$$ | |
264 | @end tex | |
b4012b75 UD |
265 | @end deftypefun |
266 | ||
267 | @comment complex.h | |
268 | @comment ISO | |
269 | @deftypefun {complex double} ctan (complex double @var{z}) | |
4260bc74 UD |
270 | @comment complex.h |
271 | @comment ISO | |
779ae82e | 272 | @deftypefunx {complex float} ctanf (complex float @var{z}) |
4260bc74 UD |
273 | @comment complex.h |
274 | @comment ISO | |
779ae82e | 275 | @deftypefunx {complex long double} ctanl (complex long double @var{z}) |
7a68c94a | 276 | These functions return the complex tangent of @var{z}. |
b4012b75 UD |
277 | The mathematical definition of the complex tangent is |
278 | ||
4c78249d | 279 | @ifnottex |
7a68c94a | 280 | @math{tan (z) = -i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))} |
4c78249d | 281 | @end ifnottex |
779ae82e | 282 | @tex |
7a68c94a | 283 | $$\tan(z) = -i \cdot {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$ |
779ae82e | 284 | @end tex |
7a68c94a UD |
285 | |
286 | @noindent | |
287 | The complex tangent has poles at @math{pi/2 + 2n}, where @math{n} is an | |
288 | integer. @code{ctan} may signal overflow if @var{z} is too close to a | |
289 | pole. | |
b4012b75 UD |
290 | @end deftypefun |
291 | ||
28f540f4 RM |
292 | |
293 | @node Inverse Trig Functions | |
294 | @section Inverse Trigonometric Functions | |
6d52618b | 295 | @cindex inverse trigonometric functions |
28f540f4 RM |
296 | |
297 | These are the usual arc sine, arc cosine and arc tangent functions, | |
04b9968b | 298 | which are the inverses of the sine, cosine and tangent functions |
28f540f4 RM |
299 | respectively. |
300 | ||
301 | @comment math.h | |
f65fd747 | 302 | @comment ISO |
28f540f4 | 303 | @deftypefun double asin (double @var{x}) |
4260bc74 UD |
304 | @comment math.h |
305 | @comment ISO | |
779ae82e | 306 | @deftypefunx float asinf (float @var{x}) |
4260bc74 UD |
307 | @comment math.h |
308 | @comment ISO | |
779ae82e | 309 | @deftypefunx {long double} asinl (long double @var{x}) |
b4012b75 | 310 | These functions compute the arc sine of @var{x}---that is, the value whose |
28f540f4 RM |
311 | sine is @var{x}. The value is in units of radians. Mathematically, |
312 | there are infinitely many such values; the one actually returned is the | |
313 | one between @code{-pi/2} and @code{pi/2} (inclusive). | |
314 | ||
7a68c94a UD |
315 | The arc sine function is defined mathematically only |
316 | over the domain @code{-1} to @code{1}. If @var{x} is outside the | |
317 | domain, @code{asin} signals a domain error. | |
28f540f4 RM |
318 | @end deftypefun |
319 | ||
320 | @comment math.h | |
f65fd747 | 321 | @comment ISO |
28f540f4 | 322 | @deftypefun double acos (double @var{x}) |
4260bc74 UD |
323 | @comment math.h |
324 | @comment ISO | |
779ae82e | 325 | @deftypefunx float acosf (float @var{x}) |
4260bc74 UD |
326 | @comment math.h |
327 | @comment ISO | |
779ae82e | 328 | @deftypefunx {long double} acosl (long double @var{x}) |
b4012b75 | 329 | These functions compute the arc cosine of @var{x}---that is, the value |
28f540f4 RM |
330 | whose cosine is @var{x}. The value is in units of radians. |
331 | Mathematically, there are infinitely many such values; the one actually | |
332 | returned is the one between @code{0} and @code{pi} (inclusive). | |
333 | ||
7a68c94a UD |
334 | The arc cosine function is defined mathematically only |
335 | over the domain @code{-1} to @code{1}. If @var{x} is outside the | |
336 | domain, @code{acos} signals a domain error. | |
28f540f4 RM |
337 | @end deftypefun |
338 | ||
28f540f4 | 339 | @comment math.h |
f65fd747 | 340 | @comment ISO |
28f540f4 | 341 | @deftypefun double atan (double @var{x}) |
4260bc74 UD |
342 | @comment math.h |
343 | @comment ISO | |
779ae82e | 344 | @deftypefunx float atanf (float @var{x}) |
4260bc74 UD |
345 | @comment math.h |
346 | @comment ISO | |
779ae82e | 347 | @deftypefunx {long double} atanl (long double @var{x}) |
b4012b75 | 348 | These functions compute the arc tangent of @var{x}---that is, the value |
28f540f4 RM |
349 | whose tangent is @var{x}. The value is in units of radians. |
350 | Mathematically, there are infinitely many such values; the one actually | |
7a68c94a | 351 | returned is the one between @code{-pi/2} and @code{pi/2} (inclusive). |
28f540f4 RM |
352 | @end deftypefun |
353 | ||
354 | @comment math.h | |
f65fd747 | 355 | @comment ISO |
28f540f4 | 356 | @deftypefun double atan2 (double @var{y}, double @var{x}) |
4260bc74 UD |
357 | @comment math.h |
358 | @comment ISO | |
779ae82e | 359 | @deftypefunx float atan2f (float @var{y}, float @var{x}) |
4260bc74 UD |
360 | @comment math.h |
361 | @comment ISO | |
779ae82e | 362 | @deftypefunx {long double} atan2l (long double @var{y}, long double @var{x}) |
7a68c94a UD |
363 | This function computes the arc tangent of @var{y}/@var{x}, but the signs |
364 | of both arguments are used to determine the quadrant of the result, and | |
365 | @var{x} is permitted to be zero. The return value is given in radians | |
366 | and is in the range @code{-pi} to @code{pi}, inclusive. | |
28f540f4 RM |
367 | |
368 | If @var{x} and @var{y} are coordinates of a point in the plane, | |
369 | @code{atan2} returns the signed angle between the line from the origin | |
370 | to that point and the x-axis. Thus, @code{atan2} is useful for | |
371 | converting Cartesian coordinates to polar coordinates. (To compute the | |
372 | radial coordinate, use @code{hypot}; see @ref{Exponents and | |
373 | Logarithms}.) | |
374 | ||
7a68c94a UD |
375 | @c This is experimentally true. Should it be so? -zw |
376 | If both @var{x} and @var{y} are zero, @code{atan2} returns zero. | |
28f540f4 RM |
377 | @end deftypefun |
378 | ||
b4012b75 | 379 | @cindex inverse complex trigonometric functions |
ec751a23 | 380 | @w{ISO C99} defines complex versions of the inverse trig functions. |
b4012b75 UD |
381 | |
382 | @comment complex.h | |
383 | @comment ISO | |
384 | @deftypefun {complex double} casin (complex double @var{z}) | |
4260bc74 UD |
385 | @comment complex.h |
386 | @comment ISO | |
779ae82e | 387 | @deftypefunx {complex float} casinf (complex float @var{z}) |
4260bc74 UD |
388 | @comment complex.h |
389 | @comment ISO | |
779ae82e | 390 | @deftypefunx {complex long double} casinl (complex long double @var{z}) |
b4012b75 | 391 | These functions compute the complex arc sine of @var{z}---that is, the |
7a68c94a | 392 | value whose sine is @var{z}. The value returned is in radians. |
b4012b75 | 393 | |
7a68c94a UD |
394 | Unlike the real-valued functions, @code{casin} is defined for all |
395 | values of @var{z}. | |
b4012b75 UD |
396 | @end deftypefun |
397 | ||
398 | @comment complex.h | |
399 | @comment ISO | |
400 | @deftypefun {complex double} cacos (complex double @var{z}) | |
4260bc74 UD |
401 | @comment complex.h |
402 | @comment ISO | |
779ae82e | 403 | @deftypefunx {complex float} cacosf (complex float @var{z}) |
4260bc74 UD |
404 | @comment complex.h |
405 | @comment ISO | |
779ae82e | 406 | @deftypefunx {complex long double} cacosl (complex long double @var{z}) |
b4012b75 | 407 | These functions compute the complex arc cosine of @var{z}---that is, the |
7a68c94a | 408 | value whose cosine is @var{z}. The value returned is in radians. |
b4012b75 | 409 | |
7a68c94a UD |
410 | Unlike the real-valued functions, @code{cacos} is defined for all |
411 | values of @var{z}. | |
b4012b75 UD |
412 | @end deftypefun |
413 | ||
414 | ||
415 | @comment complex.h | |
416 | @comment ISO | |
417 | @deftypefun {complex double} catan (complex double @var{z}) | |
4260bc74 UD |
418 | @comment complex.h |
419 | @comment ISO | |
779ae82e | 420 | @deftypefunx {complex float} catanf (complex float @var{z}) |
4260bc74 UD |
421 | @comment complex.h |
422 | @comment ISO | |
779ae82e | 423 | @deftypefunx {complex long double} catanl (complex long double @var{z}) |
b4012b75 UD |
424 | These functions compute the complex arc tangent of @var{z}---that is, |
425 | the value whose tangent is @var{z}. The value is in units of radians. | |
426 | @end deftypefun | |
427 | ||
28f540f4 RM |
428 | |
429 | @node Exponents and Logarithms | |
430 | @section Exponentiation and Logarithms | |
431 | @cindex exponentiation functions | |
432 | @cindex power functions | |
433 | @cindex logarithm functions | |
434 | ||
435 | @comment math.h | |
f65fd747 | 436 | @comment ISO |
28f540f4 | 437 | @deftypefun double exp (double @var{x}) |
4260bc74 UD |
438 | @comment math.h |
439 | @comment ISO | |
779ae82e | 440 | @deftypefunx float expf (float @var{x}) |
4260bc74 UD |
441 | @comment math.h |
442 | @comment ISO | |
779ae82e | 443 | @deftypefunx {long double} expl (long double @var{x}) |
7a68c94a UD |
444 | These functions compute @code{e} (the base of natural logarithms) raised |
445 | to the power @var{x}. | |
28f540f4 | 446 | |
7a68c94a UD |
447 | If the magnitude of the result is too large to be representable, |
448 | @code{exp} signals overflow. | |
28f540f4 RM |
449 | @end deftypefun |
450 | ||
b4012b75 UD |
451 | @comment math.h |
452 | @comment ISO | |
04a96fd4 | 453 | @deftypefun double exp2 (double @var{x}) |
4260bc74 UD |
454 | @comment math.h |
455 | @comment ISO | |
04a96fd4 | 456 | @deftypefunx float exp2f (float @var{x}) |
4260bc74 UD |
457 | @comment math.h |
458 | @comment ISO | |
04a96fd4 | 459 | @deftypefunx {long double} exp2l (long double @var{x}) |
7a68c94a | 460 | These functions compute @code{2} raised to the power @var{x}. |
04a96fd4 | 461 | Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}. |
b4012b75 UD |
462 | @end deftypefun |
463 | ||
464 | @comment math.h | |
04a96fd4 UD |
465 | @comment GNU |
466 | @deftypefun double exp10 (double @var{x}) | |
4260bc74 UD |
467 | @comment math.h |
468 | @comment GNU | |
04a96fd4 | 469 | @deftypefunx float exp10f (float @var{x}) |
4260bc74 UD |
470 | @comment math.h |
471 | @comment GNU | |
04a96fd4 | 472 | @deftypefunx {long double} exp10l (long double @var{x}) |
4260bc74 UD |
473 | @comment math.h |
474 | @comment GNU | |
04a96fd4 | 475 | @deftypefunx double pow10 (double @var{x}) |
4260bc74 UD |
476 | @comment math.h |
477 | @comment GNU | |
04a96fd4 | 478 | @deftypefunx float pow10f (float @var{x}) |
4260bc74 UD |
479 | @comment math.h |
480 | @comment GNU | |
04a96fd4 | 481 | @deftypefunx {long double} pow10l (long double @var{x}) |
7a68c94a UD |
482 | These functions compute @code{10} raised to the power @var{x}. |
483 | Mathematically, @code{exp10 (x)} is the same as @code{exp (x * log (10))}. | |
b4012b75 | 484 | |
7a68c94a UD |
485 | These functions are GNU extensions. The name @code{exp10} is |
486 | preferred, since it is analogous to @code{exp} and @code{exp2}. | |
b4012b75 UD |
487 | @end deftypefun |
488 | ||
489 | ||
28f540f4 | 490 | @comment math.h |
f65fd747 | 491 | @comment ISO |
28f540f4 | 492 | @deftypefun double log (double @var{x}) |
4260bc74 UD |
493 | @comment math.h |
494 | @comment ISO | |
f2ea0f5b | 495 | @deftypefunx float logf (float @var{x}) |
4260bc74 UD |
496 | @comment math.h |
497 | @comment ISO | |
779ae82e | 498 | @deftypefunx {long double} logl (long double @var{x}) |
7a68c94a | 499 | These functions compute the natural logarithm of @var{x}. @code{exp (log |
28f540f4 RM |
500 | (@var{x}))} equals @var{x}, exactly in mathematics and approximately in |
501 | C. | |
502 | ||
7a68c94a UD |
503 | If @var{x} is negative, @code{log} signals a domain error. If @var{x} |
504 | is zero, it returns negative infinity; if @var{x} is too close to zero, | |
505 | it may signal overflow. | |
28f540f4 RM |
506 | @end deftypefun |
507 | ||
508 | @comment math.h | |
f65fd747 | 509 | @comment ISO |
28f540f4 | 510 | @deftypefun double log10 (double @var{x}) |
4260bc74 UD |
511 | @comment math.h |
512 | @comment ISO | |
779ae82e | 513 | @deftypefunx float log10f (float @var{x}) |
4260bc74 UD |
514 | @comment math.h |
515 | @comment ISO | |
779ae82e | 516 | @deftypefunx {long double} log10l (long double @var{x}) |
7a68c94a | 517 | These functions return the base-10 logarithm of @var{x}. |
28f540f4 | 518 | @code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}. |
7a68c94a | 519 | |
28f540f4 RM |
520 | @end deftypefun |
521 | ||
b4012b75 UD |
522 | @comment math.h |
523 | @comment ISO | |
524 | @deftypefun double log2 (double @var{x}) | |
4260bc74 UD |
525 | @comment math.h |
526 | @comment ISO | |
779ae82e | 527 | @deftypefunx float log2f (float @var{x}) |
4260bc74 UD |
528 | @comment math.h |
529 | @comment ISO | |
779ae82e | 530 | @deftypefunx {long double} log2l (long double @var{x}) |
7a68c94a | 531 | These functions return the base-2 logarithm of @var{x}. |
b4012b75 UD |
532 | @code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}. |
533 | @end deftypefun | |
534 | ||
55c14926 UD |
535 | @comment math.h |
536 | @comment ISO | |
537 | @deftypefun double logb (double @var{x}) | |
4260bc74 UD |
538 | @comment math.h |
539 | @comment ISO | |
55c14926 | 540 | @deftypefunx float logbf (float @var{x}) |
4260bc74 UD |
541 | @comment math.h |
542 | @comment ISO | |
55c14926 UD |
543 | @deftypefunx {long double} logbl (long double @var{x}) |
544 | These functions extract the exponent of @var{x} and return it as a | |
7a68c94a UD |
545 | floating-point value. If @code{FLT_RADIX} is two, @code{logb} is equal |
546 | to @code{floor (log2 (x))}, except it's probably faster. | |
55c14926 | 547 | |
04b9968b | 548 | If @var{x} is de-normalized, @code{logb} returns the exponent @var{x} |
7a68c94a UD |
549 | would have if it were normalized. If @var{x} is infinity (positive or |
550 | negative), @code{logb} returns @math{@infinity{}}. If @var{x} is zero, | |
551 | @code{logb} returns @math{@infinity{}}. It does not signal. | |
55c14926 UD |
552 | @end deftypefun |
553 | ||
554 | @comment math.h | |
555 | @comment ISO | |
556 | @deftypefun int ilogb (double @var{x}) | |
4260bc74 UD |
557 | @comment math.h |
558 | @comment ISO | |
55c14926 | 559 | @deftypefunx int ilogbf (float @var{x}) |
4260bc74 UD |
560 | @comment math.h |
561 | @comment ISO | |
55c14926 UD |
562 | @deftypefunx int ilogbl (long double @var{x}) |
563 | These functions are equivalent to the corresponding @code{logb} | |
7a68c94a UD |
564 | functions except that they return signed integer values. |
565 | @end deftypefun | |
566 | ||
567 | @noindent | |
568 | Since integers cannot represent infinity and NaN, @code{ilogb} instead | |
569 | returns an integer that can't be the exponent of a normal floating-point | |
570 | number. @file{math.h} defines constants so you can check for this. | |
571 | ||
572 | @comment math.h | |
573 | @comment ISO | |
574 | @deftypevr Macro int FP_ILOGB0 | |
575 | @code{ilogb} returns this value if its argument is @code{0}. The | |
576 | numeric value is either @code{INT_MIN} or @code{-INT_MAX}. | |
577 | ||
ec751a23 | 578 | This macro is defined in @w{ISO C99}. |
7a68c94a UD |
579 | @end deftypevr |
580 | ||
581 | @comment math.h | |
582 | @comment ISO | |
583 | @deftypevr Macro int FP_ILOGBNAN | |
584 | @code{ilogb} returns this value if its argument is @code{NaN}. The | |
585 | numeric value is either @code{INT_MIN} or @code{INT_MAX}. | |
586 | ||
ec751a23 | 587 | This macro is defined in @w{ISO C99}. |
7a68c94a UD |
588 | @end deftypevr |
589 | ||
590 | These values are system specific. They might even be the same. The | |
591 | proper way to test the result of @code{ilogb} is as follows: | |
55c14926 UD |
592 | |
593 | @smallexample | |
594 | i = ilogb (f); | |
595 | if (i == FP_ILOGB0 || i == FP_ILOGBNAN) | |
596 | @{ | |
597 | if (isnan (f)) | |
598 | @{ | |
599 | /* @r{Handle NaN.} */ | |
600 | @} | |
601 | else if (f == 0.0) | |
602 | @{ | |
603 | /* @r{Handle 0.0.} */ | |
604 | @} | |
605 | else | |
606 | @{ | |
607 | /* @r{Some other value with large exponent,} | |
608 | @r{perhaps +Inf.} */ | |
609 | @} | |
610 | @} | |
611 | @end smallexample | |
612 | ||
28f540f4 | 613 | @comment math.h |
f65fd747 | 614 | @comment ISO |
28f540f4 | 615 | @deftypefun double pow (double @var{base}, double @var{power}) |
4260bc74 UD |
616 | @comment math.h |
617 | @comment ISO | |
779ae82e | 618 | @deftypefunx float powf (float @var{base}, float @var{power}) |
4260bc74 UD |
619 | @comment math.h |
620 | @comment ISO | |
779ae82e | 621 | @deftypefunx {long double} powl (long double @var{base}, long double @var{power}) |
b4012b75 | 622 | These are general exponentiation functions, returning @var{base} raised |
28f540f4 RM |
623 | to @var{power}. |
624 | ||
7a68c94a UD |
625 | Mathematically, @code{pow} would return a complex number when @var{base} |
626 | is negative and @var{power} is not an integral value. @code{pow} can't | |
627 | do that, so instead it signals a domain error. @code{pow} may also | |
628 | underflow or overflow the destination type. | |
28f540f4 RM |
629 | @end deftypefun |
630 | ||
631 | @cindex square root function | |
632 | @comment math.h | |
f65fd747 | 633 | @comment ISO |
28f540f4 | 634 | @deftypefun double sqrt (double @var{x}) |
4260bc74 UD |
635 | @comment math.h |
636 | @comment ISO | |
779ae82e | 637 | @deftypefunx float sqrtf (float @var{x}) |
4260bc74 UD |
638 | @comment math.h |
639 | @comment ISO | |
779ae82e | 640 | @deftypefunx {long double} sqrtl (long double @var{x}) |
b4012b75 | 641 | These functions return the nonnegative square root of @var{x}. |
28f540f4 | 642 | |
7a68c94a UD |
643 | If @var{x} is negative, @code{sqrt} signals a domain error. |
644 | Mathematically, it should return a complex number. | |
28f540f4 RM |
645 | @end deftypefun |
646 | ||
647 | @cindex cube root function | |
648 | @comment math.h | |
649 | @comment BSD | |
650 | @deftypefun double cbrt (double @var{x}) | |
4260bc74 UD |
651 | @comment math.h |
652 | @comment BSD | |
779ae82e | 653 | @deftypefunx float cbrtf (float @var{x}) |
4260bc74 UD |
654 | @comment math.h |
655 | @comment BSD | |
779ae82e | 656 | @deftypefunx {long double} cbrtl (long double @var{x}) |
b4012b75 | 657 | These functions return the cube root of @var{x}. They cannot |
28f540f4 RM |
658 | fail; every representable real value has a representable real cube root. |
659 | @end deftypefun | |
660 | ||
661 | @comment math.h | |
b4012b75 | 662 | @comment ISO |
28f540f4 | 663 | @deftypefun double hypot (double @var{x}, double @var{y}) |
4260bc74 UD |
664 | @comment math.h |
665 | @comment ISO | |
779ae82e | 666 | @deftypefunx float hypotf (float @var{x}, float @var{y}) |
4260bc74 UD |
667 | @comment math.h |
668 | @comment ISO | |
779ae82e | 669 | @deftypefunx {long double} hypotl (long double @var{x}, long double @var{y}) |
b4012b75 | 670 | These functions return @code{sqrt (@var{x}*@var{x} + |
7a68c94a | 671 | @var{y}*@var{y})}. This is the length of the hypotenuse of a right |
28f540f4 | 672 | triangle with sides of length @var{x} and @var{y}, or the distance |
7a68c94a UD |
673 | of the point (@var{x}, @var{y}) from the origin. Using this function |
674 | instead of the direct formula is wise, since the error is | |
b4012b75 | 675 | much smaller. See also the function @code{cabs} in @ref{Absolute Value}. |
28f540f4 RM |
676 | @end deftypefun |
677 | ||
678 | @comment math.h | |
b4012b75 | 679 | @comment ISO |
28f540f4 | 680 | @deftypefun double expm1 (double @var{x}) |
4260bc74 UD |
681 | @comment math.h |
682 | @comment ISO | |
779ae82e | 683 | @deftypefunx float expm1f (float @var{x}) |
4260bc74 UD |
684 | @comment math.h |
685 | @comment ISO | |
779ae82e | 686 | @deftypefunx {long double} expm1l (long double @var{x}) |
b4012b75 | 687 | These functions return a value equivalent to @code{exp (@var{x}) - 1}. |
7a68c94a | 688 | They are computed in a way that is accurate even if @var{x} is |
04b9968b | 689 | near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate owing |
28f540f4 RM |
690 | to subtraction of two numbers that are nearly equal. |
691 | @end deftypefun | |
692 | ||
693 | @comment math.h | |
b4012b75 | 694 | @comment ISO |
28f540f4 | 695 | @deftypefun double log1p (double @var{x}) |
4260bc74 UD |
696 | @comment math.h |
697 | @comment ISO | |
779ae82e | 698 | @deftypefunx float log1pf (float @var{x}) |
4260bc74 UD |
699 | @comment math.h |
700 | @comment ISO | |
779ae82e | 701 | @deftypefunx {long double} log1pl (long double @var{x}) |
7a68c94a UD |
702 | These functions returns a value equivalent to @w{@code{log (1 + @var{x})}}. |
703 | They are computed in a way that is accurate even if @var{x} is | |
28f540f4 RM |
704 | near zero. |
705 | @end deftypefun | |
706 | ||
b4012b75 UD |
707 | @cindex complex exponentiation functions |
708 | @cindex complex logarithm functions | |
709 | ||
ec751a23 | 710 | @w{ISO C99} defines complex variants of some of the exponentiation and |
7a68c94a | 711 | logarithm functions. |
b4012b75 UD |
712 | |
713 | @comment complex.h | |
714 | @comment ISO | |
715 | @deftypefun {complex double} cexp (complex double @var{z}) | |
4260bc74 UD |
716 | @comment complex.h |
717 | @comment ISO | |
779ae82e | 718 | @deftypefunx {complex float} cexpf (complex float @var{z}) |
4260bc74 UD |
719 | @comment complex.h |
720 | @comment ISO | |
779ae82e | 721 | @deftypefunx {complex long double} cexpl (complex long double @var{z}) |
7a68c94a UD |
722 | These functions return @code{e} (the base of natural |
723 | logarithms) raised to the power of @var{z}. | |
04b9968b | 724 | Mathematically, this corresponds to the value |
b4012b75 | 725 | |
4c78249d | 726 | @ifnottex |
779ae82e | 727 | @math{exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))} |
4c78249d | 728 | @end ifnottex |
779ae82e | 729 | @tex |
7a68c94a | 730 | $$\exp(z) = e^z = e^{{\rm Re}\,z} (\cos ({\rm Im}\,z) + i \sin ({\rm Im}\,z))$$ |
779ae82e | 731 | @end tex |
b4012b75 UD |
732 | @end deftypefun |
733 | ||
734 | @comment complex.h | |
735 | @comment ISO | |
736 | @deftypefun {complex double} clog (complex double @var{z}) | |
4260bc74 UD |
737 | @comment complex.h |
738 | @comment ISO | |
779ae82e | 739 | @deftypefunx {complex float} clogf (complex float @var{z}) |
4260bc74 UD |
740 | @comment complex.h |
741 | @comment ISO | |
779ae82e | 742 | @deftypefunx {complex long double} clogl (complex long double @var{z}) |
7a68c94a | 743 | These functions return the natural logarithm of @var{z}. |
04b9968b | 744 | Mathematically, this corresponds to the value |
b4012b75 | 745 | |
4c78249d | 746 | @ifnottex |
779ae82e | 747 | @math{log (z) = log (cabs (z)) + I * carg (z)} |
4c78249d | 748 | @end ifnottex |
779ae82e | 749 | @tex |
7a68c94a | 750 | $$\log(z) = \log |z| + i \arg z$$ |
779ae82e | 751 | @end tex |
7a68c94a UD |
752 | |
753 | @noindent | |
754 | @code{clog} has a pole at 0, and will signal overflow if @var{z} equals | |
755 | or is very close to 0. It is well-defined for all other values of | |
756 | @var{z}. | |
b4012b75 UD |
757 | @end deftypefun |
758 | ||
dfd2257a UD |
759 | |
760 | @comment complex.h | |
761 | @comment GNU | |
762 | @deftypefun {complex double} clog10 (complex double @var{z}) | |
4260bc74 UD |
763 | @comment complex.h |
764 | @comment GNU | |
dfd2257a | 765 | @deftypefunx {complex float} clog10f (complex float @var{z}) |
4260bc74 UD |
766 | @comment complex.h |
767 | @comment GNU | |
dfd2257a UD |
768 | @deftypefunx {complex long double} clog10l (complex long double @var{z}) |
769 | These functions return the base 10 logarithm of the complex value | |
04b9968b | 770 | @var{z}. Mathematically, this corresponds to the value |
dfd2257a | 771 | |
4c78249d | 772 | @ifnottex |
dfd2257a | 773 | @math{log (z) = log10 (cabs (z)) + I * carg (z)} |
4c78249d | 774 | @end ifnottex |
dfd2257a | 775 | @tex |
7a68c94a | 776 | $$\log_{10}(z) = \log_{10}|z| + i \arg z$$ |
dfd2257a | 777 | @end tex |
dfd2257a | 778 | |
7a68c94a | 779 | These functions are GNU extensions. |
dfd2257a UD |
780 | @end deftypefun |
781 | ||
b4012b75 UD |
782 | @comment complex.h |
783 | @comment ISO | |
784 | @deftypefun {complex double} csqrt (complex double @var{z}) | |
4260bc74 UD |
785 | @comment complex.h |
786 | @comment ISO | |
779ae82e | 787 | @deftypefunx {complex float} csqrtf (complex float @var{z}) |
4260bc74 UD |
788 | @comment complex.h |
789 | @comment ISO | |
779ae82e | 790 | @deftypefunx {complex long double} csqrtl (complex long double @var{z}) |
7a68c94a UD |
791 | These functions return the complex square root of the argument @var{z}. Unlike |
792 | the real-valued functions, they are defined for all values of @var{z}. | |
b4012b75 UD |
793 | @end deftypefun |
794 | ||
795 | @comment complex.h | |
796 | @comment ISO | |
797 | @deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power}) | |
4260bc74 UD |
798 | @comment complex.h |
799 | @comment ISO | |
779ae82e | 800 | @deftypefunx {complex float} cpowf (complex float @var{base}, complex float @var{power}) |
4260bc74 UD |
801 | @comment complex.h |
802 | @comment ISO | |
779ae82e | 803 | @deftypefunx {complex long double} cpowl (complex long double @var{base}, complex long double @var{power}) |
7a68c94a UD |
804 | These functions return @var{base} raised to the power of |
805 | @var{power}. This is equivalent to @w{@code{cexp (y * clog (x))}} | |
b4012b75 UD |
806 | @end deftypefun |
807 | ||
28f540f4 RM |
808 | @node Hyperbolic Functions |
809 | @section Hyperbolic Functions | |
810 | @cindex hyperbolic functions | |
811 | ||
812 | The functions in this section are related to the exponential functions; | |
813 | see @ref{Exponents and Logarithms}. | |
814 | ||
815 | @comment math.h | |
f65fd747 | 816 | @comment ISO |
28f540f4 | 817 | @deftypefun double sinh (double @var{x}) |
4260bc74 UD |
818 | @comment math.h |
819 | @comment ISO | |
779ae82e | 820 | @deftypefunx float sinhf (float @var{x}) |
4260bc74 UD |
821 | @comment math.h |
822 | @comment ISO | |
779ae82e | 823 | @deftypefunx {long double} sinhl (long double @var{x}) |
b4012b75 | 824 | These functions return the hyperbolic sine of @var{x}, defined |
7a68c94a UD |
825 | mathematically as @w{@code{(exp (@var{x}) - exp (-@var{x})) / 2}}. They |
826 | may signal overflow if @var{x} is too large. | |
28f540f4 RM |
827 | @end deftypefun |
828 | ||
829 | @comment math.h | |
f65fd747 | 830 | @comment ISO |
28f540f4 | 831 | @deftypefun double cosh (double @var{x}) |
4260bc74 UD |
832 | @comment math.h |
833 | @comment ISO | |
779ae82e | 834 | @deftypefunx float coshf (float @var{x}) |
4260bc74 UD |
835 | @comment math.h |
836 | @comment ISO | |
779ae82e | 837 | @deftypefunx {long double} coshl (long double @var{x}) |
b4012b75 UD |
838 | These function return the hyperbolic cosine of @var{x}, |
839 | defined mathematically as @w{@code{(exp (@var{x}) + exp (-@var{x})) / 2}}. | |
7a68c94a | 840 | They may signal overflow if @var{x} is too large. |
28f540f4 RM |
841 | @end deftypefun |
842 | ||
843 | @comment math.h | |
f65fd747 | 844 | @comment ISO |
28f540f4 | 845 | @deftypefun double tanh (double @var{x}) |
4260bc74 UD |
846 | @comment math.h |
847 | @comment ISO | |
779ae82e | 848 | @deftypefunx float tanhf (float @var{x}) |
4260bc74 UD |
849 | @comment math.h |
850 | @comment ISO | |
779ae82e | 851 | @deftypefunx {long double} tanhl (long double @var{x}) |
7a68c94a UD |
852 | These functions return the hyperbolic tangent of @var{x}, |
853 | defined mathematically as @w{@code{sinh (@var{x}) / cosh (@var{x})}}. | |
854 | They may signal overflow if @var{x} is too large. | |
28f540f4 RM |
855 | @end deftypefun |
856 | ||
b4012b75 UD |
857 | @cindex hyperbolic functions |
858 | ||
7a68c94a UD |
859 | There are counterparts for the hyperbolic functions which take |
860 | complex arguments. | |
b4012b75 UD |
861 | |
862 | @comment complex.h | |
863 | @comment ISO | |
864 | @deftypefun {complex double} csinh (complex double @var{z}) | |
4260bc74 UD |
865 | @comment complex.h |
866 | @comment ISO | |
779ae82e | 867 | @deftypefunx {complex float} csinhf (complex float @var{z}) |
4260bc74 UD |
868 | @comment complex.h |
869 | @comment ISO | |
779ae82e | 870 | @deftypefunx {complex long double} csinhl (complex long double @var{z}) |
b4012b75 | 871 | These functions return the complex hyperbolic sine of @var{z}, defined |
7a68c94a | 872 | mathematically as @w{@code{(exp (@var{z}) - exp (-@var{z})) / 2}}. |
b4012b75 UD |
873 | @end deftypefun |
874 | ||
875 | @comment complex.h | |
876 | @comment ISO | |
877 | @deftypefun {complex double} ccosh (complex double @var{z}) | |
4260bc74 UD |
878 | @comment complex.h |
879 | @comment ISO | |
779ae82e | 880 | @deftypefunx {complex float} ccoshf (complex float @var{z}) |
4260bc74 UD |
881 | @comment complex.h |
882 | @comment ISO | |
779ae82e | 883 | @deftypefunx {complex long double} ccoshl (complex long double @var{z}) |
b4012b75 | 884 | These functions return the complex hyperbolic cosine of @var{z}, defined |
7a68c94a | 885 | mathematically as @w{@code{(exp (@var{z}) + exp (-@var{z})) / 2}}. |
b4012b75 UD |
886 | @end deftypefun |
887 | ||
888 | @comment complex.h | |
889 | @comment ISO | |
890 | @deftypefun {complex double} ctanh (complex double @var{z}) | |
4260bc74 UD |
891 | @comment complex.h |
892 | @comment ISO | |
779ae82e | 893 | @deftypefunx {complex float} ctanhf (complex float @var{z}) |
4260bc74 UD |
894 | @comment complex.h |
895 | @comment ISO | |
779ae82e | 896 | @deftypefunx {complex long double} ctanhl (complex long double @var{z}) |
7a68c94a UD |
897 | These functions return the complex hyperbolic tangent of @var{z}, |
898 | defined mathematically as @w{@code{csinh (@var{z}) / ccosh (@var{z})}}. | |
b4012b75 UD |
899 | @end deftypefun |
900 | ||
901 | ||
28f540f4 RM |
902 | @cindex inverse hyperbolic functions |
903 | ||
904 | @comment math.h | |
b4012b75 | 905 | @comment ISO |
28f540f4 | 906 | @deftypefun double asinh (double @var{x}) |
4260bc74 UD |
907 | @comment math.h |
908 | @comment ISO | |
779ae82e | 909 | @deftypefunx float asinhf (float @var{x}) |
4260bc74 UD |
910 | @comment math.h |
911 | @comment ISO | |
779ae82e | 912 | @deftypefunx {long double} asinhl (long double @var{x}) |
b4012b75 | 913 | These functions return the inverse hyperbolic sine of @var{x}---the |
28f540f4 RM |
914 | value whose hyperbolic sine is @var{x}. |
915 | @end deftypefun | |
916 | ||
917 | @comment math.h | |
b4012b75 | 918 | @comment ISO |
28f540f4 | 919 | @deftypefun double acosh (double @var{x}) |
4260bc74 UD |
920 | @comment math.h |
921 | @comment ISO | |
779ae82e | 922 | @deftypefunx float acoshf (float @var{x}) |
4260bc74 UD |
923 | @comment math.h |
924 | @comment ISO | |
779ae82e | 925 | @deftypefunx {long double} acoshl (long double @var{x}) |
b4012b75 | 926 | These functions return the inverse hyperbolic cosine of @var{x}---the |
28f540f4 | 927 | value whose hyperbolic cosine is @var{x}. If @var{x} is less than |
7a68c94a | 928 | @code{1}, @code{acosh} signals a domain error. |
28f540f4 RM |
929 | @end deftypefun |
930 | ||
931 | @comment math.h | |
b4012b75 | 932 | @comment ISO |
28f540f4 | 933 | @deftypefun double atanh (double @var{x}) |
4260bc74 UD |
934 | @comment math.h |
935 | @comment ISO | |
779ae82e | 936 | @deftypefunx float atanhf (float @var{x}) |
4260bc74 UD |
937 | @comment math.h |
938 | @comment ISO | |
779ae82e | 939 | @deftypefunx {long double} atanhl (long double @var{x}) |
b4012b75 | 940 | These functions return the inverse hyperbolic tangent of @var{x}---the |
28f540f4 | 941 | value whose hyperbolic tangent is @var{x}. If the absolute value of |
7a68c94a UD |
942 | @var{x} is greater than @code{1}, @code{atanh} signals a domain error; |
943 | if it is equal to 1, @code{atanh} returns infinity. | |
28f540f4 RM |
944 | @end deftypefun |
945 | ||
b4012b75 UD |
946 | @cindex inverse complex hyperbolic functions |
947 | ||
948 | @comment complex.h | |
949 | @comment ISO | |
950 | @deftypefun {complex double} casinh (complex double @var{z}) | |
4260bc74 UD |
951 | @comment complex.h |
952 | @comment ISO | |
779ae82e | 953 | @deftypefunx {complex float} casinhf (complex float @var{z}) |
4260bc74 UD |
954 | @comment complex.h |
955 | @comment ISO | |
779ae82e | 956 | @deftypefunx {complex long double} casinhl (complex long double @var{z}) |
b4012b75 UD |
957 | These functions return the inverse complex hyperbolic sine of |
958 | @var{z}---the value whose complex hyperbolic sine is @var{z}. | |
959 | @end deftypefun | |
960 | ||
961 | @comment complex.h | |
962 | @comment ISO | |
963 | @deftypefun {complex double} cacosh (complex double @var{z}) | |
4260bc74 UD |
964 | @comment complex.h |
965 | @comment ISO | |
779ae82e | 966 | @deftypefunx {complex float} cacoshf (complex float @var{z}) |
4260bc74 UD |
967 | @comment complex.h |
968 | @comment ISO | |
779ae82e | 969 | @deftypefunx {complex long double} cacoshl (complex long double @var{z}) |
b4012b75 UD |
970 | These functions return the inverse complex hyperbolic cosine of |
971 | @var{z}---the value whose complex hyperbolic cosine is @var{z}. Unlike | |
7a68c94a | 972 | the real-valued functions, there are no restrictions on the value of @var{z}. |
b4012b75 UD |
973 | @end deftypefun |
974 | ||
975 | @comment complex.h | |
976 | @comment ISO | |
977 | @deftypefun {complex double} catanh (complex double @var{z}) | |
4260bc74 UD |
978 | @comment complex.h |
979 | @comment ISO | |
779ae82e | 980 | @deftypefunx {complex float} catanhf (complex float @var{z}) |
4260bc74 UD |
981 | @comment complex.h |
982 | @comment ISO | |
779ae82e | 983 | @deftypefunx {complex long double} catanhl (complex long double @var{z}) |
b4012b75 UD |
984 | These functions return the inverse complex hyperbolic tangent of |
985 | @var{z}---the value whose complex hyperbolic tangent is @var{z}. Unlike | |
7a68c94a UD |
986 | the real-valued functions, there are no restrictions on the value of |
987 | @var{z}. | |
b4012b75 UD |
988 | @end deftypefun |
989 | ||
7a68c94a UD |
990 | @node Special Functions |
991 | @section Special Functions | |
992 | @cindex special functions | |
993 | @cindex Bessel functions | |
994 | @cindex gamma function | |
995 | ||
04b9968b | 996 | These are some more exotic mathematical functions which are sometimes |
7a68c94a UD |
997 | useful. Currently they only have real-valued versions. |
998 | ||
999 | @comment math.h | |
1000 | @comment SVID | |
1001 | @deftypefun double erf (double @var{x}) | |
4260bc74 UD |
1002 | @comment math.h |
1003 | @comment SVID | |
7a68c94a | 1004 | @deftypefunx float erff (float @var{x}) |
4260bc74 UD |
1005 | @comment math.h |
1006 | @comment SVID | |
7a68c94a UD |
1007 | @deftypefunx {long double} erfl (long double @var{x}) |
1008 | @code{erf} returns the error function of @var{x}. The error | |
1009 | function is defined as | |
1010 | @tex | |
1011 | $$\hbox{erf}(x) = {2\over\sqrt{\pi}}\cdot\int_0^x e^{-t^2} \hbox{d}t$$ | |
1012 | @end tex | |
1013 | @ifnottex | |
1014 | @smallexample | |
1015 | erf (x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt | |
1016 | @end smallexample | |
1017 | @end ifnottex | |
1018 | @end deftypefun | |
1019 | ||
1020 | @comment math.h | |
1021 | @comment SVID | |
1022 | @deftypefun double erfc (double @var{x}) | |
4260bc74 UD |
1023 | @comment math.h |
1024 | @comment SVID | |
7a68c94a | 1025 | @deftypefunx float erfcf (float @var{x}) |
4260bc74 UD |
1026 | @comment math.h |
1027 | @comment SVID | |
7a68c94a UD |
1028 | @deftypefunx {long double} erfcl (long double @var{x}) |
1029 | @code{erfc} returns @code{1.0 - erf(@var{x})}, but computed in a | |
1030 | fashion that avoids round-off error when @var{x} is large. | |
1031 | @end deftypefun | |
1032 | ||
1033 | @comment math.h | |
1034 | @comment SVID | |
1035 | @deftypefun double lgamma (double @var{x}) | |
4260bc74 UD |
1036 | @comment math.h |
1037 | @comment SVID | |
7a68c94a | 1038 | @deftypefunx float lgammaf (float @var{x}) |
4260bc74 UD |
1039 | @comment math.h |
1040 | @comment SVID | |
7a68c94a UD |
1041 | @deftypefunx {long double} lgammal (long double @var{x}) |
1042 | @code{lgamma} returns the natural logarithm of the absolute value of | |
1043 | the gamma function of @var{x}. The gamma function is defined as | |
1044 | @tex | |
1045 | $$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$ | |
1046 | @end tex | |
1047 | @ifnottex | |
1048 | @smallexample | |
1049 | gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt | |
1050 | @end smallexample | |
1051 | @end ifnottex | |
1052 | ||
1053 | @vindex signgam | |
1054 | The sign of the gamma function is stored in the global variable | |
1055 | @var{signgam}, which is declared in @file{math.h}. It is @code{1} if | |
04b9968b | 1056 | the intermediate result was positive or zero, or @code{-1} if it was |
7a68c94a UD |
1057 | negative. |
1058 | ||
e852e889 UD |
1059 | To compute the real gamma function you can use the @code{tgamma} |
1060 | function or you can compute the values as follows: | |
7a68c94a UD |
1061 | @smallexample |
1062 | lgam = lgamma(x); | |
1063 | gam = signgam*exp(lgam); | |
1064 | @end smallexample | |
1065 | ||
04b9968b | 1066 | The gamma function has singularities at the non-positive integers. |
7a68c94a UD |
1067 | @code{lgamma} will raise the zero divide exception if evaluated at a |
1068 | singularity. | |
1069 | @end deftypefun | |
1070 | ||
1071 | @comment math.h | |
1072 | @comment XPG | |
07435eb4 | 1073 | @deftypefun double lgamma_r (double @var{x}, int *@var{signp}) |
4260bc74 UD |
1074 | @comment math.h |
1075 | @comment XPG | |
07435eb4 | 1076 | @deftypefunx float lgammaf_r (float @var{x}, int *@var{signp}) |
4260bc74 UD |
1077 | @comment math.h |
1078 | @comment XPG | |
07435eb4 | 1079 | @deftypefunx {long double} lgammal_r (long double @var{x}, int *@var{signp}) |
7a68c94a UD |
1080 | @code{lgamma_r} is just like @code{lgamma}, but it stores the sign of |
1081 | the intermediate result in the variable pointed to by @var{signp} | |
04b9968b | 1082 | instead of in the @var{signgam} global. This means it is reentrant. |
7a68c94a UD |
1083 | @end deftypefun |
1084 | ||
7a68c94a UD |
1085 | @comment math.h |
1086 | @comment SVID | |
1087 | @deftypefun double gamma (double @var{x}) | |
4260bc74 UD |
1088 | @comment math.h |
1089 | @comment SVID | |
7a68c94a | 1090 | @deftypefunx float gammaf (float @var{x}) |
4260bc74 UD |
1091 | @comment math.h |
1092 | @comment SVID | |
7a68c94a | 1093 | @deftypefunx {long double} gammal (long double @var{x}) |
e852e889 UD |
1094 | These functions exist for compatibility reasons. They are equivalent to |
1095 | @code{lgamma} etc. It is better to use @code{lgamma} since for one the | |
04b9968b | 1096 | name reflects better the actual computation, moreover @code{lgamma} is |
ec751a23 | 1097 | standardized in @w{ISO C99} while @code{gamma} is not. |
e852e889 UD |
1098 | @end deftypefun |
1099 | ||
1100 | @comment math.h | |
ec751a23 | 1101 | @comment XPG, ISO |
e852e889 | 1102 | @deftypefun double tgamma (double @var{x}) |
4260bc74 | 1103 | @comment math.h |
ec751a23 | 1104 | @comment XPG, ISO |
e852e889 | 1105 | @deftypefunx float tgammaf (float @var{x}) |
4260bc74 | 1106 | @comment math.h |
ec751a23 | 1107 | @comment XPG, ISO |
e852e889 UD |
1108 | @deftypefunx {long double} tgammal (long double @var{x}) |
1109 | @code{tgamma} applies the gamma function to @var{x}. The gamma | |
1110 | function is defined as | |
1111 | @tex | |
1112 | $$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$ | |
1113 | @end tex | |
1114 | @ifnottex | |
1115 | @smallexample | |
1116 | gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt | |
1117 | @end smallexample | |
1118 | @end ifnottex | |
1119 | ||
ec751a23 | 1120 | This function was introduced in @w{ISO C99}. |
7a68c94a | 1121 | @end deftypefun |
7a68c94a UD |
1122 | |
1123 | @comment math.h | |
1124 | @comment SVID | |
1125 | @deftypefun double j0 (double @var{x}) | |
4260bc74 UD |
1126 | @comment math.h |
1127 | @comment SVID | |
7a68c94a | 1128 | @deftypefunx float j0f (float @var{x}) |
4260bc74 UD |
1129 | @comment math.h |
1130 | @comment SVID | |
7a68c94a UD |
1131 | @deftypefunx {long double} j0l (long double @var{x}) |
1132 | @code{j0} returns the Bessel function of the first kind of order 0 of | |
1133 | @var{x}. It may signal underflow if @var{x} is too large. | |
1134 | @end deftypefun | |
1135 | ||
1136 | @comment math.h | |
1137 | @comment SVID | |
1138 | @deftypefun double j1 (double @var{x}) | |
4260bc74 UD |
1139 | @comment math.h |
1140 | @comment SVID | |
7a68c94a | 1141 | @deftypefunx float j1f (float @var{x}) |
4260bc74 UD |
1142 | @comment math.h |
1143 | @comment SVID | |
7a68c94a UD |
1144 | @deftypefunx {long double} j1l (long double @var{x}) |
1145 | @code{j1} returns the Bessel function of the first kind of order 1 of | |
1146 | @var{x}. It may signal underflow if @var{x} is too large. | |
1147 | @end deftypefun | |
1148 | ||
1149 | @comment math.h | |
1150 | @comment SVID | |
1151 | @deftypefun double jn (int n, double @var{x}) | |
4260bc74 UD |
1152 | @comment math.h |
1153 | @comment SVID | |
7a68c94a | 1154 | @deftypefunx float jnf (int n, float @var{x}) |
4260bc74 UD |
1155 | @comment math.h |
1156 | @comment SVID | |
7a68c94a UD |
1157 | @deftypefunx {long double} jnl (int n, long double @var{x}) |
1158 | @code{jn} returns the Bessel function of the first kind of order | |
1159 | @var{n} of @var{x}. It may signal underflow if @var{x} is too large. | |
1160 | @end deftypefun | |
1161 | ||
1162 | @comment math.h | |
1163 | @comment SVID | |
1164 | @deftypefun double y0 (double @var{x}) | |
4260bc74 UD |
1165 | @comment math.h |
1166 | @comment SVID | |
7a68c94a | 1167 | @deftypefunx float y0f (float @var{x}) |
4260bc74 UD |
1168 | @comment math.h |
1169 | @comment SVID | |
7a68c94a UD |
1170 | @deftypefunx {long double} y0l (long double @var{x}) |
1171 | @code{y0} returns the Bessel function of the second kind of order 0 of | |
1172 | @var{x}. It may signal underflow if @var{x} is too large. If @var{x} | |
1173 | is negative, @code{y0} signals a domain error; if it is zero, | |
1174 | @code{y0} signals overflow and returns @math{-@infinity}. | |
1175 | @end deftypefun | |
1176 | ||
1177 | @comment math.h | |
1178 | @comment SVID | |
1179 | @deftypefun double y1 (double @var{x}) | |
4260bc74 UD |
1180 | @comment math.h |
1181 | @comment SVID | |
7a68c94a | 1182 | @deftypefunx float y1f (float @var{x}) |
4260bc74 UD |
1183 | @comment math.h |
1184 | @comment SVID | |
7a68c94a UD |
1185 | @deftypefunx {long double} y1l (long double @var{x}) |
1186 | @code{y1} returns the Bessel function of the second kind of order 1 of | |
1187 | @var{x}. It may signal underflow if @var{x} is too large. If @var{x} | |
1188 | is negative, @code{y1} signals a domain error; if it is zero, | |
1189 | @code{y1} signals overflow and returns @math{-@infinity}. | |
1190 | @end deftypefun | |
1191 | ||
1192 | @comment math.h | |
1193 | @comment SVID | |
1194 | @deftypefun double yn (int n, double @var{x}) | |
4260bc74 UD |
1195 | @comment math.h |
1196 | @comment SVID | |
7a68c94a | 1197 | @deftypefunx float ynf (int n, float @var{x}) |
4260bc74 UD |
1198 | @comment math.h |
1199 | @comment SVID | |
7a68c94a UD |
1200 | @deftypefunx {long double} ynl (int n, long double @var{x}) |
1201 | @code{yn} returns the Bessel function of the second kind of order @var{n} of | |
1202 | @var{x}. It may signal underflow if @var{x} is too large. If @var{x} | |
1203 | is negative, @code{yn} signals a domain error; if it is zero, | |
1204 | @code{yn} signals overflow and returns @math{-@infinity}. | |
1205 | @end deftypefun | |
55c14926 | 1206 | |
aaa1276e UD |
1207 | @node Errors in Math Functions |
1208 | @section Known Maximum Errors in Math Functions | |
1209 | @cindex math errors | |
1210 | @cindex ulps | |
1211 | ||
1212 | This section lists the known errors of the functions in the math | |
1213 | library. Errors are measured in ``units of the last place''. This is a | |
1214 | measure for the relative error. For a number @math{z} with the | |
1215 | representation @math{d.d@dots{}d@mul{}2^e} (we assume IEEE | |
1216 | floating-point numbers with base 2) the ULP is represented by | |
1217 | ||
1218 | @tex | |
ec751a23 | 1219 | $${|d.d\dots d - (z/2^e)|}\over {2^{p-1}}$$ |
aaa1276e UD |
1220 | @end tex |
1221 | @ifnottex | |
1222 | @smallexample | |
1223 | |d.d...d - (z / 2^e)| / 2^(p - 1) | |
1224 | @end smallexample | |
1225 | @end ifnottex | |
1226 | ||
1227 | @noindent | |
1228 | where @math{p} is the number of bits in the mantissa of the | |
1229 | floating-point number representation. Ideally the error for all | |
1230 | functions is always less than 0.5ulps. Using rounding bits this is also | |
1231 | possible and normally implemented for the basic operations. To achieve | |
1232 | the same for the complex math functions requires a lot more work and | |
41713d4e | 1233 | this has not yet been done. |
aaa1276e UD |
1234 | |
1235 | Therefore many of the functions in the math library have errors. The | |
1236 | table lists the maximum error for each function which is exposed by one | |
41713d4e AJ |
1237 | of the existing tests in the test suite. The table tries to cover as much |
1238 | as possible and list the actual maximum error (or at least a ballpark | |
aaa1276e UD |
1239 | figure) but this is often not achieved due to the large search space. |
1240 | ||
1241 | The table lists the ULP values for different architectures. Different | |
1242 | architectures have different results since their hardware support for | |
1243 | floating-point operations varies and also the existing hardware support | |
1244 | is different. | |
1245 | ||
41713d4e AJ |
1246 | @page |
1247 | @c This multitable does not fit on a single page | |
aaa1276e UD |
1248 | @include libm-err.texi |
1249 | ||
28f540f4 RM |
1250 | @node Pseudo-Random Numbers |
1251 | @section Pseudo-Random Numbers | |
1252 | @cindex random numbers | |
1253 | @cindex pseudo-random numbers | |
1254 | @cindex seed (for random numbers) | |
1255 | ||
1256 | This section describes the GNU facilities for generating a series of | |
1257 | pseudo-random numbers. The numbers generated are not truly random; | |
7a68c94a UD |
1258 | typically, they form a sequence that repeats periodically, with a period |
1259 | so large that you can ignore it for ordinary purposes. The random | |
1260 | number generator works by remembering a @dfn{seed} value which it uses | |
1261 | to compute the next random number and also to compute a new seed. | |
28f540f4 RM |
1262 | |
1263 | Although the generated numbers look unpredictable within one run of a | |
1264 | program, the sequence of numbers is @emph{exactly the same} from one run | |
1265 | to the next. This is because the initial seed is always the same. This | |
1266 | is convenient when you are debugging a program, but it is unhelpful if | |
7a68c94a UD |
1267 | you want the program to behave unpredictably. If you want a different |
1268 | pseudo-random series each time your program runs, you must specify a | |
1269 | different seed each time. For ordinary purposes, basing the seed on the | |
1270 | current time works well. | |
28f540f4 | 1271 | |
04b9968b | 1272 | You can obtain repeatable sequences of numbers on a particular machine type |
28f540f4 RM |
1273 | by specifying the same initial seed value for the random number |
1274 | generator. There is no standard meaning for a particular seed value; | |
1275 | the same seed, used in different C libraries or on different CPU types, | |
1276 | will give you different random numbers. | |
1277 | ||
f65fd747 | 1278 | The GNU library supports the standard @w{ISO C} random number functions |
7a68c94a UD |
1279 | plus two other sets derived from BSD and SVID. The BSD and @w{ISO C} |
1280 | functions provide identical, somewhat limited functionality. If only a | |
1281 | small number of random bits are required, we recommend you use the | |
1282 | @w{ISO C} interface, @code{rand} and @code{srand}. The SVID functions | |
1283 | provide a more flexible interface, which allows better random number | |
1284 | generator algorithms, provides more random bits (up to 48) per call, and | |
1285 | can provide random floating-point numbers. These functions are required | |
1286 | by the XPG standard and therefore will be present in all modern Unix | |
1287 | systems. | |
28f540f4 RM |
1288 | |
1289 | @menu | |
7a68c94a UD |
1290 | * ISO Random:: @code{rand} and friends. |
1291 | * BSD Random:: @code{random} and friends. | |
1292 | * SVID Random:: @code{drand48} and friends. | |
28f540f4 RM |
1293 | @end menu |
1294 | ||
f65fd747 UD |
1295 | @node ISO Random |
1296 | @subsection ISO C Random Number Functions | |
28f540f4 RM |
1297 | |
1298 | This section describes the random number functions that are part of | |
f65fd747 | 1299 | the @w{ISO C} standard. |
28f540f4 RM |
1300 | |
1301 | To use these facilities, you should include the header file | |
1302 | @file{stdlib.h} in your program. | |
1303 | @pindex stdlib.h | |
1304 | ||
1305 | @comment stdlib.h | |
f65fd747 | 1306 | @comment ISO |
28f540f4 | 1307 | @deftypevr Macro int RAND_MAX |
7a68c94a UD |
1308 | The value of this macro is an integer constant representing the largest |
1309 | value the @code{rand} function can return. In the GNU library, it is | |
1310 | @code{2147483647}, which is the largest signed integer representable in | |
1311 | 32 bits. In other libraries, it may be as low as @code{32767}. | |
28f540f4 RM |
1312 | @end deftypevr |
1313 | ||
1314 | @comment stdlib.h | |
f65fd747 | 1315 | @comment ISO |
ca34d7a7 | 1316 | @deftypefun int rand (void) |
28f540f4 | 1317 | The @code{rand} function returns the next pseudo-random number in the |
7a68c94a | 1318 | series. The value ranges from @code{0} to @code{RAND_MAX}. |
28f540f4 RM |
1319 | @end deftypefun |
1320 | ||
1321 | @comment stdlib.h | |
f65fd747 | 1322 | @comment ISO |
28f540f4 RM |
1323 | @deftypefun void srand (unsigned int @var{seed}) |
1324 | This function establishes @var{seed} as the seed for a new series of | |
1325 | pseudo-random numbers. If you call @code{rand} before a seed has been | |
1326 | established with @code{srand}, it uses the value @code{1} as a default | |
1327 | seed. | |
1328 | ||
7a68c94a UD |
1329 | To produce a different pseudo-random series each time your program is |
1330 | run, do @code{srand (time (0))}. | |
28f540f4 RM |
1331 | @end deftypefun |
1332 | ||
7a68c94a UD |
1333 | POSIX.1 extended the C standard functions to support reproducible random |
1334 | numbers in multi-threaded programs. However, the extension is badly | |
1335 | designed and unsuitable for serious work. | |
61eb22d3 UD |
1336 | |
1337 | @comment stdlib.h | |
1338 | @comment POSIX.1 | |
1339 | @deftypefun int rand_r (unsigned int *@var{seed}) | |
1340 | This function returns a random number in the range 0 to @code{RAND_MAX} | |
7a68c94a UD |
1341 | just as @code{rand} does. However, all its state is stored in the |
1342 | @var{seed} argument. This means the RNG's state can only have as many | |
1343 | bits as the type @code{unsigned int} has. This is far too few to | |
1344 | provide a good RNG. | |
61eb22d3 | 1345 | |
7a68c94a UD |
1346 | If your program requires a reentrant RNG, we recommend you use the |
1347 | reentrant GNU extensions to the SVID random number generator. The | |
1348 | POSIX.1 interface should only be used when the GNU extensions are not | |
1349 | available. | |
61eb22d3 UD |
1350 | @end deftypefun |
1351 | ||
1352 | ||
28f540f4 RM |
1353 | @node BSD Random |
1354 | @subsection BSD Random Number Functions | |
1355 | ||
1356 | This section describes a set of random number generation functions that | |
1357 | are derived from BSD. There is no advantage to using these functions | |
1358 | with the GNU C library; we support them for BSD compatibility only. | |
1359 | ||
1360 | The prototypes for these functions are in @file{stdlib.h}. | |
1361 | @pindex stdlib.h | |
1362 | ||
1363 | @comment stdlib.h | |
1364 | @comment BSD | |
0423ee17 | 1365 | @deftypefun {long int} random (void) |
28f540f4 | 1366 | This function returns the next pseudo-random number in the sequence. |
7a68c94a | 1367 | The value returned ranges from @code{0} to @code{RAND_MAX}. |
ca34d7a7 | 1368 | |
48b22986 | 1369 | @strong{NB:} Temporarily this function was defined to return a |
0423ee17 UD |
1370 | @code{int32_t} value to indicate that the return value always contains |
1371 | 32 bits even if @code{long int} is wider. The standard demands it | |
1372 | differently. Users must always be aware of the 32-bit limitation, | |
1373 | though. | |
28f540f4 RM |
1374 | @end deftypefun |
1375 | ||
1376 | @comment stdlib.h | |
1377 | @comment BSD | |
1378 | @deftypefun void srandom (unsigned int @var{seed}) | |
7a68c94a UD |
1379 | The @code{srandom} function sets the state of the random number |
1380 | generator based on the integer @var{seed}. If you supply a @var{seed} value | |
28f540f4 RM |
1381 | of @code{1}, this will cause @code{random} to reproduce the default set |
1382 | of random numbers. | |
1383 | ||
7a68c94a UD |
1384 | To produce a different set of pseudo-random numbers each time your |
1385 | program runs, do @code{srandom (time (0))}. | |
28f540f4 RM |
1386 | @end deftypefun |
1387 | ||
1388 | @comment stdlib.h | |
1389 | @comment BSD | |
1390 | @deftypefun {void *} initstate (unsigned int @var{seed}, void *@var{state}, size_t @var{size}) | |
1391 | The @code{initstate} function is used to initialize the random number | |
1392 | generator state. The argument @var{state} is an array of @var{size} | |
7a68c94a UD |
1393 | bytes, used to hold the state information. It is initialized based on |
1394 | @var{seed}. The size must be between 8 and 256 bytes, and should be a | |
1395 | power of two. The bigger the @var{state} array, the better. | |
28f540f4 RM |
1396 | |
1397 | The return value is the previous value of the state information array. | |
1398 | You can use this value later as an argument to @code{setstate} to | |
1399 | restore that state. | |
1400 | @end deftypefun | |
1401 | ||
1402 | @comment stdlib.h | |
1403 | @comment BSD | |
1404 | @deftypefun {void *} setstate (void *@var{state}) | |
1405 | The @code{setstate} function restores the random number state | |
1406 | information @var{state}. The argument must have been the result of | |
2c6fe0bd | 1407 | a previous call to @var{initstate} or @var{setstate}. |
28f540f4 RM |
1408 | |
1409 | The return value is the previous value of the state information array. | |
f2ea0f5b | 1410 | You can use this value later as an argument to @code{setstate} to |
28f540f4 | 1411 | restore that state. |
a785f6c5 UD |
1412 | |
1413 | If the function fails the return value is @code{NULL}. | |
28f540f4 | 1414 | @end deftypefun |
b4012b75 | 1415 | |
4c78249d UD |
1416 | The four functions described so far in this section all work on a state |
1417 | which is shared by all threads. The state is not directly accessible to | |
1418 | the user and can only be modified by these functions. This makes it | |
1419 | hard to deal with situations where each thread should have its own | |
1420 | pseudo-random number generator. | |
1421 | ||
1422 | The GNU C library contains four additional functions which contain the | |
1423 | state as an explicit parameter and therefore make it possible to handle | |
e2f4aa54 | 1424 | thread-local PRNGs. Beside this there is no difference. In fact, the |
4c78249d UD |
1425 | four functions already discussed are implemented internally using the |
1426 | following interfaces. | |
1427 | ||
1428 | The @file{stdlib.h} header contains a definition of the following type: | |
1429 | ||
1430 | @comment stdlib.h | |
1431 | @comment GNU | |
1432 | @deftp {Data Type} {struct random_data} | |
1433 | ||
1434 | Objects of type @code{struct random_data} contain the information | |
1435 | necessary to represent the state of the PRNG. Although a complete | |
1436 | definition of the type is present the type should be treated as opaque. | |
1437 | @end deftp | |
1438 | ||
1439 | The functions modifying the state follow exactly the already described | |
1440 | functions. | |
1441 | ||
1442 | @comment stdlib.h | |
1443 | @comment GNU | |
1444 | @deftypefun int random_r (struct random_data *restrict @var{buf}, int32_t *restrict @var{result}) | |
1445 | The @code{random_r} function behaves exactly like the @code{random} | |
1446 | function except that it uses and modifies the state in the object | |
1447 | pointed to by the first parameter instead of the global state. | |
1448 | @end deftypefun | |
1449 | ||
1450 | @comment stdlib.h | |
1451 | @comment GNU | |
1452 | @deftypefun int srandom_r (unsigned int @var{seed}, struct random_data *@var{buf}) | |
1453 | The @code{srandom_r} function behaves exactly like the @code{srandom} | |
1454 | function except that it uses and modifies the state in the object | |
1455 | pointed to by the second parameter instead of the global state. | |
1456 | @end deftypefun | |
1457 | ||
1458 | @comment stdlib.h | |
1459 | @comment GNU | |
1460 | @deftypefun int initstate_r (unsigned int @var{seed}, char *restrict @var{statebuf}, size_t @var{statelen}, struct random_data *restrict @var{buf}) | |
1461 | The @code{initstate_r} function behaves exactly like the @code{initstate} | |
1462 | function except that it uses and modifies the state in the object | |
1463 | pointed to by the fourth parameter instead of the global state. | |
1464 | @end deftypefun | |
1465 | ||
1466 | @comment stdlib.h | |
1467 | @comment GNU | |
1468 | @deftypefun int setstate_r (char *restrict @var{statebuf}, struct random_data *restrict @var{buf}) | |
1469 | The @code{setstate_r} function behaves exactly like the @code{setstate} | |
1470 | function except that it uses and modifies the state in the object | |
1471 | pointed to by the first parameter instead of the global state. | |
1472 | @end deftypefun | |
1473 | ||
b4012b75 UD |
1474 | @node SVID Random |
1475 | @subsection SVID Random Number Function | |
1476 | ||
1477 | The C library on SVID systems contains yet another kind of random number | |
1478 | generator functions. They use a state of 48 bits of data. The user can | |
7a68c94a | 1479 | choose among a collection of functions which return the random bits |
b4012b75 UD |
1480 | in different forms. |
1481 | ||
04b9968b | 1482 | Generally there are two kinds of function. The first uses a state of |
b4012b75 | 1483 | the random number generator which is shared among several functions and |
04b9968b UD |
1484 | by all threads of the process. The second requires the user to handle |
1485 | the state. | |
b4012b75 UD |
1486 | |
1487 | All functions have in common that they use the same congruential | |
1488 | formula with the same constants. The formula is | |
1489 | ||
1490 | @smallexample | |
1491 | Y = (a * X + c) mod m | |
1492 | @end smallexample | |
1493 | ||
1494 | @noindent | |
1495 | where @var{X} is the state of the generator at the beginning and | |
1496 | @var{Y} the state at the end. @code{a} and @code{c} are constants | |
04b9968b | 1497 | determining the way the generator works. By default they are |
b4012b75 UD |
1498 | |
1499 | @smallexample | |
1500 | a = 0x5DEECE66D = 25214903917 | |
1501 | c = 0xb = 11 | |
1502 | @end smallexample | |
1503 | ||
1504 | @noindent | |
1505 | but they can also be changed by the user. @code{m} is of course 2^48 | |
04b9968b | 1506 | since the state consists of a 48-bit array. |
b4012b75 | 1507 | |
f2615995 UD |
1508 | The prototypes for these functions are in @file{stdlib.h}. |
1509 | @pindex stdlib.h | |
1510 | ||
b4012b75 UD |
1511 | |
1512 | @comment stdlib.h | |
1513 | @comment SVID | |
55c14926 | 1514 | @deftypefun double drand48 (void) |
b4012b75 UD |
1515 | This function returns a @code{double} value in the range of @code{0.0} |
1516 | to @code{1.0} (exclusive). The random bits are determined by the global | |
1517 | state of the random number generator in the C library. | |
1518 | ||
04b9968b | 1519 | Since the @code{double} type according to @w{IEEE 754} has a 52-bit |
b4012b75 UD |
1520 | mantissa this means 4 bits are not initialized by the random number |
1521 | generator. These are (of course) chosen to be the least significant | |
1522 | bits and they are initialized to @code{0}. | |
1523 | @end deftypefun | |
1524 | ||
1525 | @comment stdlib.h | |
1526 | @comment SVID | |
1527 | @deftypefun double erand48 (unsigned short int @var{xsubi}[3]) | |
1528 | This function returns a @code{double} value in the range of @code{0.0} | |
04b9968b | 1529 | to @code{1.0} (exclusive), similarly to @code{drand48}. The argument is |
b4012b75 UD |
1530 | an array describing the state of the random number generator. |
1531 | ||
1532 | This function can be called subsequently since it updates the array to | |
1533 | guarantee random numbers. The array should have been initialized before | |
04b9968b | 1534 | initial use to obtain reproducible results. |
b4012b75 UD |
1535 | @end deftypefun |
1536 | ||
1537 | @comment stdlib.h | |
1538 | @comment SVID | |
55c14926 | 1539 | @deftypefun {long int} lrand48 (void) |
04b9968b | 1540 | The @code{lrand48} function returns an integer value in the range of |
b4012b75 | 1541 | @code{0} to @code{2^31} (exclusive). Even if the size of the @code{long |
04b9968b | 1542 | int} type can take more than 32 bits, no higher numbers are returned. |
b4012b75 UD |
1543 | The random bits are determined by the global state of the random number |
1544 | generator in the C library. | |
1545 | @end deftypefun | |
1546 | ||
1547 | @comment stdlib.h | |
1548 | @comment SVID | |
1549 | @deftypefun {long int} nrand48 (unsigned short int @var{xsubi}[3]) | |
1550 | This function is similar to the @code{lrand48} function in that it | |
1551 | returns a number in the range of @code{0} to @code{2^31} (exclusive) but | |
1552 | the state of the random number generator used to produce the random bits | |
1553 | is determined by the array provided as the parameter to the function. | |
1554 | ||
04b9968b UD |
1555 | The numbers in the array are updated afterwards so that subsequent calls |
1556 | to this function yield different results (as is expected of a random | |
1557 | number generator). The array should have been initialized before the | |
1558 | first call to obtain reproducible results. | |
b4012b75 UD |
1559 | @end deftypefun |
1560 | ||
1561 | @comment stdlib.h | |
1562 | @comment SVID | |
55c14926 | 1563 | @deftypefun {long int} mrand48 (void) |
b4012b75 UD |
1564 | The @code{mrand48} function is similar to @code{lrand48}. The only |
1565 | difference is that the numbers returned are in the range @code{-2^31} to | |
1566 | @code{2^31} (exclusive). | |
1567 | @end deftypefun | |
1568 | ||
1569 | @comment stdlib.h | |
1570 | @comment SVID | |
1571 | @deftypefun {long int} jrand48 (unsigned short int @var{xsubi}[3]) | |
1572 | The @code{jrand48} function is similar to @code{nrand48}. The only | |
1573 | difference is that the numbers returned are in the range @code{-2^31} to | |
1574 | @code{2^31} (exclusive). For the @code{xsubi} parameter the same | |
1575 | requirements are necessary. | |
1576 | @end deftypefun | |
1577 | ||
1578 | The internal state of the random number generator can be initialized in | |
04b9968b | 1579 | several ways. The methods differ in the completeness of the |
b4012b75 UD |
1580 | information provided. |
1581 | ||
1582 | @comment stdlib.h | |
1583 | @comment SVID | |
04b9968b | 1584 | @deftypefun void srand48 (long int @var{seedval}) |
b4012b75 | 1585 | The @code{srand48} function sets the most significant 32 bits of the |
04b9968b | 1586 | internal state of the random number generator to the least |
f2ea0f5b UD |
1587 | significant 32 bits of the @var{seedval} parameter. The lower 16 bits |
1588 | are initialized to the value @code{0x330E}. Even if the @code{long | |
04b9968b | 1589 | int} type contains more than 32 bits only the lower 32 bits are used. |
b4012b75 | 1590 | |
04b9968b UD |
1591 | Owing to this limitation, initialization of the state of this |
1592 | function is not very useful. But it makes it easy to use a construct | |
b4012b75 UD |
1593 | like @code{srand48 (time (0))}. |
1594 | ||
1595 | A side-effect of this function is that the values @code{a} and @code{c} | |
1596 | from the internal state, which are used in the congruential formula, | |
1597 | are reset to the default values given above. This is of importance once | |
04b9968b | 1598 | the user has called the @code{lcong48} function (see below). |
b4012b75 UD |
1599 | @end deftypefun |
1600 | ||
1601 | @comment stdlib.h | |
1602 | @comment SVID | |
1603 | @deftypefun {unsigned short int *} seed48 (unsigned short int @var{seed16v}[3]) | |
1604 | The @code{seed48} function initializes all 48 bits of the state of the | |
04b9968b | 1605 | internal random number generator from the contents of the parameter |
b4012b75 UD |
1606 | @var{seed16v}. Here the lower 16 bits of the first element of |
1607 | @var{see16v} initialize the least significant 16 bits of the internal | |
1608 | state, the lower 16 bits of @code{@var{seed16v}[1]} initialize the mid-order | |
1609 | 16 bits of the state and the 16 lower bits of @code{@var{seed16v}[2]} | |
1610 | initialize the most significant 16 bits of the state. | |
1611 | ||
1612 | Unlike @code{srand48} this function lets the user initialize all 48 bits | |
1613 | of the state. | |
1614 | ||
1615 | The value returned by @code{seed48} is a pointer to an array containing | |
1616 | the values of the internal state before the change. This might be | |
1617 | useful to restart the random number generator at a certain state. | |
04b9968b | 1618 | Otherwise the value can simply be ignored. |
b4012b75 UD |
1619 | |
1620 | As for @code{srand48}, the values @code{a} and @code{c} from the | |
1621 | congruential formula are reset to the default values. | |
1622 | @end deftypefun | |
1623 | ||
1624 | There is one more function to initialize the random number generator | |
04b9968b UD |
1625 | which enables you to specify even more information by allowing you to |
1626 | change the parameters in the congruential formula. | |
b4012b75 UD |
1627 | |
1628 | @comment stdlib.h | |
1629 | @comment SVID | |
1630 | @deftypefun void lcong48 (unsigned short int @var{param}[7]) | |
1631 | The @code{lcong48} function allows the user to change the complete state | |
1632 | of the random number generator. Unlike @code{srand48} and | |
1633 | @code{seed48}, this function also changes the constants in the | |
1634 | congruential formula. | |
1635 | ||
1636 | From the seven elements in the array @var{param} the least significant | |
1637 | 16 bits of the entries @code{@var{param}[0]} to @code{@var{param}[2]} | |
04b9968b | 1638 | determine the initial state, the least significant 16 bits of |
b4012b75 | 1639 | @code{@var{param}[3]} to @code{@var{param}[5]} determine the 48 bit |
04b9968b | 1640 | constant @code{a} and @code{@var{param}[6]} determines the 16-bit value |
b4012b75 UD |
1641 | @code{c}. |
1642 | @end deftypefun | |
1643 | ||
1644 | All the above functions have in common that they use the global | |
1645 | parameters for the congruential formula. In multi-threaded programs it | |
1646 | might sometimes be useful to have different parameters in different | |
1647 | threads. For this reason all the above functions have a counterpart | |
1648 | which works on a description of the random number generator in the | |
1649 | user-supplied buffer instead of the global state. | |
1650 | ||
1651 | Please note that it is no problem if several threads use the global | |
1652 | state if all threads use the functions which take a pointer to an array | |
1653 | containing the state. The random numbers are computed following the | |
1654 | same loop but if the state in the array is different all threads will | |
04b9968b | 1655 | obtain an individual random number generator. |
b4012b75 | 1656 | |
04b9968b UD |
1657 | The user-supplied buffer must be of type @code{struct drand48_data}. |
1658 | This type should be regarded as opaque and not manipulated directly. | |
b4012b75 UD |
1659 | |
1660 | @comment stdlib.h | |
1661 | @comment GNU | |
1662 | @deftypefun int drand48_r (struct drand48_data *@var{buffer}, double *@var{result}) | |
1663 | This function is equivalent to the @code{drand48} function with the | |
04b9968b UD |
1664 | difference that it does not modify the global random number generator |
1665 | parameters but instead the parameters in the buffer supplied through the | |
1666 | pointer @var{buffer}. The random number is returned in the variable | |
1667 | pointed to by @var{result}. | |
b4012b75 | 1668 | |
04b9968b | 1669 | The return value of the function indicates whether the call succeeded. |
b4012b75 UD |
1670 | If the value is less than @code{0} an error occurred and @var{errno} is |
1671 | set to indicate the problem. | |
1672 | ||
1673 | This function is a GNU extension and should not be used in portable | |
1674 | programs. | |
1675 | @end deftypefun | |
1676 | ||
1677 | @comment stdlib.h | |
1678 | @comment GNU | |
1679 | @deftypefun int erand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, double *@var{result}) | |
04b9968b UD |
1680 | The @code{erand48_r} function works like @code{erand48}, but in addition |
1681 | it takes an argument @var{buffer} which describes the random number | |
1682 | generator. The state of the random number generator is taken from the | |
1683 | @code{xsubi} array, the parameters for the congruential formula from the | |
1684 | global random number generator data. The random number is returned in | |
1685 | the variable pointed to by @var{result}. | |
b4012b75 | 1686 | |
04b9968b | 1687 | The return value is non-negative if the call succeeded. |
b4012b75 UD |
1688 | |
1689 | This function is a GNU extension and should not be used in portable | |
1690 | programs. | |
1691 | @end deftypefun | |
1692 | ||
1693 | @comment stdlib.h | |
1694 | @comment GNU | |
1695 | @deftypefun int lrand48_r (struct drand48_data *@var{buffer}, double *@var{result}) | |
04b9968b UD |
1696 | This function is similar to @code{lrand48}, but in addition it takes a |
1697 | pointer to a buffer describing the state of the random number generator | |
1698 | just like @code{drand48}. | |
b4012b75 UD |
1699 | |
1700 | If the return value of the function is non-negative the variable pointed | |
1701 | to by @var{result} contains the result. Otherwise an error occurred. | |
1702 | ||
1703 | This function is a GNU extension and should not be used in portable | |
1704 | programs. | |
1705 | @end deftypefun | |
1706 | ||
1707 | @comment stdlib.h | |
1708 | @comment GNU | |
1709 | @deftypefun int nrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result}) | |
1710 | The @code{nrand48_r} function works like @code{nrand48} in that it | |
04b9968b | 1711 | produces a random number in the range @code{0} to @code{2^31}. But instead |
b4012b75 UD |
1712 | of using the global parameters for the congruential formula it uses the |
1713 | information from the buffer pointed to by @var{buffer}. The state is | |
1714 | described by the values in @var{xsubi}. | |
1715 | ||
1716 | If the return value is non-negative the variable pointed to by | |
1717 | @var{result} contains the result. | |
1718 | ||
1719 | This function is a GNU extension and should not be used in portable | |
1720 | programs. | |
1721 | @end deftypefun | |
1722 | ||
1723 | @comment stdlib.h | |
1724 | @comment GNU | |
1725 | @deftypefun int mrand48_r (struct drand48_data *@var{buffer}, double *@var{result}) | |
04b9968b UD |
1726 | This function is similar to @code{mrand48} but like the other reentrant |
1727 | functions it uses the random number generator described by the value in | |
b4012b75 UD |
1728 | the buffer pointed to by @var{buffer}. |
1729 | ||
1730 | If the return value is non-negative the variable pointed to by | |
1731 | @var{result} contains the result. | |
1732 | ||
1733 | This function is a GNU extension and should not be used in portable | |
1734 | programs. | |
1735 | @end deftypefun | |
1736 | ||
1737 | @comment stdlib.h | |
1738 | @comment GNU | |
1739 | @deftypefun int jrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result}) | |
04b9968b | 1740 | The @code{jrand48_r} function is similar to @code{jrand48}. Like the |
b4012b75 UD |
1741 | other reentrant functions of this function family it uses the |
1742 | congruential formula parameters from the buffer pointed to by | |
1743 | @var{buffer}. | |
1744 | ||
1745 | If the return value is non-negative the variable pointed to by | |
1746 | @var{result} contains the result. | |
1747 | ||
1748 | This function is a GNU extension and should not be used in portable | |
1749 | programs. | |
1750 | @end deftypefun | |
1751 | ||
04b9968b UD |
1752 | Before any of the above functions are used the buffer of type |
1753 | @code{struct drand48_data} should be initialized. The easiest way to do | |
1754 | this is to fill the whole buffer with null bytes, e.g. by | |
b4012b75 UD |
1755 | |
1756 | @smallexample | |
1757 | memset (buffer, '\0', sizeof (struct drand48_data)); | |
1758 | @end smallexample | |
1759 | ||
1760 | @noindent | |
f2ea0f5b | 1761 | Using any of the reentrant functions of this family now will |
b4012b75 UD |
1762 | automatically initialize the random number generator to the default |
1763 | values for the state and the parameters of the congruential formula. | |
1764 | ||
04b9968b | 1765 | The other possibility is to use any of the functions which explicitly |
b4012b75 | 1766 | initialize the buffer. Though it might be obvious how to initialize the |
04b9968b | 1767 | buffer from looking at the parameter to the function, it is highly |
b4012b75 UD |
1768 | recommended to use these functions since the result might not always be |
1769 | what you expect. | |
1770 | ||
1771 | @comment stdlib.h | |
1772 | @comment GNU | |
1773 | @deftypefun int srand48_r (long int @var{seedval}, struct drand48_data *@var{buffer}) | |
1774 | The description of the random number generator represented by the | |
04b9968b | 1775 | information in @var{buffer} is initialized similarly to what the function |
f2ea0f5b UD |
1776 | @code{srand48} does. The state is initialized from the parameter |
1777 | @var{seedval} and the parameters for the congruential formula are | |
04b9968b | 1778 | initialized to their default values. |
b4012b75 UD |
1779 | |
1780 | If the return value is non-negative the function call succeeded. | |
1781 | ||
1782 | This function is a GNU extension and should not be used in portable | |
1783 | programs. | |
1784 | @end deftypefun | |
1785 | ||
1786 | @comment stdlib.h | |
1787 | @comment GNU | |
1788 | @deftypefun int seed48_r (unsigned short int @var{seed16v}[3], struct drand48_data *@var{buffer}) | |
1789 | This function is similar to @code{srand48_r} but like @code{seed48} it | |
1790 | initializes all 48 bits of the state from the parameter @var{seed16v}. | |
1791 | ||
1792 | If the return value is non-negative the function call succeeded. It | |
1793 | does not return a pointer to the previous state of the random number | |
04b9968b UD |
1794 | generator like the @code{seed48} function does. If the user wants to |
1795 | preserve the state for a later re-run s/he can copy the whole buffer | |
b4012b75 UD |
1796 | pointed to by @var{buffer}. |
1797 | ||
1798 | This function is a GNU extension and should not be used in portable | |
1799 | programs. | |
1800 | @end deftypefun | |
1801 | ||
1802 | @comment stdlib.h | |
1803 | @comment GNU | |
1804 | @deftypefun int lcong48_r (unsigned short int @var{param}[7], struct drand48_data *@var{buffer}) | |
1805 | This function initializes all aspects of the random number generator | |
04b9968b UD |
1806 | described in @var{buffer} with the data in @var{param}. Here it is |
1807 | especially true that the function does more than just copying the | |
1808 | contents of @var{param} and @var{buffer}. More work is required and | |
1809 | therefore it is important to use this function rather than initializing | |
1810 | the random number generator directly. | |
b4012b75 UD |
1811 | |
1812 | If the return value is non-negative the function call succeeded. | |
1813 | ||
1814 | This function is a GNU extension and should not be used in portable | |
1815 | programs. | |
1816 | @end deftypefun | |
7a68c94a UD |
1817 | |
1818 | @node FP Function Optimizations | |
1819 | @section Is Fast Code or Small Code preferred? | |
1820 | @cindex Optimization | |
1821 | ||
04b9968b UD |
1822 | If an application uses many floating point functions it is often the case |
1823 | that the cost of the function calls themselves is not negligible. | |
1824 | Modern processors can often execute the operations themselves | |
1825 | very fast, but the function call disrupts the instruction pipeline. | |
7a68c94a UD |
1826 | |
1827 | For this reason the GNU C Library provides optimizations for many of the | |
04b9968b UD |
1828 | frequently-used math functions. When GNU CC is used and the user |
1829 | activates the optimizer, several new inline functions and macros are | |
7a68c94a | 1830 | defined. These new functions and macros have the same names as the |
04b9968b | 1831 | library functions and so are used instead of the latter. In the case of |
7a68c94a | 1832 | inline functions the compiler will decide whether it is reasonable to |
04b9968b | 1833 | use them, and this decision is usually correct. |
7a68c94a | 1834 | |
04b9968b UD |
1835 | This means that no calls to the library functions may be necessary, and |
1836 | can increase the speed of generated code significantly. The drawback is | |
1837 | that code size will increase, and the increase is not always negligible. | |
7a68c94a | 1838 | |
378fbeb4 UD |
1839 | There are two kind of inline functions: Those that give the same result |
1840 | as the library functions and others that might not set @code{errno} and | |
1841 | might have a reduced precision and/or argument range in comparison with | |
1842 | the library functions. The latter inline functions are only available | |
1843 | if the flag @code{-ffast-math} is given to GNU CC. | |
aa847ee5 | 1844 | |
7a68c94a UD |
1845 | In cases where the inline functions and macros are not wanted the symbol |
1846 | @code{__NO_MATH_INLINES} should be defined before any system header is | |
04b9968b UD |
1847 | included. This will ensure that only library functions are used. Of |
1848 | course, it can be determined for each file in the project whether | |
1849 | giving this option is preferable or not. | |
1850 | ||
1851 | Not all hardware implements the entire @w{IEEE 754} standard, and even | |
1852 | if it does there may be a substantial performance penalty for using some | |
1853 | of its features. For example, enabling traps on some processors forces | |
1854 | the FPU to run un-pipelined, which can more than double calculation time. | |
7a68c94a | 1855 | @c ***Add explanation of -lieee, -mieee. |