summary refs log tree commit diff
path: root/manual/arith.texi
blob: 7879a77b7c9a9010a5428ff1c5b3229847422722 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
@node Arithmetic, Date and Time, Mathematics, Top
@c %MENU% Low level arithmetic functions
@chapter Arithmetic Functions

This chapter contains information about functions for doing basic
arithmetic operations, such as splitting a float into its integer and
fractional parts or retrieving the imaginary part of a complex value.
These functions are declared in the header files @file{math.h} and
@file{complex.h}.

@menu
* Floating Point Numbers::      Basic concepts.  IEEE 754.
* Floating Point Classes::      The five kinds of floating-point number.
* Floating Point Errors::       When something goes wrong in a calculation.
* Rounding::                    Controlling how results are rounded.
* Control Functions::           Saving and restoring the FPU's state.
* Arithmetic Functions::        Fundamental operations provided by the library.
* Complex Numbers::             The types.  Writing complex constants.
* Operations on Complex::       Projection, conjugation, decomposition.
* Integer Division::            Integer division with guaranteed rounding.
* Parsing of Numbers::          Converting strings to numbers.
* System V Number Conversion::  An archaic way to convert numbers to strings.
@end menu

@node Floating Point Numbers
@section Floating Point Numbers
@cindex floating point
@cindex IEEE 754
@cindex IEEE floating point

Most computer hardware has support for two different kinds of numbers:
integers (@math{@dots{}-3, -2, -1, 0, 1, 2, 3@dots{}}) and
floating-point numbers.  Floating-point numbers have three parts: the
@dfn{mantissa}, the @dfn{exponent}, and the @dfn{sign bit}.  The real
number represented by a floating-point value is given by
@tex
$(s \mathrel? -1 \mathrel: 1) \cdot 2^e \cdot M$
@end tex
@ifnottex
@math{(s ? -1 : 1) @mul{} 2^e @mul{} M}
@end ifnottex
where @math{s} is the sign bit, @math{e} the exponent, and @math{M}
the mantissa.  @xref{Floating Point Concepts}, for details.  (It is
possible to have a different @dfn{base} for the exponent, but all modern
hardware uses @math{2}.)

Floating-point numbers can represent a finite subset of the real
numbers.  While this subset is large enough for most purposes, it is
important to remember that the only reals that can be represented
exactly are rational numbers that have a terminating binary expansion
shorter than the width of the mantissa.  Even simple fractions such as
@math{1/5} can only be approximated by floating point.

Mathematical operations and functions frequently need to produce values
that are not representable.  Often these values can be approximated
closely enough for practical purposes, but sometimes they can't.
Historically there was no way to tell when the results of a calculation
were inaccurate.  Modern computers implement the @w{IEEE 754} standard
for numerical computations, which defines a framework for indicating to
the program when the results of calculation are not trustworthy.  This
framework consists of a set of @dfn{exceptions} that indicate why a
result could not be represented, and the special values @dfn{infinity}
and @dfn{not a number} (NaN).

@node Floating Point Classes
@section Floating-Point Number Classification Functions
@cindex floating-point classes
@cindex classes, floating-point
@pindex math.h

@w{ISO C 9x} defines macros that let you determine what sort of
floating-point number a variable holds.

@comment math.h
@comment ISO
@deftypefn {Macro} int fpclassify (@emph{float-type} @var{x})
This is a generic macro which works on all floating-point types and
which returns a value of type @code{int}.  The possible values are:

@vtable @code
@item FP_NAN
The floating-point number @var{x} is ``Not a Number'' (@pxref{Infinity
and NaN})
@item FP_INFINITE
The value of @var{x} is either plus or minus infinity (@pxref{Infinity
and NaN})
@item FP_ZERO
The value of @var{x} is zero.  In floating-point formats like @w{IEEE
754}, where zero can be signed, this value is also returned if
@var{x} is negative zero.
@item FP_SUBNORMAL
Numbers whose absolute value is too small to be represented in the
normal format are represented in an alternate, @dfn{denormalized} format
(@pxref{Floating Point Concepts}).  This format is less precise but can
represent values closer to zero.  @code{fpclassify} returns this value
for values of @var{x} in this alternate format.
@item FP_NORMAL
This value is returned for all other values of @var{x}.  It indicates
that there is nothing special about the number.
@end vtable

@end deftypefn

@code{fpclassify} is most useful if more than one property of a number
must be tested.  There are more specific macros which only test one
property at a time.  Generally these macros execute faster than
@code{fpclassify}, since there is special hardware support for them.
You should therefore use the specific macros whenever possible.

@comment math.h
@comment ISO
@deftypefn {Macro} int isfinite (@emph{float-type} @var{x})
This macro returns a nonzero value if @var{x} is finite: not plus or
minus infinity, and not NaN.  It is equivalent to

@smallexample
(fpclassify (x) != FP_NAN && fpclassify (x) != FP_INFINITE)
@end smallexample

@code{isfinite} is implemented as a macro which accepts any
floating-point type.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn {Macro} int isnormal (@emph{float-type} @var{x})
This macro returns a nonzero value if @var{x} is finite and normalized.
It is equivalent to

@smallexample
(fpclassify (x) == FP_NORMAL)
@end smallexample
@end deftypefn

@comment math.h
@comment ISO
@deftypefn {Macro} int isnan (@emph{float-type} @var{x})
This macro returns a nonzero value if @var{x} is NaN.  It is equivalent
to

@smallexample
(fpclassify (x) == FP_NAN)
@end smallexample
@end deftypefn

Another set of floating-point classification functions was provided by
BSD.  The GNU C library also supports these functions; however, we
recommend that you use the C9x macros in new code.  Those are standard
and will be available more widely.  Also, since they are macros, you do
not have to worry about the type of their argument.

@comment math.h
@comment BSD
@deftypefun int isinf (double @var{x})
@comment math.h
@comment BSD
@deftypefunx int isinff (float @var{x})
@comment math.h
@comment BSD
@deftypefunx int isinfl (long double @var{x})
This function returns @code{-1} if @var{x} represents negative infinity,
@code{1} if @var{x} represents positive infinity, and @code{0} otherwise.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun int isnan (double @var{x})
@comment math.h
@comment BSD
@deftypefunx int isnanf (float @var{x})
@comment math.h
@comment BSD
@deftypefunx int isnanl (long double @var{x})
This function returns a nonzero value if @var{x} is a ``not a number''
value, and zero otherwise.

@strong{Note:} The @code{isnan} macro defined by @w{ISO C 9x} overrides
the BSD function.  This is normally not a problem, because the two
routines behave identically.  However, if you really need to get the BSD
function for some reason, you can write

@smallexample
(isnan) (x)
@end smallexample
@end deftypefun

@comment math.h
@comment BSD
@deftypefun int finite (double @var{x})
@comment math.h
@comment BSD
@deftypefunx int finitef (float @var{x})
@comment math.h
@comment BSD
@deftypefunx int finitel (long double @var{x})
This function returns a nonzero value if @var{x} is finite or a ``not a
number'' value, and zero otherwise.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double infnan (int @var{error})
This function is provided for compatibility with BSD.  Its argument is
an error code, @code{EDOM} or @code{ERANGE}; @code{infnan} returns the
value that a math function would return if it set @code{errno} to that
value.  @xref{Math Error Reporting}.  @code{-ERANGE} is also acceptable
as an argument, and corresponds to @code{-HUGE_VAL} as a value.

In the BSD library, on certain machines, @code{infnan} raises a fatal
signal in all cases.  The GNU library does not do likewise, because that
does not fit the @w{ISO C} specification.
@end deftypefun

@strong{Portability Note:} The functions listed in this section are BSD
extensions.


@node Floating Point Errors
@section Errors in Floating-Point Calculations

@menu
* FP Exceptions::               IEEE 754 math exceptions and how to detect them.
* Infinity and NaN::            Special values returned by calculations.
* Status bit operations::       Checking for exceptions after the fact.
* Math Error Reporting::        How the math functions report errors.
@end menu

@node FP Exceptions
@subsection FP Exceptions
@cindex exception
@cindex signal
@cindex zero divide
@cindex division by zero
@cindex inexact exception
@cindex invalid exception
@cindex overflow exception
@cindex underflow exception

The @w{IEEE 754} standard defines five @dfn{exceptions} that can occur
during a calculation.  Each corresponds to a particular sort of error,
such as overflow.

When exceptions occur (when exceptions are @dfn{raised}, in the language
of the standard), one of two things can happen.  By default the
exception is simply noted in the floating-point @dfn{status word}, and
the program continues as if nothing had happened.  The operation
produces a default value, which depends on the exception (see the table
below).  Your program can check the status word to find out which
exceptions happened.

Alternatively, you can enable @dfn{traps} for exceptions.  In that case,
when an exception is raised, your program will receive the @code{SIGFPE}
signal.  The default action for this signal is to terminate the
program.  @xref{Signal Handling}, for how you can change the effect of
the signal.

@findex matherr
In the System V math library, the user-defined function @code{matherr}
is called when certain exceptions occur inside math library functions.
However, the Unix98 standard deprecates this interface.  We support it
for historical compatibility, but recommend that you do not use it in
new programs.

@noindent
The exceptions defined in @w{IEEE 754} are:

@table @samp
@item Invalid Operation
This exception is raised if the given operands are invalid for the
operation to be performed.  Examples are
(see @w{IEEE 754}, @w{section 7}):
@enumerate
@item
Addition or subtraction: @math{@infinity{} - @infinity{}}.  (But
@math{@infinity{} + @infinity{} = @infinity{}}).
@item
Multiplication: @math{0 @mul{} @infinity{}}.
@item
Division: @math{0/0} or @math{@infinity{}/@infinity{}}.
@item
Remainder: @math{x} REM @math{y}, where @math{y} is zero or @math{x} is
infinite.
@item
Square root if the operand is less then zero.  More generally, any
mathematical function evaluated outside its domain produces this
exception.
@item
Conversion of a floating-point number to an integer or decimal
string, when the number cannot be represented in the target format (due
to overflow, infinity, or NaN).
@item
Conversion of an unrecognizable input string.
@item
Comparison via predicates involving @math{<} or @math{>}, when one or
other of the operands is NaN.  You can prevent this exception by using
the unordered comparison functions instead; see @ref{FP Comparison Functions}.
@end enumerate

If the exception does not trap, the result of the operation is NaN.

@item Division by Zero
This exception is raised when a finite nonzero number is divided
by zero.  If no trap occurs the result is either @math{+@infinity{}} or
@math{-@infinity{}}, depending on the signs of the operands.

@item Overflow
This exception is raised whenever the result cannot be represented
as a finite value in the precision format of the destination.  If no trap
occurs the result depends on the sign of the intermediate result and the
current rounding mode (@w{IEEE 754}, @w{section 7.3}):
@enumerate
@item
Round to nearest carries all overflows to @math{@infinity{}}
with the sign of the intermediate result.
@item
Round toward @math{0} carries all overflows to the largest representable
finite number with the sign of the intermediate result.
@item
Round toward @math{-@infinity{}} carries positive overflows to the
largest representable finite number and negative overflows to
@math{-@infinity{}}.

@item
Round toward @math{@infinity{}} carries negative overflows to the
most negative representable finite number and positive overflows
to @math{@infinity{}}.
@end enumerate

Whenever the overflow exception is raised, the inexact exception is also
raised.

@item Underflow
The underflow exception is raised when an intermediate result is too
small to be calculated accurately, or if the operation's result rounded
to the destination precision is too small to be normalized.

When no trap is installed for the underflow exception, underflow is
signaled (via the underflow flag) only when both tininess and loss of
accuracy have been detected.  If no trap handler is installed the
operation continues with an imprecise small value, or zero if the
destination precision cannot hold the small exact result.

@item Inexact
This exception is signalled if a rounded result is not exact (such as
when calculating the square root of two) or a result overflows without
an overflow trap.
@end table

@node Infinity and NaN
@subsection Infinity and NaN
@cindex infinity
@cindex not a number
@cindex NaN

@w{IEEE 754} floating point numbers can represent positive or negative
infinity, and @dfn{NaN} (not a number).  These three values arise from
calculations whose result is undefined or cannot be represented
accurately.  You can also deliberately set a floating-point variable to
any of them, which is sometimes useful.  Some examples of calculations
that produce infinity or NaN:

@ifnottex
@smallexample
@math{1/0 = @infinity{}}
@math{log (0) = -@infinity{}}
@math{sqrt (-1) = NaN}
@end smallexample
@end ifnottex
@tex
$${1\over0} = \infty$$
$$\log 0 = -\infty$$
$$\sqrt{-1} = \hbox{NaN}$$
@end tex

When a calculation produces any of these values, an exception also
occurs; see @ref{FP Exceptions}.

The basic operations and math functions all accept infinity and NaN and
produce sensible output.  Infinities propagate through calculations as
one would expect: for example, @math{2 + @infinity{} = @infinity{}},
@math{4/@infinity{} = 0}, atan @math{(@infinity{}) = @pi{}/2}.  NaN, on
the other hand, infects any calculation that involves it.  Unless the
calculation would produce the same result no matter what real value
replaced NaN, the result is NaN.

In comparison operations, positive infinity is larger than all values
except itself and NaN, and negative infinity is smaller than all values
except itself and NaN.  NaN is @dfn{unordered}: it is not equal to,
greater than, or less than anything, @emph{including itself}. @code{x ==
x} is false if the value of @code{x} is NaN.  You can use this to test
whether a value is NaN or not, but the recommended way to test for NaN
is with the @code{isnan} function (@pxref{Floating Point Classes}).  In
addition, @code{<}, @code{>}, @code{<=}, and @code{>=} will raise an
exception when applied to NaNs.

@file{math.h} defines macros that allow you to explicitly set a variable
to infinity or NaN.

@comment math.h
@comment ISO
@deftypevr Macro float INFINITY
An expression representing positive infinity.  It is equal to the value
produced  by mathematical operations like @code{1.0 / 0.0}.
@code{-INFINITY} represents negative infinity.

You can test whether a floating-point value is infinite by comparing it
to this macro.  However, this is not recommended; you should use the
@code{isfinite} macro instead.  @xref{Floating Point Classes}.

This macro was introduced in the @w{ISO C 9X} standard.
@end deftypevr

@comment math.h
@comment GNU
@deftypevr Macro float NAN
An expression representing a value which is ``not a number''.  This
macro is a GNU extension, available only on machines that support the
``not a number'' value---that is to say, on all machines that support
IEEE floating point.

You can use @samp{#ifdef NAN} to test whether the machine supports
NaN.  (Of course, you must arrange for GNU extensions to be visible,
such as by defining @code{_GNU_SOURCE}, and then you must include
@file{math.h}.)
@end deftypevr

@w{IEEE 754} also allows for another unusual value: negative zero.  This
value is produced when you divide a positive number by negative
infinity, or when a negative result is smaller than the limits of
representation.  Negative zero behaves identically to zero in all
calculations, unless you explicitly test the sign bit with
@code{signbit} or @code{copysign}.

@node Status bit operations
@subsection Examining the FPU status word

@w{ISO C 9x} defines functions to query and manipulate the
floating-point status word.  You can use these functions to check for
untrapped exceptions when it's convenient, rather than worrying about
them in the middle of a calculation.

These constants represent the various @w{IEEE 754} exceptions.  Not all
FPUs report all the different exceptions.  Each constant is defined if
and only if the FPU you are compiling for supports that exception, so
you can test for FPU support with @samp{#ifdef}.  They are defined in
@file{fenv.h}.

@vtable @code
@comment fenv.h
@comment ISO
@item FE_INEXACT
 The inexact exception.
@comment fenv.h
@comment ISO
@item FE_DIVBYZERO
 The divide by zero exception.
@comment fenv.h
@comment ISO
@item FE_UNDERFLOW
 The underflow exception.
@comment fenv.h
@comment ISO
@item FE_OVERFLOW
 The overflow exception.
@comment fenv.h
@comment ISO
@item FE_INVALID
 The invalid exception.
@end vtable

The macro @code{FE_ALL_EXCEPT} is the bitwise OR of all exception macros
which are supported by the FP implementation.

These functions allow you to clear exception flags, test for exceptions,
and save and restore the set of exceptions flagged.

@comment fenv.h
@comment ISO
@deftypefun void feclearexcept (int @var{excepts})
This function clears all of the supported exception flags indicated by
@var{excepts}.
@end deftypefun

@comment fenv.h
@comment ISO
@deftypefun int fetestexcept (int @var{excepts})
Test whether the exception flags indicated by the parameter @var{except}
are currently set.  If any of them are, a nonzero value is returned
which specifies which exceptions are set.  Otherwise the result is zero.
@end deftypefun

To understand these functions, imagine that the status word is an
integer variable named @var{status}.  @code{feclearexcept} is then
equivalent to @samp{status &= ~excepts} and @code{fetestexcept} is
equivalent to @samp{(status & excepts)}.  The actual implementation may
be very different, of course.

Exception flags are only cleared when the program explicitly requests it,
by calling @code{feclearexcept}.  If you want to check for exceptions
from a set of calculations, you should clear all the flags first.  Here
is a simple example of the way to use @code{fetestexcept}:

@smallexample
@{
  double f;
  int raised;
  feclearexcept (FE_ALL_EXCEPT);
  f = compute ();
  raised = fetestexcept (FE_OVERFLOW | FE_INVALID);
  if (raised & FE_OVERFLOW) @{ /* ... */ @}
  if (raised & FE_INVALID) @{ /* ... */ @}
  /* ... */
@}
@end smallexample

You cannot explicitly set bits in the status word.  You can, however,
save the entire status word and restore it later.  This is done with the
following functions:

@comment fenv.h
@comment ISO
@deftypefun void fegetexceptflag (fexcept_t *@var{flagp}, int @var{excepts})
This function stores in the variable pointed to by @var{flagp} an
implementation-defined value representing the current setting of the
exception flags indicated by @var{excepts}.
@end deftypefun

@comment fenv.h
@comment ISO
@deftypefun void fesetexceptflag (const fexcept_t *@var{flagp}, int
@var{excepts})
This function restores the flags for the exceptions indicated by
@var{excepts} to the values stored in the variable pointed to by
@var{flagp}.
@end deftypefun

Note that the value stored in @code{fexcept_t} bears no resemblance to
the bit mask returned by @code{fetestexcept}.  The type may not even be
an integer.  Do not attempt to modify an @code{fexcept_t} variable.

@node Math Error Reporting
@subsection Error Reporting by Mathematical Functions
@cindex errors, mathematical
@cindex domain error
@cindex range error

Many of the math functions are defined only over a subset of the real or
complex numbers.  Even if they are mathematically defined, their result
may be larger or smaller than the range representable by their return
type.  These are known as @dfn{domain errors}, @dfn{overflows}, and
@dfn{underflows}, respectively.  Math functions do several things when
one of these errors occurs.  In this manual we will refer to the
complete response as @dfn{signalling} a domain error, overflow, or
underflow.

When a math function suffers a domain error, it raises the invalid
exception and returns NaN.  It also sets @var{errno} to @code{EDOM};
this is for compatibility with old systems that do not support @w{IEEE
754} exception handling.  Likewise, when overflow occurs, math
functions raise the overflow exception and return @math{@infinity{}} or
@math{-@infinity{}} as appropriate.  They also set @var{errno} to
@code{ERANGE}.  When underflow occurs, the underflow exception is
raised, and zero (appropriately signed) is returned.  @var{errno} may be
set to @code{ERANGE}, but this is not guaranteed.

Some of the math functions are defined mathematically to result in a
complex value over parts of their domains.  The most familiar example of
this is taking the square root of a negative number.  The complex math
functions, such as @code{csqrt}, will return the appropriate complex value
in this case.  The real-valued functions, such as @code{sqrt}, will
signal a domain error.

Some older hardware does not support infinities.  On that hardware,
overflows instead return a particular very large number (usually the
largest representable number).  @file{math.h} defines macros you can use
to test for overflow on both old and new hardware.

@comment math.h
@comment ISO
@deftypevr Macro double HUGE_VAL
@comment math.h
@comment ISO
@deftypevrx Macro float HUGE_VALF
@comment math.h
@comment ISO
@deftypevrx Macro {long double} HUGE_VALL
An expression representing a particular very large number.  On machines
that use @w{IEEE 754} floating point format, @code{HUGE_VAL} is infinity.
On other machines, it's typically the largest positive number that can
be represented.

Mathematical functions return the appropriately typed version of
@code{HUGE_VAL} or @code{@minus{}HUGE_VAL} when the result is too large
to be represented.
@end deftypevr

@node Rounding
@section Rounding Modes

Floating-point calculations are carried out internally with extra
precision, and then rounded to fit into the destination type.  This
ensures that results are as precise as the input data.  @w{IEEE 754}
defines four possible rounding modes:

@table @asis
@item Round to nearest.
This is the default mode.  It should be used unless there is a specific
need for one of the others.  In this mode results are rounded to the
nearest representable value.  If the result is midway between two
representable values, the even representable is chosen. @dfn{Even} here
means the lowest-order bit is zero.  This rounding mode prevents
statistical bias and guarantees numeric stability: round-off errors in a
lengthy calculation will remain smaller than half of @code{FLT_EPSILON}.

@c @item Round toward @math{+@infinity{}}
@item Round toward plus Infinity.
All results are rounded to the smallest representable value
which is greater than the result.

@c @item Round toward @math{-@infinity{}}
@item Round toward minus Infinity.
All results are rounded to the largest representable value which is less
than the result.

@item Round toward zero.
All results are rounded to the largest representable value whose
magnitude is less than that of the result.  In other words, if the
result is negative it is rounded up; if it is positive, it is rounded
down.
@end table

@noindent
@file{fenv.h} defines constants which you can use to refer to the
various rounding modes.  Each one will be defined if and only if the FPU
supports the corresponding rounding mode.

@table @code
@comment fenv.h
@comment ISO
@vindex FE_TONEAREST
@item FE_TONEAREST
Round to nearest.

@comment fenv.h
@comment ISO
@vindex FE_UPWARD
@item FE_UPWARD
Round toward @math{+@infinity{}}.

@comment fenv.h
@comment ISO
@vindex FE_DOWNWARD
@item FE_DOWNWARD
Round toward @math{-@infinity{}}.

@comment fenv.h
@comment ISO
@vindex FE_TOWARDZERO
@item FE_TOWARDZERO
Round toward zero.
@end table

Underflow is an unusual case.  Normally, @w{IEEE 754} floating point
numbers are always normalized (@pxref{Floating Point Concepts}).
Numbers smaller than @math{2^r} (where @math{r} is the minimum exponent,
@code{FLT_MIN_RADIX-1} for @var{float}) cannot be represented as
normalized numbers.  Rounding all such numbers to zero or @math{2^r}
would cause some algorithms to fail at 0.  Therefore, they are left in
denormalized form.  That produces loss of precision, since some bits of
the mantissa are stolen to indicate the decimal point.

If a result is too small to be represented as a denormalized number, it
is rounded to zero.  However, the sign of the result is preserved; if
the calculation was negative, the result is @dfn{negative zero}.
Negative zero can also result from some operations on infinity, such as
@math{4/-@infinity{}}.  Negative zero behaves identically to zero except
when the @code{copysign} or @code{signbit} functions are used to check
the sign bit directly.

At any time one of the above four rounding modes is selected.  You can
find out which one with this function:

@comment fenv.h
@comment ISO
@deftypefun int fegetround (void)
Returns the currently selected rounding mode, represented by one of the
values of the defined rounding mode macros.
@end deftypefun

@noindent
To change the rounding mode, use this function:

@comment fenv.h
@comment ISO
@deftypefun int fesetround (int @var{round})
Changes the currently selected rounding mode to @var{round}.  If
@var{round} does not correspond to one of the supported rounding modes
nothing is changed.  @code{fesetround} returns a nonzero value if it
changed the rounding mode, zero if the mode is not supported.
@end deftypefun

You should avoid changing the rounding mode if possible.  It can be an
expensive operation; also, some hardware requires you to compile your
program differently for it to work.  The resulting code may run slower.
See your compiler documentation for details.
@c This section used to claim that functions existed to round one number
@c in a specific fashion.  I can't find any functions in the library
@c that do that. -zw

@node Control Functions
@section Floating-Point Control Functions

@w{IEEE 754} floating-point implementations allow the programmer to
decide whether traps will occur for each of the exceptions, by setting
bits in the @dfn{control word}.  In C, traps result in the program
receiving the @code{SIGFPE} signal; see @ref{Signal Handling}.

@strong{Note:} @w{IEEE 754} says that trap handlers are given details of
the exceptional situation, and can set the result value.  C signals do
not provide any mechanism to pass this information back and forth.
Trapping exceptions in C is therefore not very useful.

It is sometimes necessary to save the state of the floating-point unit
while you perform some calculation.  The library provides functions
which save and restore the exception flags, the set of exceptions that
generate traps, and the rounding mode.  This information is known as the
@dfn{floating-point environment}.

The functions to save and restore the floating-point environment all use
a variable of type @code{fenv_t} to store information.  This type is
defined in @file{fenv.h}.  Its size and contents are
implementation-defined.  You should not attempt to manipulate a variable
of this type directly.

To save the state of the FPU, use one of these functions:

@comment fenv.h
@comment ISO
@deftypefun void fegetenv (fenv_t *@var{envp})
Store the floating-point environment in the variable pointed to by
@var{envp}.
@end deftypefun

@comment fenv.h
@comment ISO
@deftypefun int feholdexcept (fenv_t *@var{envp})
Store the current floating-point environment in the object pointed to by
@var{envp}.  Then clear all exception flags, and set the FPU to trap no
exceptions.  Not all FPUs support trapping no exceptions; if
@code{feholdexcept} cannot set this mode, it returns zero.  If it
succeeds, it returns a nonzero value.
@end deftypefun

The functions which restore the floating-point environment can take two
kinds of arguments:

@itemize @bullet
@item
Pointers to @code{fenv_t} objects, which were initialized previously by a
call to @code{fegetenv} or @code{feholdexcept}.
@item
@vindex FE_DFL_ENV
The special macro @code{FE_DFL_ENV} which represents the floating-point
environment as it was available at program start.
@item
Implementation defined macros with names starting with @code{FE_}.

@vindex FE_NOMASK_ENV
If possible, the GNU C Library defines a macro @code{FE_NOMASK_ENV}
which represents an environment where every exception raised causes a
trap to occur.  You can test for this macro using @code{#ifdef}.  It is
only defined if @code{_GNU_SOURCE} is defined.

Some platforms might define other predefined environments.
@end itemize

@noindent
To set the floating-point environment, you can use either of these
functions:

@comment fenv.h
@comment ISO
@deftypefun void fesetenv (const fenv_t *@var{envp})
Set the floating-point environment to that described by @var{envp}.
@end deftypefun

@comment fenv.h
@comment ISO
@deftypefun void feupdateenv (const fenv_t *@var{envp})
Like @code{fesetenv}, this function sets the floating-point environment
to that described by @var{envp}.  However, if any exceptions were
flagged in the status word before @code{feupdateenv} was called, they
remain flagged after the call.  In other words, after @code{feupdateenv}
is called, the status word is the bitwise OR of the previous status word
and the one saved in @var{envp}.
@end deftypefun

@node Arithmetic Functions
@section Arithmetic Functions

The C library provides functions to do basic operations on
floating-point numbers.  These include absolute value, maximum and minimum,
normalization, bit twiddling, rounding, and a few others.

@menu
* Absolute Value::              Absolute values of integers and floats.
* Normalization Functions::     Extracting exponents and putting them back.
* Rounding Functions::          Rounding floats to integers.
* Remainder Functions::         Remainders on division, precisely defined.
* FP Bit Twiddling::            Sign bit adjustment.  Adding epsilon.
* FP Comparison Functions::     Comparisons without risk of exceptions.
* Misc FP Arithmetic::          Max, min, positive difference, multiply-add.
@end menu

@node Absolute Value
@subsection Absolute Value
@cindex absolute value functions

These functions are provided for obtaining the @dfn{absolute value} (or
@dfn{magnitude}) of a number.  The absolute value of a real number
@var{x} is @var{x} if @var{x} is positive, @minus{}@var{x} if @var{x} is
negative.  For a complex number @var{z}, whose real part is @var{x} and
whose imaginary part is @var{y}, the absolute value is @w{@code{sqrt
(@var{x}*@var{x} + @var{y}*@var{y})}}.

@pindex math.h
@pindex stdlib.h
Prototypes for @code{abs}, @code{labs} and @code{llabs} are in @file{stdlib.h};
@code{imaxabs} is declared in @file{inttypes.h};
@code{fabs}, @code{fabsf} and @code{fabsl} are declared in @file{math.h}.
@code{cabs}, @code{cabsf} and @code{cabsl} are declared in @file{complex.h}.

@comment stdlib.h
@comment ISO
@deftypefun int abs (int @var{number})
@comment stdlib.h
@comment ISO
@deftypefunx {long int} labs (long int @var{number})
@comment stdlib.h
@comment ISO
@deftypefunx {long long int} llabs (long long int @var{number})
@comment inttypes.h
@comment ISO
@deftypefunx intmax_t imaxabs (intmax_t @var{number})
These functions return the absolute value of @var{number}.

Most computers use a two's complement integer representation, in which
the absolute value of @code{INT_MIN} (the smallest possible @code{int})
cannot be represented; thus, @w{@code{abs (INT_MIN)}} is not defined.

@code{llabs} and @code{imaxdiv} are new to @w{ISO C 9x}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fabs (double @var{number})
@comment math.h
@comment ISO
@deftypefunx float fabsf (float @var{number})
@comment math.h
@comment ISO
@deftypefunx {long double} fabsl (long double @var{number})
This function returns the absolute value of the floating-point number
@var{number}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double cabs (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx float cabsf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {long double} cabsl (complex long double @var{z})
These functions return the absolute  value of the complex number @var{z}
(@pxref{Complex Numbers}).  The absolute value of a complex number is:

@smallexample
sqrt (creal (@var{z}) * creal (@var{z}) + cimag (@var{z}) * cimag (@var{z}))
@end smallexample

This function should always be used instead of the direct formula
because it takes special care to avoid losing precision.  It may also
take advantage of hardware support for this operation. See @code{hypot}
in @ref{Exponents and Logarithms}.
@end deftypefun

@node Normalization Functions
@subsection Normalization Functions
@cindex normalization functions (floating-point)

The functions described in this section are primarily provided as a way
to efficiently perform certain low-level manipulations on floating point
numbers that are represented internally using a binary radix;
see @ref{Floating Point Concepts}.  These functions are required to
have equivalent behavior even if the representation does not use a radix
of 2, but of course they are unlikely to be particularly efficient in
those cases.

@pindex math.h
All these functions are declared in @file{math.h}.

@comment math.h
@comment ISO
@deftypefun double frexp (double @var{value}, int *@var{exponent})
@comment math.h
@comment ISO
@deftypefunx float frexpf (float @var{value}, int *@var{exponent})
@comment math.h
@comment ISO
@deftypefunx {long double} frexpl (long double @var{value}, int *@var{exponent})
These functions are used to split the number @var{value}
into a normalized fraction and an exponent.

If the argument @var{value} is not zero, the return value is @var{value}
times a power of two, and is always in the range 1/2 (inclusive) to 1
(exclusive).  The corresponding exponent is stored in
@code{*@var{exponent}}; the return value multiplied by 2 raised to this
exponent equals the original number @var{value}.

For example, @code{frexp (12.8, &exponent)} returns @code{0.8} and
stores @code{4} in @code{exponent}.

If @var{value} is zero, then the return value is zero and
zero is stored in @code{*@var{exponent}}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double ldexp (double @var{value}, int @var{exponent})
@comment math.h
@comment ISO
@deftypefunx float ldexpf (float @var{value}, int @var{exponent})
@comment math.h
@comment ISO
@deftypefunx {long double} ldexpl (long double @var{value}, int @var{exponent})
These functions return the result of multiplying the floating-point
number @var{value} by 2 raised to the power @var{exponent}.  (It can
be used to reassemble floating-point numbers that were taken apart
by @code{frexp}.)

For example, @code{ldexp (0.8, 4)} returns @code{12.8}.
@end deftypefun

The following functions, which come from BSD, provide facilities
equivalent to those of @code{ldexp} and @code{frexp}.

@comment math.h
@comment BSD
@deftypefun double logb (double @var{x})
@comment math.h
@comment BSD
@deftypefunx float logbf (float @var{x})
@comment math.h
@comment BSD
@deftypefunx {long double} logbl (long double @var{x})
These functions return the integer part of the base-2 logarithm of
@var{x}, an integer value represented in type @code{double}.  This is
the highest integer power of @code{2} contained in @var{x}.  The sign of
@var{x} is ignored.  For example, @code{logb (3.5)} is @code{1.0} and
@code{logb (4.0)} is @code{2.0}.

When @code{2} raised to this power is divided into @var{x}, it gives a
quotient between @code{1} (inclusive) and @code{2} (exclusive).

If @var{x} is zero, the return value is minus infinity if the machine
supports infinities, and a very small number if it does not.  If @var{x}
is infinity, the return value is infinity.

For finite @var{x}, the value returned by @code{logb} is one less than
the value that @code{frexp} would store into @code{*@var{exponent}}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double scalb (double @var{value}, int @var{exponent})
@comment math.h
@comment BSD
@deftypefunx float scalbf (float @var{value}, int @var{exponent})
@comment math.h
@comment BSD
@deftypefunx {long double} scalbl (long double @var{value}, int @var{exponent})
The @code{scalb} function is the BSD name for @code{ldexp}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun {long long int} scalbn (double @var{x}, int n)
@comment math.h
@comment BSD
@deftypefunx {long long int} scalbnf (float @var{x}, int n)
@comment math.h
@comment BSD
@deftypefunx {long long int} scalbnl (long double @var{x}, int n)
@code{scalbn} is identical to @code{scalb}, except that the exponent
@var{n} is an @code{int} instead of a floating-point number.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun {long long int} scalbln (double @var{x}, long int n)
@comment math.h
@comment BSD
@deftypefunx {long long int} scalblnf (float @var{x}, long int n)
@comment math.h
@comment BSD
@deftypefunx {long long int} scalblnl (long double @var{x}, long int n)
@code{scalbln} is identical to @code{scalb}, except that the exponent
@var{n} is a @code{long int} instead of a floating-point number.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun {long long int} significand (double @var{x})
@comment math.h
@comment BSD
@deftypefunx {long long int} significandf (float @var{x})
@comment math.h
@comment BSD
@deftypefunx {long long int} significandl (long double @var{x})
@code{significand} returns the mantissa of @var{x} scaled to the range
@math{[1, 2)}.
It is equivalent to @w{@code{scalb (@var{x}, (double) -ilogb (@var{x}))}}.

This function exists mainly for use in certain standardized tests
of @w{IEEE 754} conformance.
@end deftypefun

@node Rounding Functions
@subsection Rounding Functions
@cindex converting floats to integers

@pindex math.h
The functions listed here perform operations such as rounding and
truncation of floating-point values. Some of these functions convert
floating point numbers to integer values.  They are all declared in
@file{math.h}.

You can also convert floating-point numbers to integers simply by
casting them to @code{int}.  This discards the fractional part,
effectively rounding towards zero.  However, this only works if the
result can actually be represented as an @code{int}---for very large
numbers, this is impossible.  The functions listed here return the
result as a @code{double} instead to get around this problem.

@comment math.h
@comment ISO
@deftypefun double ceil (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float ceilf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} ceill (long double @var{x})
These functions round @var{x} upwards to the nearest integer,
returning that value as a @code{double}.  Thus, @code{ceil (1.5)}
is @code{2.0}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double floor (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float floorf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} floorl (long double @var{x})
These functions round @var{x} downwards to the nearest
integer, returning that value as a @code{double}.  Thus, @code{floor
(1.5)} is @code{1.0} and @code{floor (-1.5)} is @code{-2.0}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double trunc (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float truncf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} truncl (long double @var{x})
@code{trunc} is another name for @code{floor}
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double rint (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float rintf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} rintl (long double @var{x})
These functions round @var{x} to an integer value according to the
current rounding mode.  @xref{Floating Point Parameters}, for
information about the various rounding modes.  The default
rounding mode is to round to the nearest integer; some machines
support other modes, but round-to-nearest is always used unless
you explicitly select another.

If @var{x} was not initially an integer, these functions raise the
inexact exception.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double nearbyint (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float nearbyintf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} nearbyintl (long double @var{x})
These functions return the same value as the @code{rint} functions, but
do not raise the inexact exception if @var{x} is not an integer.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double round (double @var{x})
@comment math.h
@comment ISO
@deftypefunx float roundf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long double} roundl (long double @var{x})
These functions are similar to @code{rint}, but they round halfway
cases away from zero instead of to the nearest even integer.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun {long int} lrint (double @var{x})
@comment math.h
@comment ISO
@deftypefunx {long int} lrintf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long int} lrintl (long double @var{x})
These functions are just like @code{rint}, but they return a
@code{long int} instead of a floating-point number.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun {long long int} llrint (double @var{x})
@comment math.h
@comment ISO
@deftypefunx {long long int} llrintf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long long int} llrintl (long double @var{x})
These functions are just like @code{rint}, but they return a
@code{long long int} instead of a floating-point number.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun {long int} lround (double @var{x})
@comment math.h
@comment ISO
@deftypefunx {long int} lroundf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long int} lroundl (long double @var{x})
These functions are just like @code{round}, but they return a
@code{long int} instead of a floating-point number.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun {long long int} llround (double @var{x})
@comment math.h
@comment ISO
@deftypefunx {long long int} llroundf (float @var{x})
@comment math.h
@comment ISO
@deftypefunx {long long int} llroundl (long double @var{x})
These functions are just like @code{round}, but they return a
@code{long long int} instead of a floating-point number.
@end deftypefun


@comment math.h
@comment ISO
@deftypefun double modf (double @var{value}, double *@var{integer-part})
@comment math.h
@comment ISO
@deftypefunx float modff (float @var{value}, float *@var{integer-part})
@comment math.h
@comment ISO
@deftypefunx {long double} modfl (long double @var{value}, long double *@var{integer-part})
These functions break the argument @var{value} into an integer part and a
fractional part (between @code{-1} and @code{1}, exclusive).  Their sum
equals @var{value}.  Each of the parts has the same sign as @var{value},
and the integer part is always rounded toward zero.

@code{modf} stores the integer part in @code{*@var{integer-part}}, and
returns the fractional part.  For example, @code{modf (2.5, &intpart)}
returns @code{0.5} and stores @code{2.0} into @code{intpart}.
@end deftypefun

@node Remainder Functions
@subsection Remainder Functions

The functions in this section compute the remainder on division of two
floating-point numbers.  Each is a little different; pick the one that
suits your problem.

@comment math.h
@comment ISO
@deftypefun double fmod (double @var{numerator}, double @var{denominator})
@comment math.h
@comment ISO
@deftypefunx float fmodf (float @var{numerator}, float @var{denominator})
@comment math.h
@comment ISO
@deftypefunx {long double} fmodl (long double @var{numerator}, long double @var{denominator})
These functions compute the remainder from the division of
@var{numerator} by @var{denominator}.  Specifically, the return value is
@code{@var{numerator} - @w{@var{n} * @var{denominator}}}, where @var{n}
is the quotient of @var{numerator} divided by @var{denominator}, rounded
towards zero to an integer.  Thus, @w{@code{fmod (6.5, 2.3)}} returns
@code{1.9}, which is @code{6.5} minus @code{4.6}.

The result has the same sign as the @var{numerator} and has magnitude
less than the magnitude of the @var{denominator}.

If @var{denominator} is zero, @code{fmod} signals a domain error.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double drem (double @var{numerator}, double @var{denominator})
@comment math.h
@comment BSD
@deftypefunx float dremf (float @var{numerator}, float @var{denominator})
@comment math.h
@comment BSD
@deftypefunx {long double} dreml (long double @var{numerator}, long double @var{denominator})
These functions are like @code{fmod} except that they rounds the
internal quotient @var{n} to the nearest integer instead of towards zero
to an integer.  For example, @code{drem (6.5, 2.3)} returns @code{-0.4},
which is @code{6.5} minus @code{6.9}.

The absolute value of the result is less than or equal to half the
absolute value of the @var{denominator}.  The difference between
@code{fmod (@var{numerator}, @var{denominator})} and @code{drem
(@var{numerator}, @var{denominator})} is always either
@var{denominator}, minus @var{denominator}, or zero.

If @var{denominator} is zero, @code{drem} signals a domain error.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double remainder (double @var{numerator}, double @var{denominator})
@comment math.h
@comment BSD
@deftypefunx float remainderf (float @var{numerator}, float @var{denominator})
@comment math.h
@comment BSD
@deftypefunx {long double} remainderl (long double @var{numerator}, long double @var{denominator})
This function is another name for @code{drem}.
@end deftypefun

@node FP Bit Twiddling
@subsection Setting and modifying single bits of FP values
@cindex FP arithmetic

There are some operations that are too complicated or expensive to
perform by hand on floating-point numbers.  @w{ISO C 9x} defines
functions to do these operations, which mostly involve changing single
bits.

@comment math.h
@comment ISO
@deftypefun double copysign (double @var{x}, double @var{y})
@comment math.h
@comment ISO
@deftypefunx float copysignf (float @var{x}, float @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} copysignl (long double @var{x}, long double @var{y})
These functions return @var{x} but with the sign of @var{y}.  They work
even if @var{x} or @var{y} are NaN or zero.  Both of these can carry a
sign (although not all implementations support it) and this is one of
the few operations that can tell the difference.

@code{copysign} never raises an exception.
@c except signalling NaNs

This function is defined in @w{IEC 559} (and the appendix with
recommended functions in @w{IEEE 754}/@w{IEEE 854}).
@end deftypefun

@comment math.h
@comment ISO
@deftypefun int signbit (@emph{float-type} @var{x})
@code{signbit} is a generic macro which can work on all floating-point
types.  It returns a nonzero value if the value of @var{x} has its sign
bit set.

This is not the same as @code{x < 0.0}, because @w{IEEE 754} floating
point allows zero to be signed.  The comparison @code{-0.0 < 0.0} is
false, but @code{signbit (-0.0)} will return a nonzero value.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double nextafter (double @var{x}, double @var{y})
@comment math.h
@comment ISO
@deftypefunx float nextafterf (float @var{x}, float @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} nextafterl (long double @var{x}, long double @var{y})
The @code{nextafter} function returns the next representable neighbor of
@var{x} in the direction towards @var{y}.  The size of the step between
@var{x} and the result depends on the type of the result.  If
@math{@var{x} = @var{y}} the function simply returns @var{x}.  If either
value is @code{NaN}, @code{NaN} is returned.  Otherwise
a value corresponding to the value of the least significant bit in the
mantissa is added or subtracted, depending on the direction.
@code{nextafter} will signal overflow or underflow if the result goes
outside of the range of normalized numbers.

This function is defined in @w{IEC 559} (and the appendix with
recommended functions in @w{IEEE 754}/@w{IEEE 854}).
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double nexttoward (double @var{x}, long double @var{y})
@comment math.h
@comment ISO
@deftypefunx float nexttowardf (float @var{x}, long double @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} nexttowardl (long double @var{x}, long double @var{y})
These functions are identical to the corresponding versions of
@code{nextafter} except that their second argument is a @code{long
double}.
@end deftypefun

@cindex NaN
@comment math.h
@comment ISO
@deftypefun double nan (const char *@var{tagp})
@comment math.h
@comment ISO
@deftypefunx float nanf (const char *@var{tagp})
@comment math.h
@comment ISO
@deftypefunx {long double} nanl (const char *@var{tagp})
The @code{nan} function returns a representation of NaN, provided that
NaN is supported by the target platform.
@code{nan ("@var{n-char-sequence}")} is equivalent to
@code{strtod ("NAN(@var{n-char-sequence})")}.

The argument @var{tagp} is used in an unspecified manner.  On @w{IEEE
754} systems, there are many representations of NaN, and @var{tagp}
selects one.  On other systems it may do nothing.
@end deftypefun

@node FP Comparison Functions
@subsection Floating-Point Comparison Functions
@cindex unordered comparison

The standard C comparison operators provoke exceptions when one or other
of the operands is NaN.  For example,

@smallexample
int v = a < 1.0;
@end smallexample

@noindent
will raise an exception if @var{a} is NaN.  (This does @emph{not}
happen with @code{==} and @code{!=}; those merely return false and true,
respectively, when NaN is examined.)  Frequently this exception is
undesirable.  @w{ISO C 9x} therefore defines comparison functions that
do not raise exceptions when NaN is examined.  All of the functions are
implemented as macros which allow their arguments to be of any
floating-point type.  The macros are guaranteed to evaluate their
arguments only once.

@comment math.h
@comment ISO
@deftypefn Macro int isgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether the argument @var{x} is greater than
@var{y}.  It is equivalent to @code{(@var{x}) > (@var{y})}, but no
exception is raised if @var{x} or @var{y} are NaN.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn Macro int isgreaterequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether the argument @var{x} is greater than or
equal to @var{y}.  It is equivalent to @code{(@var{x}) >= (@var{y})}, but no
exception is raised if @var{x} or @var{y} are NaN.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn Macro int isless (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether the argument @var{x} is less than @var{y}.
It is equivalent to @code{(@var{x}) < (@var{y})}, but no exception is
raised if @var{x} or @var{y} are NaN.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn Macro int islessequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether the argument @var{x} is less than or equal
to @var{y}.  It is equivalent to @code{(@var{x}) <= (@var{y})}, but no
exception is raised if @var{x} or @var{y} are NaN.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn Macro int islessgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether the argument @var{x} is less or greater
than @var{y}.  It is equivalent to @code{(@var{x}) < (@var{y}) ||
(@var{x}) > (@var{y})} (although it only evaluates @var{x} and @var{y}
once), but no exception is raised if @var{x} or @var{y} are NaN.

This macro is not equivalent to @code{@var{x} != @var{y}}, because that
expression is true if @var{x} or @var{y} are NaN.
@end deftypefn

@comment math.h
@comment ISO
@deftypefn Macro int isunordered (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
This macro determines whether its arguments are unordered.  In other
words, it is true if @var{x} or @var{y} are NaN, and false otherwise.
@end deftypefn

Not all machines provide hardware support for these operations.  On
machines that don't, the macros can be very slow.  Therefore, you should
not use these functions when NaN is not a concern.

@strong{Note:} There are no macros @code{isequal} or @code{isunequal}.
They are unnecessary, because the @code{==} and @code{!=} operators do
@emph{not} throw an exception if one or both of the operands are NaN.

@node Misc FP Arithmetic
@subsection Miscellaneous FP arithmetic functions
@cindex minimum
@cindex maximum
@cindex positive difference
@cindex multiply-add

The functions in this section perform miscellaneous but common
operations that are awkward to express with C operators.  On some
processors these functions can use special machine instructions to
perform these operations faster than the equivalent C code.

@comment math.h
@comment ISO
@deftypefun double fmin (double @var{x}, double @var{y})
@comment math.h
@comment ISO
@deftypefunx float fminf (float @var{x}, float @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} fminl (long double @var{x}, long double @var{y})
The @code{fmin} function returns the lesser of the two values @var{x}
and @var{y}.  It is similar to the expression
@smallexample
((x) < (y) ? (x) : (y))
@end smallexample
except that @var{x} and @var{y} are only evaluated once.

If an argument is NaN, the other argument is returned.  If both arguments
are NaN, NaN is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fmax (double @var{x}, double @var{y})
@comment math.h
@comment ISO
@deftypefunx float fmaxf (float @var{x}, float @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} fmaxl (long double @var{x}, long double @var{y})
The @code{fmax} function returns the greater of the two values @var{x}
and @var{y}.

If an argument is NaN, the other argument is returned.  If both arguments
are NaN, NaN is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fdim (double @var{x}, double @var{y})
@comment math.h
@comment ISO
@deftypefunx float fdimf (float @var{x}, float @var{y})
@comment math.h
@comment ISO
@deftypefunx {long double} fdiml (long double @var{x}, long double @var{y})
The @code{fdim} function returns the positive difference between
@var{x} and @var{y}.  The positive difference is @math{@var{x} -
@var{y}} if @var{x} is greater than @var{y}, and @math{0} otherwise.

If @var{x}, @var{y}, or both are NaN, NaN is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fma (double @var{x}, double @var{y}, double @var{z})
@comment math.h
@comment ISO
@deftypefunx float fmaf (float @var{x}, float @var{y}, float @var{z})
@comment math.h
@comment ISO
@deftypefunx {long double} fmal (long double @var{x}, long double @var{y}, long double @var{z})
@cindex butterfly
The @code{fma} function performs floating-point multiply-add.  This is
the operation @math{(@var{x} @mul{} @var{y}) + @var{z}}, but the
intermediate result is not rounded to the destination type.  This can
sometimes improve the precision of a calculation.

This function was introduced because some processors have a special
instruction to perform multiply-add.  The C compiler cannot use it
directly, because the expression @samp{x*y + z} is defined to round the
intermediate result.  @code{fma} lets you choose when you want to round
only once.

@vindex FP_FAST_FMA
On processors which do not implement multiply-add in hardware,
@code{fma} can be very slow since it must avoid intermediate rounding.
@file{math.h} defines the symbols @code{FP_FAST_FMA},
@code{FP_FAST_FMAF}, and @code{FP_FAST_FMAL} when the corresponding
version of @code{fma} is no slower than the expression @samp{x*y + z}.
In the GNU C library, this always means the operation is implemented in
hardware.
@end deftypefun

@node Complex Numbers
@section Complex Numbers
@pindex complex.h
@cindex complex numbers

@w{ISO C 9x} introduces support for complex numbers in C.  This is done
with a new type qualifier, @code{complex}.  It is a keyword if and only
if @file{complex.h} has been included.  There are three complex types,
corresponding to the three real types:  @code{float complex},
@code{double complex}, and @code{long double complex}.

To construct complex numbers you need a way to indicate the imaginary
part of a number.  There is no standard notation for an imaginary
floating point constant.  Instead, @file{complex.h} defines two macros
that can be used to create complex numbers.

@deftypevr Macro {const float complex} _Complex_I
This macro is a representation of the complex number ``@math{0+1i}''.
Multiplying a real floating-point value by @code{_Complex_I} gives a
complex number whose value is purely imaginary.  You can use this to
construct complex constants:

@smallexample
@math{3.0 + 4.0i} = @code{3.0 + 4.0 * _Complex_I}
@end smallexample

Note that @code{_Complex_I * _Complex_I} has the value @code{-1}, but
the type of that value is @code{complex}.
@end deftypevr

@c Put this back in when gcc supports _Imaginary_I.  It's too confusing.
@ignore
@noindent
Without an optimizing compiler this is more expensive than the use of
@code{_Imaginary_I} but with is better than nothing.  You can avoid all
the hassles if you use the @code{I} macro below if the name is not
problem.

@deftypevr Macro {const float imaginary} _Imaginary_I
This macro is a representation of the value ``@math{1i}''.  I.e., it is
the value for which

@smallexample
_Imaginary_I * _Imaginary_I = -1
@end smallexample

@noindent
The result is not of type @code{float imaginary} but instead @code{float}.
One can use it to easily construct complex number like in

@smallexample
3.0 - _Imaginary_I * 4.0
@end smallexample

@noindent
which results in the complex number with a real part of 3.0 and a
imaginary part -4.0.
@end deftypevr
@end ignore

@noindent
@code{_Complex_I} is a bit of a mouthful.  @file{complex.h} also defines
a shorter name for the same constant.

@deftypevr Macro {const float complex} I
This macro has exactly the same value as @code{_Complex_I}.  Most of the
time it is preferable.  However, it causes problems if you want to use
the identifier @code{I} for something else.  You can safely write

@smallexample
#include <complex.h>
#undef I
@end smallexample

@noindent
if you need @code{I} for your own purposes.  (In that case we recommend
you also define some other short name for @code{_Complex_I}, such as
@code{J}.)

@ignore
If the implementation does not support the @code{imaginary} types
@code{I} is defined as @code{_Complex_I} which is the second best
solution.  It still can be used in the same way but requires a most
clever compiler to get the same results.
@end ignore
@end deftypevr

@node Operations on Complex
@section Projections, Conjugates, and Decomposing of Complex Numbers
@cindex project complex numbers
@cindex conjugate complex numbers
@cindex decompose complex numbers
@pindex complex.h

@w{ISO C 9x} also defines functions that perform basic operations on
complex numbers, such as decomposition and conjugation.  The prototypes
for all these functions are in @file{complex.h}.  All functions are
available in three variants, one for each of the three complex types.

@comment complex.h
@comment ISO
@deftypefun double creal (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx float crealf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {long double} creall (complex long double @var{z})
These functions return the real part of the complex number @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double cimag (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx float cimagf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {long double} cimagl (complex long double @var{z})
These functions return the imaginary part of the complex number @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} conj (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx {complex float} conjf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {complex long double} conjl (complex long double @var{z})
These functions return the conjugate value of the complex number
@var{z}.  The conjugate of a complex number has the same real part and a
negated imaginary part.  In other words, @samp{conj(a + bi) = a + -bi}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double carg (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx float cargf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {long double} cargl (complex long double @var{z})
These functions return the argument of the complex number @var{z}.
The argument of a complex number is the angle in the complex plane
between the positive real axis and a line passing through zero and the
number.  This angle is measured in the usual fashion and ranges from @math{0}
to @math{2@pi{}}.

@code{carg} has a branch cut along the positive real axis.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cproj (complex double @var{z})
@comment complex.h
@comment ISO
@deftypefunx {complex float} cprojf (complex float @var{z})
@comment complex.h
@comment ISO
@deftypefunx {complex long double} cprojl (complex long double @var{z})
These functions return the projection of the complex value @var{z} onto
the Riemann sphere.  Values with a infinite imaginary part are projected
to positive infinity on the real axis, even if the real part is NaN.  If
the real part is infinite, the result is equivalent to

@smallexample
INFINITY + I * copysign (0.0, cimag (z))
@end smallexample
@end deftypefun

@node Integer Division
@section Integer Division
@cindex integer division functions

This section describes functions for performing integer division.  These
functions are redundant when GNU CC is used, because in GNU C the
@samp{/} operator always rounds towards zero.  But in other C
implementations, @samp{/} may round differently with negative arguments.
@code{div} and @code{ldiv} are useful because they specify how to round
the quotient: towards zero.  The remainder has the same sign as the
numerator.

These functions are specified to return a result @var{r} such that the value
@code{@var{r}.quot*@var{denominator} + @var{r}.rem} equals
@var{numerator}.

@pindex stdlib.h
To use these facilities, you should include the header file
@file{stdlib.h} in your program.

@comment stdlib.h
@comment ISO
@deftp {Data Type} div_t
This is a structure type used to hold the result returned by the @code{div}
function.  It has the following members:

@table @code
@item int quot
The quotient from the division.

@item int rem
The remainder from the division.
@end table
@end deftp

@comment stdlib.h
@comment ISO
@deftypefun div_t div (int @var{numerator}, int @var{denominator})
This function @code{div} computes the quotient and remainder from
the division of @var{numerator} by @var{denominator}, returning the
result in a structure of type @code{div_t}.

If the result cannot be represented (as in a division by zero), the
behavior is undefined.

Here is an example, albeit not a very useful one.

@smallexample
div_t result;
result = div (20, -6);
@end smallexample

@noindent
Now @code{result.quot} is @code{-3} and @code{result.rem} is @code{2}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftp {Data Type} ldiv_t
This is a structure type used to hold the result returned by the @code{ldiv}
function.  It has the following members:

@table @code
@item long int quot
The quotient from the division.

@item long int rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{long int} rather than @code{int}.)
@end deftp

@comment stdlib.h
@comment ISO
@deftypefun ldiv_t ldiv (long int @var{numerator}, long int @var{denominator})
The @code{ldiv} function is similar to @code{div}, except that the
arguments are of type @code{long int} and the result is returned as a
structure of type @code{ldiv_t}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftp {Data Type} lldiv_t
This is a structure type used to hold the result returned by the @code{lldiv}
function.  It has the following members:

@table @code
@item long long int quot
The quotient from the division.

@item long long int rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{long long int} rather than @code{int}.)
@end deftp

@comment stdlib.h
@comment ISO
@deftypefun lldiv_t lldiv (long long int @var{numerator}, long long int @var{denominator})
The @code{lldiv} function is like the @code{div} function, but the
arguments are of type @code{long long int} and the result is returned as
a structure of type @code{lldiv_t}.

The @code{lldiv} function was added in @w{ISO C 9x}.
@end deftypefun

@comment inttypes.h
@comment ISO
@deftp {Data Type} imaxdiv_t
This is a structure type used to hold the result returned by the @code{imaxdiv}
function.  It has the following members:

@table @code
@item intmax_t quot
The quotient from the division.

@item intmax_t rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{intmax_t} rather than @code{int}.)
@end deftp

@comment inttypes.h
@comment ISO
@deftypefun imaxdiv_t imaxdiv (intmax_t @var{numerator}, intmax_t @var{denominator})
The @code{imaxdiv} function is like the @code{div} function, but the
arguments are of type @code{intmax_t} and the result is returned as
a structure of type @code{imaxdiv_t}.

The @code{imaxdiv} function was added in @w{ISO C 9x}.
@end deftypefun


@node Parsing of Numbers
@section Parsing of Numbers
@cindex parsing numbers (in formatted input)
@cindex converting strings to numbers
@cindex number syntax, parsing
@cindex syntax, for reading numbers

This section describes functions for ``reading'' integer and
floating-point numbers from a string.  It may be more convenient in some
cases to use @code{sscanf} or one of the related functions; see
@ref{Formatted Input}.  But often you can make a program more robust by
finding the tokens in the string by hand, then converting the numbers
one by one.

@menu
* Parsing of Integers::         Functions for conversion of integer values.
* Parsing of Floats::           Functions for conversion of floating-point
				 values.
@end menu

@node Parsing of Integers
@subsection Parsing of Integers

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ISO
@deftypefun {long int} strtol (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtol} (``string-to-long'') function converts the initial
part of @var{string} to a signed integer, which is returned as a value
of type @code{long int}.

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item
An optional plus or minus sign (@samp{+} or @samp{-}).

@item
A nonempty sequence of digits in the radix specified by @var{base}.

If @var{base} is zero, decimal radix is assumed unless the series of
digits begins with @samp{0} (specifying octal radix), or @samp{0x} or
@samp{0X} (specifying hexadecimal radix); in other words, the same
syntax used for integer constants in C.

Otherwise @var{base} must have a value between @code{2} and @code{35}.
If @var{base} is @code{16}, the digits may optionally be preceded by
@samp{0x} or @samp{0X}.  If base has no legal value the value returned
is @code{0l} and the global variable @code{errno} is set to @code{EINVAL}.

@item
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, @code{strtol} stores a pointer to this tail in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for an integer in the
specified @var{base}, no conversion is performed.  In this case,
@code{strtol} returns a value of zero and the value stored in
@code{*@var{tailptr}} is the value of @var{string}.

In a locale other than the standard @code{"C"} locale, this function
may recognize additional implementation-dependent syntax.

If the string has valid syntax for an integer but the value is not
representable because of overflow, @code{strtol} returns either
@code{LONG_MAX} or @code{LONG_MIN} (@pxref{Range of Type}), as
appropriate for the sign of the value.  It also sets @code{errno}
to @code{ERANGE} to indicate there was overflow.

You should not check for errors by examining the return value of
@code{strtol}, because the string might be a valid representation of
@code{0l}, @code{LONG_MAX}, or @code{LONG_MIN}.  Instead, check whether
@var{tailptr} points to what you expect after the number
(e.g. @code{'\0'} if the string should end after the number).  You also
need to clear @var{errno} before the call and check it afterward, in
case there was overflow.

There is an example at the end of this section.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {unsigned long int} strtoul (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoul} (``string-to-unsigned-long'') function is like
@code{strtol} except it returns an @code{unsigned long int} value.  If
the number has a leading @samp{-} sign, the return value is negated.
The syntax is the same as described above for @code{strtol}.  The value
returned on overflow is @code{ULONG_MAX} (@pxref{Range of
Type}).

@code{strtoul} sets @var{errno} to @code{EINVAL} if @var{base} is out of
range, or @code{ERANGE} on overflow.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long long int} strtoll (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoll} function is like @code{strtol} except that it returns
a @code{long long int} value, and accepts numbers with a correspondingly
larger range.

If the string has valid syntax for an integer but the value is not
representable because of overflow, @code{strtoll} returns either
@code{LONG_LONG_MAX} or @code{LONG_LONG_MIN} (@pxref{Range of Type}), as
appropriate for the sign of the value.  It also sets @code{errno} to
@code{ERANGE} to indicate there was overflow.

The @code{strtoll} function was introduced in @w{ISO C 9x}.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {long long int} strtoq (const char *@var{string}, char **@var{tailptr}, int @var{base})
@code{strtoq} (``string-to-quad-word'') is the BSD name for @code{strtoll}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {unsigned long long int} strtoull (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoull} function is like @code{strtoul} except that it
returns an @code{unsigned long long int}.  The value returned on overflow
is @code{ULONG_LONG_MAX} (@pxref{Range of Type}).

The @code{strtoull} function was introduced in @w{ISO C 9x}.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {unsigned long long int} strtouq (const char *@var{string}, char **@var{tailptr}, int @var{base})
@code{strtouq} is the BSD name for @code{strtoull}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long int} atol (const char *@var{string})
This function is similar to the @code{strtol} function with a @var{base}
argument of @code{10}, except that it need not detect overflow errors.
The @code{atol} function is provided mostly for compatibility with
existing code; using @code{strtol} is more robust.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun int atoi (const char *@var{string})
This function is like @code{atol}, except that it returns an @code{int}.
The @code{atoi} function is also considered obsolete; use @code{strtol}
instead.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long long int} atoll (const char *@var{string})
This function is similar to @code{atol}, except it returns a @code{long
long int}.

The @code{atoll} function was introduced in @w{ISO C 9x}.  It too is
obsolete (despite having just been added); use @code{strtoll} instead.
@end deftypefun

@c !!! please fact check this paragraph -zw
@findex strtol_l
@findex strtoul_l
@findex strtoll_l
@findex strtoull_l
@cindex parsing numbers and locales
@cindex locales, parsing numbers and
Some locales specify a printed syntax for numbers other than the one
that these functions understand.  If you need to read numbers formatted
in some other locale, you can use the @code{strtoX_l} functions.  Each
of the @code{strtoX} functions has a counterpart with @samp{_l} added to
its name.  The @samp{_l} counterparts take an additional argument: a
pointer to an @code{locale_t} structure, which describes how the numbers
to be read are formatted.  @xref{Locales}.

@strong{Portability Note:} These functions are all GNU extensions.  You
can also use @code{scanf} or its relatives, which have the @samp{'} flag
for parsing numeric input according to the current locale
(@pxref{Numeric Input Conversions}).  This feature is standard.

Here is a function which parses a string as a sequence of integers and
returns the sum of them:

@smallexample
int
sum_ints_from_string (char *string)
@{
  int sum = 0;

  while (1) @{
    char *tail;
    int next;

    /* @r{Skip whitespace by hand, to detect the end.}  */
    while (isspace (*string)) string++;
    if (*string == 0)
      break;

    /* @r{There is more nonwhitespace,}  */
    /* @r{so it ought to be another number.}  */
    errno = 0;
    /* @r{Parse it.}  */
    next = strtol (string, &tail, 0);
    /* @r{Add it in, if not overflow.}  */
    if (errno)
      printf ("Overflow\n");
    else
      sum += next;
    /* @r{Advance past it.}  */
    string = tail;
  @}

  return sum;
@}
@end smallexample

@node Parsing of Floats
@subsection Parsing of Floats

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ISO
@deftypefun double strtod (const char *@var{string}, char **@var{tailptr})
The @code{strtod} (``string-to-double'') function converts the initial
part of @var{string} to a floating-point number, which is returned as a
value of type @code{double}.

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item
An optional plus or minus sign (@samp{+} or @samp{-}).

@item
A nonempty sequence of digits optionally containing a decimal-point
character---normally @samp{.}, but it depends on the locale
(@pxref{General Numeric}).

@item
An optional exponent part, consisting of a character @samp{e} or
@samp{E}, an optional sign, and a sequence of digits.

@item
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, a pointer to this tail of the string is stored in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for a floating-point
number, no conversion is performed.  In this case, @code{strtod} returns
a value of zero and the value returned in @code{*@var{tailptr}} is the
value of @var{string}.

In a locale other than the standard @code{"C"} or @code{"POSIX"} locales,
this function may recognize additional locale-dependent syntax.

If the string has valid syntax for a floating-point number but the value
is outside the range of a @code{double}, @code{strtod} will signal
overflow or underflow as described in @ref{Math Error Reporting}.

@code{strtod} recognizes four special input strings.  The strings
@code{"inf"} and @code{"infinity"} are converted to @math{@infinity{}},
or to the largest representable value if the floating-point format
doesn't support infinities.  You can prepend a @code{"+"} or @code{"-"}
to specify the sign.  Case is ignored when scanning these strings.

The strings @code{"nan"} and @code{"nan(@var{chars...})"} are converted
to NaN.  Again, case is ignored.  If @var{chars...} are provided, they
are used in some unspecified fashion to select a particular
representation of NaN (there can be several).

Since zero is a valid result as well as the value returned on error, you
should check for errors in the same way as for @code{strtol}, by
examining @var{errno} and @var{tailptr}.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun float strtof (const char *@var{string}, char **@var{tailptr})
@comment stdlib.h
@comment GNU
@deftypefunx {long double} strtold (const char *@var{string}, char **@var{tailptr})
These functions are analogous to @code{strtod}, but return @code{float}
and @code{long double} values respectively.  They report errors in the
same way as @code{strtod}.  @code{strtof} can be substantially faster
than @code{strtod}, but has less precision; conversely, @code{strtold}
can be much slower but has more precision (on systems where @code{long
double} is a separate type).

These functions are GNU extensions.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun double atof (const char *@var{string})
This function is similar to the @code{strtod} function, except that it
need not detect overflow and underflow errors.  The @code{atof} function
is provided mostly for compatibility with existing code; using
@code{strtod} is more robust.
@end deftypefun

The GNU C library also provides @samp{_l} versions of thse functions,
which take an additional argument, the locale to use in conversion.
@xref{Parsing of Integers}.

@node System V Number Conversion
@section Old-fashioned System V number-to-string functions

The old @w{System V} C library provided three functions to convert
numbers to strings, with unusual and hard-to-use semantics.  The GNU C
library also provides these functions and some natural extensions.

These functions are only available in glibc and on systems descended
from AT&T Unix.  Therefore, unless these functions do precisely what you
need, it is better to use @code{sprintf}, which is standard.

All these functions are defined in @file{stdlib.h}.

@comment stdlib.h
@comment SVID, Unix98
@deftypefun {char *} ecvt (double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg})
The function @code{ecvt} converts the floating-point number @var{value}
to a string with at most @var{ndigit} decimal digits.
The returned string contains no decimal point or sign. The first
digit of the string is non-zero (unless @var{value} is actually zero)
and the last digit is rounded to nearest.  @var{decpt} is set to the
index in the string of the first digit after the decimal point.
@var{neg} is set to a nonzero value if @var{value} is negative, zero
otherwise.

If @var{ndigit} decimal digits would exceed the precision of a
@code{double} it is reduced to a system-specific value.

The returned string is statically allocated and overwritten by each call
to @code{ecvt}.

If @var{value} is zero, it's implementation defined whether @var{decpt} is
@code{0} or @code{1}.

For example: @code{ecvt (12.3, 5, &decpt, &neg)} returns @code{"12300"}
and sets @var{decpt} to @code{2} and @var{neg} to @code{0}.
@end deftypefun

@comment stdlib.h
@comment SVID, Unix98
@deftypefun {char *} fcvt (double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg})
The function @code{fcvt} is like @code{ecvt}, but @var{ndigit} specifies
the number of digits after the decimal point.  If @var{ndigit} is less
than zero, @var{value} is rounded to the @math{@var{ndigit}+1}'th place to the
left of the decimal point.  For example, if @var{ndigit} is @code{-1},
@var{value} will be rounded to the nearest 10.  If @var{ndigit} is
negative and larger than the number of digits to the left of the decimal
point in @var{value}, @var{value} will be rounded to one significant digit.

If @var{ndigit} decimal digits would exceed the precision of a
@code{double} it is reduced to a system-specific value.

The returned string is statically allocated and overwritten by each call
to @code{fcvt}.
@end deftypefun

@comment stdlib.h
@comment SVID, Unix98
@deftypefun {char *} gcvt (double @var{value}, int @var{ndigit}, char *@var{buf})
@code{gcvt} is functionally equivalent to @samp{sprintf(buf, "%*g",
ndigit, value}.  It is provided only for compatibility's sake.  It
returns @var{buf}.

If @var{ndigit} decimal digits would exceed the precision of a
@code{double} it is reduced to a system-specific value.
@end deftypefun

As extensions, the GNU C library provides versions of these three
functions that take @code{long double} arguments.

@comment stdlib.h
@comment GNU
@deftypefun {char *} qecvt (long double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg})
This function is equivalent to @code{ecvt} except that it takes a
@code{long double} for the first parameter and that @var{ndigit} is
restricted by the precision of a @code{long double}.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {char *} qfcvt (long double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg})
This function is equivalent to @code{fcvt} except that it
takes a @code{long double} for the first parameter and that @var{ndigit} is
restricted by the precision of a @code{long double}.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {char *} qgcvt (long double @var{value}, int @var{ndigit}, char *@var{buf})
This function is equivalent to @code{gcvt} except that it takes a
@code{long double} for the first parameter and that @var{ndigit} is
restricted by the precision of a @code{long double}.
@end deftypefun


@cindex gcvt_r
The @code{ecvt} and @code{fcvt} functions, and their @code{long double}
equivalents, all return a string located in a static buffer which is
overwritten by the next call to the function.  The GNU C library
provides another set of extended functions which write the converted
string into a user-supplied buffer.  These have the conventional
@code{_r} suffix.

@code{gcvt_r} is not necessary, because @code{gcvt} already uses a
user-supplied buffer.

@comment stdlib.h
@comment GNU
@deftypefun {char *} ecvt_r (double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
The @code{ecvt_r} function is the same as @code{ecvt}, except
that it places its result into the user-specified buffer pointed to by
@var{buf}, with length @var{len}.

This function is a GNU extension.
@end deftypefun

@comment stdlib.h
@comment SVID, Unix98
@deftypefun {char *} fcvt_r (double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
The @code{fcvt_r} function is the same as @code{fcvt}, except
that it places its result into the user-specified buffer pointed to by
@var{buf}, with length @var{len}.

This function is a GNU extension.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {char *} qecvt_r (long double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
The @code{qecvt_r} function is the same as @code{qecvt}, except
that it places its result into the user-specified buffer pointed to by
@var{buf}, with length @var{len}.

This function is a GNU extension.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {char *} qfcvt_r (long double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
The @code{qfcvt_r} function is the same as @code{qfcvt}, except
that it places its result into the user-specified buffer pointed to by
@var{buf}, with length @var{len}.

This function is a GNU extension.
@end deftypefun