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
/*! `fun`damental `ty`pes

This crate provides trait unification of the Rust fundamental numbers, allowing
users to declare the behavior they want from a number without committing to a
single particular numeric type.

The number types can be categorized along two axes: behavior and width. Traits
for each axis and group on that axis are provided:

## Numeric Categories

The most general category is represented by the trait [`IsNumber`]. It is
implemented by all the numeric fundamentals, and includes only the traits that
they all implement. This is an already-large amount: basic memory management,
comparison, rendering, and numeric arithmetic.

The numbers are then split into [`IsInteger`] and [`IsFloat`]. The former fills
out the API of `f32` and `f64`, while the latter covers all of the `iN` and `uN`
numbers.

Lastly, [`IsInteger`] splits further, into [`IsSigned`] and [`IsUnsigned`].
These provide the last specializations unique to the differences between `iN`
and `uN`.

## Width Categories

Every number implements the trait `IsN` for the `N` of its bit width. `isize`
and `usize` implement the trait that matches their width on the target platform.

In addition, the trait groups `AtLeastN` and `AtMostN` enable clamping the range
of acceptable widths to lower or upper bounds. These traits are equivalent to
`mem::size_of::<T>() >= N` and `mem::size_of::<T>() <= N`, respectively.

[`IsFloat`]: trait.IsFloat.html
[`IsInteger`]: trait.IsInteger.html
[`IsNumber`]: trait.IsNumber.html
[`IsSigned`]: trait.IsSigned.html
[`IsUnsigned`]: trait.IsUnsigned.html
!*/

#![cfg_attr(not(feature = "std"), no_std)]
#![deny(unconditional_recursion)]

use core::{
	convert::{
		TryFrom,
		TryInto,
	},
	fmt::{
		Binary,
		Debug,
		Display,
		LowerExp,
		LowerHex,
		Octal,
		UpperExp,
		UpperHex,
	},
	hash::Hash,
	iter::{
		Product,
		Sum,
	},
	num::{
		FpCategory,
		ParseIntError,
	},
	ops::{
		Add,
		AddAssign,
		BitAnd,
		BitAndAssign,
		BitOr,
		BitOrAssign,
		BitXor,
		BitXorAssign,
		Div,
		DivAssign,
		Mul,
		MulAssign,
		Neg,
		Not,
		Rem,
		RemAssign,
		Shl,
		ShlAssign,
		Shr,
		ShrAssign,
		Sub,
		SubAssign,
	},
	str::FromStr,
};

/// Declare that a type is an abstract number.
///
/// This unifies all of the signed-integer, unsigned-integer, and floating-point
/// types.
pub trait IsNumber: Sized
	+ Send
	+ Sync
	+ Unpin
	+ Clone
	+ Copy
	+ Default
	+ FromStr
	//  cmp
	+ PartialEq<Self>
	+ PartialOrd<Self>
	//  fmt
	+ Debug
	+ Display
	//  iter
	+ Product<Self>
	+ for<'a> Product<&'a Self>
	+ Sum<Self>
	+ for<'a> Sum<&'a Self>
	//  numeric ops
	+ Add<Self, Output = Self>
	+ for<'a> Add<&'a Self, Output = Self>
	+ AddAssign<Self>
	+ for<'a> AddAssign<&'a Self>
	+ Sub<Self, Output = Self>
	+ for<'a> Sub<&'a Self, Output = Self>
	+ SubAssign<Self>
	+ for<'a> SubAssign<&'a Self>
	+ Mul<Self, Output = Self>
	+ for<'a> Mul<&'a Self, Output = Self>
	+ MulAssign<Self>
	+ for<'a> MulAssign<&'a Self>
	+ Div<Self, Output = Self>
	+ for<'a> Div<&'a Self, Output = Self>
	+ DivAssign<Self>
	+ for<'a> DivAssign<&'a Self>
	+ Rem<Self, Output = Self>
	+ for<'a> Rem<&'a Self, Output = Self>
	+ RemAssign<Self>
	+ for<'a> RemAssign<&'a Self>
{
	type Bytes;

	/// Return the memory representation of this number as a byte array in
	/// big-endian (network) byte order.
	fn to_be_bytes(self) -> Self::Bytes;

	/// Return the memory representation of this number as a byte array in
	/// little-endian byte order.
	fn to_le_bytes(self) -> Self::Bytes;

	/// Return the memory representation of this number as a byte array in
	/// native byte order.
	fn to_ne_bytes(self) -> Self::Bytes;

	/// Create a numeric value from its representation as a byte array in big
	/// endian.
	fn from_be_bytes(bytes: Self::Bytes) -> Self;

	/// Create a numeric value from its representation as a byte array in little
	/// endian.
	fn from_le_bytes(bytes: Self::Bytes) -> Self;

	/// Create a numeric value from its memory representation as a byte array in
	/// native endianness.
	fn from_ne_bytes(bytes: Self::Bytes) -> Self;
}

/// Declare that a type is a fixed-point integer.
///
/// This unifies all of the signed and unsigned integral types.
pub trait IsInteger:
	IsNumber
	+ Hash
	+ Eq
	+ Ord
	+ Binary
	+ LowerHex
	+ UpperHex
	+ Octal
	+ BitAnd<Self, Output = Self>
	+ for<'a> BitAnd<&'a Self, Output = Self>
	+ BitAndAssign<Self>
	+ for<'a> BitAndAssign<&'a Self>
	+ BitOr<Self, Output = Self>
	+ for<'a> BitOr<&'a Self, Output = Self>
	+ BitOrAssign<Self>
	+ for<'a> BitOrAssign<&'a Self>
	+ BitXor<Self, Output = Self>
	+ for<'a> BitXor<&'a Self, Output = Self>
	+ BitXorAssign<Self>
	+ for<'a> BitXorAssign<&'a Self>
	+ Not<Output = Self>
	+ TryFrom<i8>
	+ TryFrom<u8>
	+ TryFrom<i16>
	+ TryFrom<u16>
	+ TryFrom<i32>
	+ TryFrom<u32>
	+ TryFrom<i64>
	+ TryFrom<u64>
	+ TryFrom<i128>
	+ TryFrom<u128>
	+ TryFrom<isize>
	+ TryFrom<usize>
	+ TryInto<i8>
	+ TryInto<u8>
	+ TryInto<i16>
	+ TryInto<u16>
	+ TryInto<i32>
	+ TryInto<u32>
	+ TryInto<i64>
	+ TryInto<u64>
	+ TryInto<i128>
	+ TryInto<u128>
	+ TryInto<isize>
	+ TryInto<usize>
	+ Shl<i8, Output = Self>
	+ for<'a> Shl<&'a i8, Output = Self>
	+ ShlAssign<i8>
	+ for<'a> ShlAssign<&'a i8>
	+ Shr<i8, Output = Self>
	+ for<'a> Shr<&'a i8, Output = Self>
	+ ShrAssign<i8>
	+ for<'a> ShrAssign<&'a i8>
	+ Shl<u8, Output = Self>
	+ for<'a> Shl<&'a u8, Output = Self>
	+ ShlAssign<u8>
	+ for<'a> ShlAssign<&'a u8>
	+ Shr<u8, Output = Self>
	+ for<'a> Shr<&'a u8, Output = Self>
	+ ShrAssign<u8>
	+ for<'a> ShrAssign<&'a u8>
	+ Shl<i16, Output = Self>
	+ for<'a> Shl<&'a i16, Output = Self>
	+ ShlAssign<i16>
	+ for<'a> ShlAssign<&'a i16>
	+ Shr<i16, Output = Self>
	+ for<'a> Shr<&'a i16, Output = Self>
	+ ShrAssign<i16>
	+ for<'a> ShrAssign<&'a i16>
	+ Shl<u16, Output = Self>
	+ for<'a> Shl<&'a u16, Output = Self>
	+ ShlAssign<u16>
	+ for<'a> ShlAssign<&'a u16>
	+ Shr<u16, Output = Self>
	+ for<'a> Shr<&'a u16, Output = Self>
	+ ShrAssign<u16>
	+ for<'a> ShrAssign<&'a u16>
	+ Shl<i32, Output = Self>
	+ for<'a> Shl<&'a i32, Output = Self>
	+ ShlAssign<i32>
	+ for<'a> ShlAssign<&'a i32>
	+ Shr<i32, Output = Self>
	+ for<'a> Shr<&'a i32, Output = Self>
	+ ShrAssign<i32>
	+ for<'a> ShrAssign<&'a i32>
	+ Shl<u32, Output = Self>
	+ for<'a> Shl<&'a u32, Output = Self>
	+ ShlAssign<u32>
	+ for<'a> ShlAssign<&'a u32>
	+ Shr<u32, Output = Self>
	+ for<'a> Shr<&'a u32, Output = Self>
	+ ShrAssign<u32>
	+ for<'a> ShrAssign<&'a u32>
	+ Shl<i64, Output = Self>
	+ for<'a> Shl<&'a i64, Output = Self>
	+ ShlAssign<i64>
	+ for<'a> ShlAssign<&'a i64>
	+ Shr<i64, Output = Self>
	+ for<'a> Shr<&'a i64, Output = Self>
	+ ShrAssign<i64>
	+ for<'a> ShrAssign<&'a i64>
	+ Shl<u64, Output = Self>
	+ for<'a> Shl<&'a u64, Output = Self>
	+ ShlAssign<u64>
	+ for<'a> ShlAssign<&'a u64>
	+ Shr<u64, Output = Self>
	+ for<'a> Shr<&'a u64, Output = Self>
	+ ShrAssign<u64>
	+ for<'a> ShrAssign<&'a u64>
	+ Shl<i128, Output = Self>
	+ for<'a> Shl<&'a i128, Output = Self>
	+ ShlAssign<i128>
	+ for<'a> ShlAssign<&'a i128>
	+ Shr<i128, Output = Self>
	+ for<'a> Shr<&'a i128, Output = Self>
	+ ShrAssign<i128>
	+ for<'a> ShrAssign<&'a i128>
	+ Shl<u128, Output = Self>
	+ for<'a> Shl<&'a u128, Output = Self>
	+ ShlAssign<u128>
	+ for<'a> ShlAssign<&'a u128>
	+ Shr<u128, Output = Self>
	+ for<'a> Shr<&'a u128, Output = Self>
	+ ShrAssign<u128>
	+ for<'a> ShrAssign<&'a u128>
	+ Shl<isize, Output = Self>
	+ for<'a> Shl<&'a isize, Output = Self>
	+ ShlAssign<isize>
	+ for<'a> ShlAssign<&'a isize>
	+ Shr<isize, Output = Self>
	+ for<'a> Shr<&'a isize, Output = Self>
	+ ShrAssign<isize>
	+ for<'a> ShrAssign<&'a isize>
	+ Shl<usize, Output = Self>
	+ for<'a> Shl<&'a usize, Output = Self>
	+ ShlAssign<usize>
	+ for<'a> ShlAssign<&'a usize>
	+ Shr<usize, Output = Self>
	+ for<'a> Shr<&'a usize, Output = Self>
	+ ShrAssign<usize>
	+ for<'a> ShrAssign<&'a usize>
{
	/// The type’s zero value.
	const ZERO: Self;

	/// The type’s minimum value. This is zero for unsigned integers.
	const MIN: Self;

	/// The type’s maximum value.
	const MAX: Self;

	/// Returns the smallest value that can be represented by this integer type.
	fn min_value() -> Self;

	/// Returns the largest value that can be represented by this integer type.
	fn max_value() -> Self;

	/// Converts a string slice in a given base to an integer.
	///
	/// The string is expected to be an optional `+` or `-` sign followed by
	/// digits. Leading and trailing whitespace represent an error. Digits are a
	/// subset of these characters, depending on `radix`:
	///
	/// - `0-9`
	/// - `a-z`
	/// - `A-Z`
	///
	/// # Panics
	///
	/// This function panics if `radix` is not in the range from 2 to 36.
	fn from_str_radix(src: &str, radix: u32) -> Result<Self, ParseIntError>;

	/// Returns the number of ones in the binary representation of `self`.
	fn count_ones(self) -> u32;

	/// Returns the number of zeros in the binary representation of `self`.
	fn count_zeros(self) -> u32;

	/// Returns the number of leading zeros in the binary representation of
	/// `self`.
	fn leading_zeros(self) -> u32;

	/// Returns the number of trailing zeros in the binary representation of
	/// `self`.
	fn trailing_zeros(self) -> u32;

	/// Returns the number of leading ones in the binary representation of
	/// `self`.
	fn leading_ones(self) -> u32;

	/// Returns the number of trailing ones in the binary representation of
	/// `self`.
	fn trailing_ones(self) -> u32;

	/// Shifts the bits to the left by a specified amount, `n`, wrapping the
	/// truncated bits to the end of the resulting integer.
	///
	/// Please note this isn’t the same operation as the `<<` shifting operator!
	fn rotate_left(self, n: u32) -> Self;

	/// Shifts the bits to the right by a specified amount, `n`, wrapping the
	/// truncated bits to the beginning of the resulting integer.
	///
	/// Please note this isn’t the same operation as the `>>` shifting operator!
	fn rotate_right(self, n: u32) -> Self;

	/// Reverses the byte order of the integer.
	fn swap_bytes(self) -> Self;

	/// Reverses the bit pattern of the integer.
	fn reverse_bits(self) -> Self;

	/// Converts an integer from big endian to the target’s endianness.
	///
	/// On big endian this is a no-op. On little endian the bytes are swapped.
	fn from_be(self) -> Self;

	/// Converts an integer frm little endian to the target’s endianness.
	///
	/// On little endian this is a no-op. On big endian the bytes are swapped.
	fn from_le(self) -> Self;

	/// Converts `self` to big endian from the target’s endianness.
	///
	/// On big endian this is a no-op. On little endian the bytes are swapped.
	fn to_be(self) -> Self;

	/// Converts `self` to little endian from the target’s endianness.
	///
	/// On little endian this is a no-op. On big endian the bytes are swapped.
	fn to_le(self) -> Self;

	/// Checked integer addition. Computes `self + rhs`, returning `None` if
	/// overflow occurred.
	fn checked_add(self, rhs: Self) -> Option<Self>;

	/// Checked integer subtraction. Computes `self - rhs`, returning `None` if
	/// overflow occurred.
	fn checked_sub(self, rhs: Self) -> Option<Self>;

	/// Checked integer multiplication. Computes `self * rhs`, returning `None`
	/// if overflow occurred.
	fn checked_mul(self, rhs: Self) -> Option<Self>;

	/// Checked integer division. Computes `self / rhs`, returning `None` if
	/// `rhs == 0` or the division results in overflow.
	fn checked_div(self, rhs: Self) -> Option<Self>;

	/// Checked Euclidean division. Computes `self.div_euclid(rhs)`, returning
	/// `None` if `rhs == 0` or the division results in overflow.
	fn checked_div_euclid(self, rhs: Self) -> Option<Self>;

	/// Checked integer remainder. Computes `self % rhs`, returning `None` if
	/// `rhs == 0` or the division results in overflow.
	fn checked_rem(self, rhs: Self) -> Option<Self>;

	/// Checked Euclidean remainder. Computes `self.rem_euclid(rhs)`, returning
	/// `None` if `rhs == 0` or the division results in overflow.
	fn checked_rem_euclid(self, rhs: Self) -> Option<Self>;

	/// Checked negation. Computes `-self`, returning `None` if `self == MIN`.
	///
	/// Note that negating any positive integer will overflow.
	fn checked_neg(self) -> Option<Self>;

	/// Checked shift left. Computes `self << rhs`, returning `None` if `rhs` is
	/// larger than or equal to the number of bits in `self`.
	fn checked_shl(self, rhs: u32) -> Option<Self>;

	/// Checked shift right. Computes `self >> rhs`, returning `None` if `rhs`
	/// is larger than or equal to the number of bits in `self`.
	fn checked_shr(self, rhs: u32) -> Option<Self>;

	/// Checked exponentiation. Computes `self.pow(exp)`, returning `None` if
	/// overflow occurred.
	fn checked_pow(self, rhs: u32) -> Option<Self>;

	/// Saturating integer addition. Computes `self + rhs`, saturating at the
	/// numeric bounds instead of overflowing.
	fn saturating_add(self, rhs: Self) -> Self;

	/// Saturating integer subtraction. Computes `self - rhs`, saturating at the
	/// numeric bounds instead of overflowing.
	fn saturating_sub(self, rhs: Self) -> Self;

	/// Saturating integer multiplication. Computes `self * rhs`, saturating at
	/// the numeric bounds instead of overflowing.
	fn saturating_mul(self, rhs: Self) -> Self;

	/// Saturating integer exponentiation. Computes `self.pow(exp)`, saturating
	/// at the numeric bounds instead of overflowing.
	fn saturating_pow(self, rhs: u32) -> Self;

	/// Wrapping (modular) addition. Computes `self + rhs`, wrapping around at
	/// the boundary of the type.
	fn wrapping_add(self, rhs: Self) -> Self;

	/// Wrapping (modular) subtraction. Computes `self - rhs`, wrapping around
	/// at the boundary of the type.
	fn wrapping_sub(self, rhs: Self) -> Self;

	/// Wrapping (modular) multiplication. Computes `self * rhs`, wrapping
	/// around at the boundary of the type.
	fn wrapping_mul(self, rhs: Self) -> Self;

	/// Wrapping (modular) division. Computes `self / rhs`, wrapping around at
	/// the boundary of the type.
	///
	/// # Signed Integers
	///
	/// The only case where such wrapping can occur is when one divides
	/// `MIN / -1` on a signed type (where `MIN` is the negative minimal value
	/// for the type); this is equivalent to `-MIN`, a positive value that is
	/// too large to represent in the type. In such a case, this function
	/// returns `MIN` itself.
	///
	/// # Unsigned Integers
	///
	/// Wrapping (modular) division. Computes `self / rhs`. Wrapped division on
	/// unsigned types is just normal division. There’s no way wrapping could
	/// ever happen. This function exists, so that all operations are accounted
	/// for in the wrapping operations.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn wrapping_div(self, rhs: Self) -> Self;

	/// Wrapping Eulidean division. Computes `self.div_euclid(rhs)`, wrapping
	/// around at the boundary of the type.
	///
	/// # Signed Types
	///
	/// Wrapping will only occur in `MIN / -1` on a signed type (where `MIN` is
	/// the negative minimal value for the type). This is equivalent to `-MIN`,
	/// a positive value that is too large to represent in the type. In this
	/// case, this method returns `MIN` itself.
	///
	/// # Unsigned Types
	///
	/// Wrapped division on unsigned types is just normal division. There’s no
	/// way wrapping could ever happen. This function exists, so that all
	/// operations are accounted for in the wrapping operations. Since, for the
	/// positive integers, all common definitions of division are equal, this is
	/// exactly equal to `self.wrapping_div(rhs)`.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn wrapping_div_euclid(self, rhs: Self) -> Self;

	/// Wrapping (modular) remainder. Computes `self % rhs`, wrapping around at
	/// the boundary of the type.
	///
	/// # Signed Integers
	///
	/// Such wrap-around never actually occurs mathematically; implementation
	/// artifacts make `x % y` invalid for `MIN / -1` on a signed type (where
	/// `MIN` is the negative minimal value). In such a case, this function
	/// returns `0`.
	///
	/// # Unsigned Integers
	///
	/// Wrapped remainder calculation on unsigned types is just the regular
	/// remainder calculation. There’s no way wrapping could ever happen. This
	/// function exists, so that all operations are accounted for in the
	/// wrapping operations.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn wrapping_rem(self, rhs: Self) -> Self;

	/// Wrapping Euclidean remainder. Computes `self.rem_euclid(rhs)`, wrapping
	/// around at the boundary of the type.
	///
	/// # Signed Integers
	///
	/// Wrapping will only occur in `MIN % -1` on a signed type (where `MIN` is
	/// the negative minimal value for the type). In this case, this method
	/// returns 0.
	///
	/// # Unsigned Integers
	///
	/// Wrapped modulo calculation on unsigned types is just the regular
	/// remainder calculation. There’s no way wrapping could ever happen. This
	/// function exists, so that all operations are accounted for in the
	/// wrapping operations. Since, for the positive integers, all common
	/// definitions of division are equal, this is exactly equal to
	/// `self.wrapping_rem(rhs)`.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn wrapping_rem_euclid(self, rhs: Self) -> Self;

	/// Wrapping (modular) negation. Computes `-self`, wrapping around at the
	/// boundary of the type.
	///
	/// # Signed Integers
	///
	/// The  only case where such wrapping can occur is when one negates `MIN`
	/// on a signed type (where `MIN` is the negative minimal value for the
	/// type); this is a positive value that is too large to represent in the
	/// type. In such a case, this function returns `MIN` itself.
	///
	/// # Unsigned Integers
	///
	/// Since unsigned types do not have negative equivalents all applications
	/// of this function will wrap (except for `-0`). For values smaller than
	/// the corresponding signed type’s maximum the result is the same as
	/// casting the corresponding signed value. Any larger values are equivalent
	/// to `MAX + 1 - (val - MAX - 1)` where `MAX` is the corresponding signed
	/// type’s maximum.
	fn wrapping_neg(self) -> Self;

	/// Panic-free bitwise shift-left; yields `self << mask(rhs)`, where `mask`
	/// removes any high-order bits of `rhs` that would cause the shift to
	/// exceed the bitwidth of the type.
	///
	/// Note that this is not the same as a rotate-left; the RHS of a wrapping
	/// shift-left is restricted to the range of the type, rather than the bits
	/// shifted out of the LHS being returned to the other end. The primitive
	/// integer types all implement a `rotate_left` function, which may be what
	/// you want instead.
	fn wrapping_shl(self, rhs: u32) -> Self;

	/// Panic-free bitwise shift-right; yields `self >> mask(rhs)`, where `mask`
	/// removes any high-order bits of `rhs` that would cause the shift to
	/// exceed the bitwidth of the type.
	///
	/// Note that this is not the same as a rotate-right; the RHS of a wrapping
	/// shift-right is restricted to the range of the type, rather than the bits
	/// shifted out of the LHS being returned to the other end. The primitive
	/// integer types all implement a `rotate_right` function, which may be what
	/// you want instead.
	fn wrapping_shr(self, rhs: u32) -> Self;

	/// Wrapping (modular) exponentiation. Computes `self.pow(exp)`, wrapping
	/// around at the boundary of the type.
	fn wrapping_pow(self, rhs: u32) -> Self;

	/// Calculates `self + rhs`
	///
	/// Returns a tuple of the addition along with a boolean indicating whether
	/// an arithmetic overflow would occur. If an overflow would have occurred
	/// then the wrapped value is returned.
	fn overflowing_add(self, rhs: Self) -> (Self, bool);

	/// Calculates `self - rhs`
	///
	/// Returns a tuple of the subtraction along with a boolean indicating
	/// whether an arithmetic overflow would occur. If an overflow would have
	/// occurred then the wrapped value is returned.
	fn overflowing_sub(self, rhs: Self) -> (Self, bool);

	/// Calculates the multiplication of `self` and `rhs`.
	///
	/// Returns a tuple of the multiplication along with a boolean indicating
	/// whether an arithmetic overflow would occur. If an overflow would have
	/// occurred then the wrapped value is returned.
	fn overflowing_mul(self, rhs: Self) -> (Self, bool);

	/// Calculates the divisor when `self` is divided by `rhs`.
	///
	/// Returns a tuple of the divisor along with a boolean indicating whether
	/// an arithmetic overflow would occur. If an overflow would occur then self
	/// is returned.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn overflowing_div(self, rhs: Self) -> (Self, bool);

	/// Calculates the quotient of Euclidean division `self.div_euclid(rhs)`.
	///
	/// Returns a tuple of the divisor along with a boolean indicating whether
	/// an arithmetic overflow would occur. If an overflow would occur then self
	/// is returned.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn overflowing_div_euclid(self, rhs: Self) -> (Self, bool);

	/// Calculates the remainder when `self` is divided by `rhs`.
	///
	/// Returns a tuple of the remainder after dividing along with a boolean
	/// indicating whether an arithmetic overflow would occur. If an overflow
	/// would occur then 0 is returned.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0.
	fn overflowing_rem(self, rhs: Self) -> (Self, bool);

	/// Overflowing Euclidean remainder. Calculates `self.rem_euclid(rhs)`.
	///
	/// Returns a tuple of the remainder after dividing along with a boolean
	/// indicating whether an arithmetic overflow would occur. If an overflow
	/// would occur then 0 is returned.
	///
	/// # Panics
	///
	/// This function will panic if rhs is 0.
	fn overflowing_rem_euclid(self, rhs: Self) -> (Self, bool);

	/// Negates self, overflowing if this is equal to the minimum value.
	///
	/// Returns a tuple of the negated version of self along with a boolean
	/// indicating whether an overflow happened. If `self` is the minimum value
	/// (e.g., `i32::MIN` for values of type `i32`), then the minimum value will
	/// be returned again and `true` will be returned for an overflow happening.
	fn overflowing_neg(self) -> (Self, bool);

	/// Shifts self left by `rhs` bits.
	///
	/// Returns a tuple of the shifted version of self along with a boolean
	/// indicating whether the shift value was larger than or equal to the
	/// number of bits. If the shift value is too large, then value is masked
	/// (N-1) where N is the number of bits, and this value is then used to
	/// perform the shift.
	fn overflowing_shl(self, rhs: u32) -> (Self, bool);

	/// Shifts self right by `rhs` bits.
	///
	/// Returns a tuple of the shifted version of self along with a boolean
	/// indicating whether the shift value was larger than or equal to the
	/// number of bits. If the shift value is too large, then value is masked
	/// (N-1) where N is the number of bits, and this value is then used to
	/// perform the shift.
	fn overflowing_shr(self, rhs: u32) -> (Self, bool);

	/// Raises self to the power of `exp`, using exponentiation by squaring.
	///
	/// Returns a tuple of the exponentiation along with a bool indicating
	/// whether an overflow happened.
	fn overflowing_pow(self, rhs: u32) -> (Self, bool);

	/// Raises self to the power of `exp`, using exponentiation by squaring.
	fn pow(self, rhs: u32) -> Self;

	/// Calculates the quotient of Euclidean division of self by rhs.
	///
	/// This computes the integer `n` such that
	/// `self = n * rhs + self.rem_euclid(rhs)`, with
	/// `0 <= self.rem_euclid(rhs) < rhs`.
	///
	/// In other words, the result is `self / rhs` rounded to the integer `n`
	/// such that `self >= n * rhs`. If `self > 0`, this is equal to round
	/// towards zero (the default in Rust); if `self < 0`, this is equal to
	/// round towards +/- infinity.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0 or the division results in
	/// overflow.
	fn div_euclid(self, rhs: Self) -> Self;

	/// Calculates the least nonnegative remainder of `self (mod rhs)`.
	///
	/// This is done as if by the Euclidean division algorithm -- given
	/// `r = self.rem_euclid(rhs)`, `self = rhs * self.div_euclid(rhs) + r`, and
	/// `0 <= r < abs(rhs)`.
	///
	/// # Panics
	///
	/// This function will panic if `rhs` is 0 or the division results in
	/// overflow.
	fn rem_euclid(self, rhs: Self) -> Self;
}

/// Declare that a type is a signed integer.
pub trait IsSigned: IsInteger + Neg {
	/// Checked absolute value. Computes `self.abs()`, returning `None` if
	/// `self == MIN`.
	fn checked_abs(self) -> Option<Self>;

	/// Wrapping (modular) absolute value. Computes `self.abs()`, wrapping
	/// around at the boundary of the type.
	///
	/// The only case where such wrapping can occur is when one takes the
	/// absolute value of the negative minimal value for the type this is a
	/// positive value that is too large to represent in the type. In such a
	/// case, this function returns `MIN` itself.
	fn wrapping_abs(self) -> Self;

	/// Computes the absolute value of `self`.
	///
	/// Returns a tuple of the absolute version of self along with a boolean
	/// indicating whether an overflow happened. If self is the minimum value
	/// (e.g., iN::MIN for values of type iN), then the minimum value will be
	/// returned again and true will be returned for an overflow happening.
	fn overflowing_abs(self) -> (Self, bool);

	//// Computes the absolute value of self.
	///
	/// # Overflow behavior
	///
	/// The absolute value of `iN::min_value()` cannot be represented as an
	/// `iN`, and attempting to calculate it will cause an overflow. This means
	/// that code in debug mode will trigger a panic on this case and optimized
	/// code will return `iN::min_value()` without a panic.
	fn abs(self) -> Self;

	/// Returns a number representing sign of `self`.
	///
	/// - `0` if the number is zero
	/// - `1` if the number is positive
	/// - `-1` if the number is negative
	fn signum(self) -> Self;

	/// Returns `true` if `self` is positive and `false` if the number is zero
	/// or negative.
	fn is_positive(self) -> bool;

	/// Returns `true` if `self` is negative and `false` if the number is zero
	/// or positive.
	fn is_negative(self) -> bool;
}

/// Declare that a type is an unsigned integer.
pub trait IsUnsigned: IsInteger {
	/// Returns `true` if and only if `self == 2^k` for some `k`.
	fn is_power_of_two(self) -> bool;

	/// Returns the smallest power of two greater than or equal to `self`.
	///
	/// When return value overflows (i.e., `self > (1 << (N-1))` for type `uN`),
	/// it panics in debug mode and return value is wrapped to 0 in release mode
	/// (the only situation in which method can return 0).
	fn next_power_of_two(self) -> Self;

	/// Returns the smallest power of two greater than or equal to `n`. If the
	/// next power of two is greater than the type’s maximum value, `None` is
	/// returned, otherwise the power of two is wrapped in `Some`.
	fn checked_next_power_of_two(self) -> Option<Self>;
}

/// Declare that a type is a floating-point number.
pub trait IsFloat:
	IsNumber
	+ LowerExp
	+ UpperExp
	+ Neg
	+ From<f32>
	+ From<i8>
	+ From<i16>
	+ From<u8>
	+ From<u16>
{
	/// The unsigned integer type of the same width as `Self`.
	type Raw: IsUnsigned;

	/// The radix or base of the internal representation of `f32`.
	const RADIX: u32;

	/// Number of significant digits in base 2.
	const MANTISSA_DIGITS: u32;

	/// Approximate number of significant digits in base 10.
	const DIGITS: u32;

	/// [Machine epsilon] value for `f32`.
	///
	/// This is the difference between `1.0` and the next larger representable
	/// number.
	///
	/// [Machine epsilon]: https://en.wikipedia.org/wiki/Machine_epsilon
	const EPSILON: Self;

	/// Smallest finite `f32` value.
	const MIN: Self;

	/// Smallest positive normal `f32` value.
	const MIN_POSITIVE: Self;

	/// Largest finite `f32` value.
	const MAX: Self;

	/// One greater than the minimum possible normal power of 2 exponent.
	const MIN_EXP: i32;

	/// Maximum possible power of 2 exponent.
	const MAX_EXP: i32;

	/// Minimum possible normal power of 10 exponent.
	const MIN_10_EXP: i32;

	/// Maximum possible power of 10 exponent.
	const MAX_10_EXP: i32;

	/// Not a Number (NaN).
	const NAN: Self;

	/// Infinity (∞).
	const INFINITY: Self;

	/// Negative infinity (−∞).
	const NEG_INFINITY: Self;

	/// Archimedes' constant (π)
	const PI: Self;

	/// π/2
	const FRAC_PI_2: Self;

	/// π/3
	const FRAC_PI_3: Self;

	/// π/4
	const FRAC_PI_4: Self;

	/// π/6
	const FRAC_PI_6: Self;

	/// π/8
	const FRAC_PI_8: Self;

	/// 1/π
	const FRAC_1_PI: Self;

	/// 2/π
	const FRAC_2_PI: Self;

	/// 2/sqrt(π)
	const FRAC_2_SQRT_PI: Self;

	/// sqrt(2)
	const SQRT_2: Self;

	/// 1/sqrt(2)
	const FRAC_1_SQRT_2: Self;

	/// Euler’s number (e)
	const E: Self;

	/// log<sub>2</sub>(e)
	const LOG2_E: Self;

	/// log<sub>10</sub>(e)
	const LOG10_E: Self;

	/// ln(2)
	const LN_2: Self;

	/// ln(10)
	const LN_10: Self;

	//  These functions are only available in `libstd`, because they rely on the
	//  system math library `libm` which is not provided by `libcore`.

	/// Returns the largest integer less than or equal to a number.
	#[cfg(feature = "std")]
	fn floor(self) -> Self;

	/// Returns the smallest integer greater than or equal to a number.
	#[cfg(feature = "std")]
	fn ceil(self) -> Self;

	/// Returns the nearest integer to a number. Round half-way cases away from
	/// `0.0`.
	#[cfg(feature = "std")]
	fn round(self) -> Self;

	/// Returns the integer part of a number.
	#[cfg(feature = "std")]
	fn trunc(self) -> Self;

	/// Returns the fractional part of a number.
	#[cfg(feature = "std")]
	fn fract(self) -> Self;

	/// Computes the absolute value of `self`. Returns `NAN` if the
	/// number is `NAN`.
	#[cfg(feature = "std")]
	fn abs(self) -> Self;

	/// Returns a number that represents the sign of `self`.
	///
	/// - `1.0` if the number is positive, `+0.0` or `INFINITY`
	/// - `-1.0` if the number is negative, `-0.0` or `NEG_INFINITY`
	/// - `NAN` if the number is `NAN`
	#[cfg(feature = "std")]
	fn signum(self) -> Self;

	/// Returns a number composed of the magnitude of `self` and the sign of
	/// `sign`.
	///
	/// Equal to `self` if the sign of `self` and `sign` are the same, otherwise
	/// equal to `-self`. If `self` is a `NAN`, then a `NAN` with the sign of
	/// `sign` is returned.
	#[cfg(feature = "std")]
	fn copysign(self, sign: Self) -> Self;

	/// Fused multiply-add. Computes `(self * a) + b` with only one rounding
	/// error, yielding a more accurate result than an unfused multiply-add.
	///
	/// Using `mul_add` can be more performant than an unfused multiply-add if
	/// the target architecture has a dedicated `fma` CPU instruction.
	#[cfg(feature = "std")]
	fn mul_add(self, a: Self, b: Self) -> Self;

	/// Calculates Euclidean division, the matching method for `rem_euclid`.
	///
	/// This computes the integer `n` such that
	/// `self = n * rhs + self.rem_euclid(rhs)`.
	/// In other words, the result is `self / rhs` rounded to the integer `n`
	/// such that `self >= n * rhs`.
	#[cfg(feature = "std")]
	fn div_euclid(self, rhs: Self) -> Self;

	/// Calculates the least nonnegative remainder of `self (mod rhs)`.
	///
	/// In particular, the return value `r` satisfies `0.0 <= r < rhs.abs()` in
	/// most cases. However, due to a floating point round-off error it can
	/// result in `r == rhs.abs()`, violating the mathematical definition, if
	/// `self` is much smaller than `rhs.abs()` in magnitude and `self < 0.0`.
	/// This result is not an element of the function's codomain, but it is the
	/// closest floating point number in the real numbers and thus fulfills the
	/// property `self == self.div_euclid(rhs) * rhs + self.rem_euclid(rhs)`
	/// approximatively.
	#[cfg(feature = "std")]
	fn rem_euclid(self, rhs: Self) -> Self;

	/// Raises a number to an integer power.
	///
	/// Using this function is generally faster than using `powf`
	#[cfg(feature = "std")]
	fn powi(self, n: i32) -> Self;

	/// Raises a number to a floating point power.
	#[cfg(feature = "std")]
	fn powf(self, n: Self) -> Self;

	/// Returns the square root of a number.
	///
	/// Returns NaN if `self` is a negative number.
	#[cfg(feature = "std")]
	fn sqrt(self) -> Self;

	/// Returns `e^(self)`, (the exponential function).
	#[cfg(feature = "std")]
	fn exp(self) -> Self;

	/// Returns `2^(self)`.
	#[cfg(feature = "std")]
	fn exp2(self) -> Self;

	/// Returns the natural logarithm of the number.
	#[cfg(feature = "std")]
	fn ln(self) -> Self;

	/// Returns the logarithm of the number with respect to an arbitrary base.
	///
	/// The result may not be correctly rounded owing to implementation details;
	/// `self.log2()` can produce more accurate results for base 2, and
	/// `self.log10()` can produce more accurate results for base 10.
	#[cfg(feature = "std")]
	fn log(self, base: Self) -> Self;

	/// Returns the base 2 logarithm of the number.
	#[cfg(feature = "std")]
	fn log2(self) -> Self;

	/// Returns the base 10 logarithm of the number.
	#[cfg(feature = "std")]
	fn log10(self) -> Self;

	/// Returns the cubic root of a number.
	#[cfg(feature = "std")]
	fn cbrt(self) -> Self;

	/// Computes the sine of a number (in radians).
	#[cfg(feature = "std")]
	fn hypot(self, other: Self) -> Self;

	/// Computes the sine of a number (in radians).
	#[cfg(feature = "std")]
	fn sin(self) -> Self;

	/// Computes the cosine of a number (in radians).
	#[cfg(feature = "std")]
	fn cos(self) -> Self;

	/// Computes the tangent of a number (in radians).
	#[cfg(feature = "std")]
	fn tan(self) -> Self;

	/// Computes the arcsine of a number. Return value is in radians in the
	/// range [-pi/2, pi/2] or NaN if the number is outside the range [-1, 1].
	#[cfg(feature = "std")]
	fn asin(self) -> Self;

	/// Computes the arccosine of a number. Return value is in radians in the
	/// range [0, pi] or NaN if the number is outside the range [-1, 1].
	#[cfg(feature = "std")]
	fn acos(self) -> Self;

	/// Computes the arctangent of a number. Return value is in radians in the
	/// range [-pi/2, pi/2];
	#[cfg(feature = "std")]
	fn atan(self) -> Self;

	/// Computes the four quadrant arctangent of `self` (`y`) and `other` (`x`)
	/// in radians.
	///
	/// - `x = 0`, `y = 0`: `0`
	/// - `x >= 0`: `arctan(y/x)` -> `[-pi/2, pi/2]`
	/// - `y >= 0`: `arctan(y/x) + pi` -> `(pi/2, pi]`
	/// - `y < 0`: `arctan(y/x) - pi` -> `(-pi, -pi/2)`
	#[cfg(feature = "std")]
	fn atan2(self, other: Self) -> Self;

	/// Simultaneously computes the sine and cosine of the number, `x`. Returns
	/// `(sin(x), cos(x))`.
	#[cfg(feature = "std")]
	fn sin_cos(self) -> (Self, Self);

	/// Returns `e^(self) - 1` in a way that is accurate even if the number is
	/// close to zero.
	#[cfg(feature = "std")]
	fn exp_m1(self) -> Self;

	/// Returns `ln(1+n)` (natural logarithm) more accurately than if the
	/// operations were performed separately.
	#[cfg(feature = "std")]
	fn ln_1p(self) -> Self;

	/// Hyperbolic sine function.
	#[cfg(feature = "std")]
	fn sinh(self) -> Self;

	/// Hyperbolic cosine function.
	#[cfg(feature = "std")]
	fn cosh(self) -> Self;

	/// Hyperbolic tangent function.
	#[cfg(feature = "std")]
	fn tanh(self) -> Self;

	/// Inverse hyperbolic sine function.
	#[cfg(feature = "std")]
	fn asinh(self) -> Self;

	/// Inverse hyperbolic cosine function.
	#[cfg(feature = "std")]
	fn acosh(self) -> Self;

	/// Inverse hyperbolic tangent function.
	#[cfg(feature = "std")]
	fn atanh(self) -> Self;

	/// Returns `true` if this value is `NaN`.
	fn is_nan(self) -> bool;

	/// Returns `true` if this value is positive infinity or negative infinity,
	/// and `false` otherwise.
	fn is_infinite(self) -> bool;

	/// Returns `true` if this number is neither infinite nor `NaN`.
	fn is_finite(self) -> bool;

	/// Returns `true` if the number is neither zero, infinite, [subnormal], or
	/// `NaN`.
	///
	/// [subnormal]: https://en.wixipedia.org/wiki/Denormal_number
	fn is_normal(self) -> bool;

	/// Returns the floating point category of the number. If only one property
	/// is going to be tested, it is generally faster to use the specific
	/// predicate instead.
	fn classify(self) -> FpCategory;

	/// Returns `true` if `self` has a positive sign, including `+0.0`, `NaN`s
	/// with positive sign bit and positive infinity.
	fn is_sign_positive(self) -> bool;

	/// Returns `true` if `self` has a negative sign, including `-0.0`, `NaN`s
	/// with negative sign bit and negative infinity.
	fn is_sign_negative(self) -> bool;

	/// Takes the reciprocal (inverse) of a number, `1/x`.
	fn recip(self) -> Self;

	/// Converts radians to degrees.
	fn to_degrees(self) -> Self;

	/// Converts degrees to radians.
	fn to_radians(self) -> Self;

	/// Returns the maximum of the two numbers.
	fn max(self, other: Self) -> Self;

	/// Returns the minimum of the two numbers.
	fn min(self, other: Self) -> Self;

	/// Raw transmutation to `u32`.
	///
	/// This is currently identical to `transmute::<f32, u32>(self)` on all
	/// platforms.
	///
	/// See `from_bits` for some discussion of the portability of this operation
	/// (there are almost no issues).
	///
	/// Note that this function is distinct from `as` casting, which attempts to
	/// preserve the *numeric* value, and not the bitwise value.
	fn to_bits(self) -> Self::Raw;

	/// Raw transmutation from `u32`.
	///
	/// This is currently identical to `transmute::<u32, f32>(v)` on all
	/// platforms. It turns out this is incredibly portable, for two reasons:
	///
	/// - Floats and Ints have the same endianness on all supported platforms.
	/// - IEEE-754 very precisely specifies the bit layout of floats.
	///
	/// However there is one caveat: prior to the 2008 version of IEEE-754, how
	/// to interpret the NaN signaling bit wasn't actually specified. Most
	/// platforms (notably x86 and ARM) picked the interpretation that was
	/// ultimately standardized in 2008, but some didn't (notably MIPS). As a
	/// result, all signaling NaNs on MIPS are quiet NaNs on x86, and
	/// vice-versa.
	///
	/// Rather than trying to preserve signaling-ness cross-platform, this
	/// implementation favors preserving the exact bits. This means that
	/// any payloads encoded in NaNs will be preserved even if the result of
	/// this method is sent over the network from an x86 machine to a MIPS one.
	///
	/// If the results of this method are only manipulated by the same
	/// architecture that produced them, then there is no portability concern.
	///
	/// If the input isn't NaN, then there is no portability concern.
	///
	/// If you don't care about signalingness (very likely), then there is no
	/// portability concern.
	///
	/// Note that this function is distinct from `as` casting, which attempts to
	/// preserve the *numeric* value, and not the bitwise value.
	fn from_bits(bits: Self::Raw) -> Self;
}

/// Declare that a type is exactly eight bits wide.
pub trait Is8: IsNumber {}

/// Declare that a type is exactly sixteen bits wide.
pub trait Is16: IsNumber {}

/// Declare that a type is exactly thirty-two bits wide.
pub trait Is32: IsNumber {}

/// Declare that a type is exactly sixty-four bits wide.
pub trait Is64: IsNumber {}

/// Declare that a type is exactly one hundred twenty-eight bits wide.
pub trait Is128: IsNumber {}

/// Declare that a type is eight or more bits wide.
pub trait AtLeast8: IsNumber {}

/// Declare that a type is sixteen or more bits wide.
pub trait AtLeast16: IsNumber {}

/// Declare that a type is thirty-two or more bits wide.
pub trait AtLeast32: IsNumber {}

/// Declare that a type is sixty-four or more bits wide.
pub trait AtLeast64: IsNumber {}

/// Declare that a type is one hundred twenty-eight or more bits wide.
pub trait AtLeast128: IsNumber {}

/// Declare that a type is eight or fewer bits wide.
pub trait AtMost8: IsNumber {}

/// Declare that a type is sixteen or fewer bits wide.
pub trait AtMost16: IsNumber {}

/// Declare that a type is thirty-two or fewer bits wide.
pub trait AtMost32: IsNumber {}

/// Declare that a type is sixty-four or fewer bits wide.
pub trait AtMost64: IsNumber {}

/// Declare that a type is one hundred twenty-eight or fewer bits wide.
pub trait AtMost128: IsNumber {}

macro_rules! func {
	( $name:ident ( self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		fn $name ( self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( self $(, $arg )* )}
	};
	( $name:ident ( &self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		fn $name ( &self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( &self $(, $arg )* )}
	};
	( $name:ident ( &mut self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		fn $name ( &mut self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( &mut self $(, $arg )* )}
	};
	( $name:ident ( $($arg:ident : $t:ty),* ) $( -> $ret:ty )? ) => {
		fn $name ( $($arg : $t ),* ) $( -> $ret )? { <Self>:: $name ( $( $arg ),* )}
	};
}

macro_rules! stdfunc {
	( $name:ident ( self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		#[cfg(feature = "std")]
		fn $name ( self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( self $(, $arg )* )}
	};
	( $name:ident ( &self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		#[cfg(feature = "std")]
		fn $name ( &self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( &self $(, $arg )* )}
	};
	( $name:ident ( &mut self $(, $arg:ident : $t:ty)* ) $( -> $ret:ty )? ) => {
		#[cfg(feature = "std")]
		fn $name ( &mut self $(, $arg : $t )* ) $( -> $ret )? { <Self>:: $name ( &mut self $(, $arg )* )}
	};
	( $name:ident ( $($arg:ident : $t:ty),* ) $( -> $ret:ty )? ) => {
		#[cfg(feature = "std")]
		fn $name ( $($arg : $t ),* ) $( -> $ret )? { <Self>:: $name ( $( $arg ),* )}
	};
}

macro_rules! impl_for {
	( IsNumber => $($t:ty),+ $(,)? ) => { $(
		impl IsNumber for $t {
			type Bytes = [u8; core::mem::size_of::<Self>()];

			func!(to_be_bytes(self) -> Self::Bytes);
			func!(to_le_bytes(self) -> Self::Bytes);
			func!(to_ne_bytes(self) -> Self::Bytes);
			func!(from_be_bytes(bytes: Self::Bytes) -> Self);
			func!(from_le_bytes(bytes: Self::Bytes) -> Self);
			func!(from_ne_bytes(bytes: Self::Bytes) -> Self);
		}
	)+ };
	( IsInteger => $($t:ty),+ $(,)? ) => { $(
		impl IsInteger for $t {
			const ZERO: Self = 0;
			const MIN: Self = <Self>::min_value();
			const MAX: Self = <Self>::max_value();

			func!(min_value() -> Self);
			func!(max_value() -> Self);
			func!(from_str_radix(src: &str, radix: u32) -> Result<Self, ParseIntError>);
			func!(count_ones(self) -> u32);
			func!(count_zeros(self) -> u32);
			func!(leading_zeros(self) -> u32);
			func!(trailing_zeros(self) -> u32);
			func!(leading_ones(self) -> u32);
			func!(trailing_ones(self) -> u32);
			func!(rotate_left(self, n: u32) -> Self);
			func!(rotate_right(self, n: u32) -> Self);
			func!(swap_bytes(self) -> Self);
			func!(reverse_bits(self) -> Self);
			func!(from_be(self) -> Self);
			func!(from_le(self) -> Self);
			func!(to_be(self) -> Self);
			func!(to_le(self) -> Self);
			func!(checked_add(self, rhs: Self) -> Option<Self>);
			func!(checked_sub(self, rhs: Self) -> Option<Self>);
			func!(checked_mul(self, rhs: Self) -> Option<Self>);
			func!(checked_div(self, rhs: Self) -> Option<Self>);
			func!(checked_div_euclid(self, rhs: Self) -> Option<Self>);
			func!(checked_rem(self, rhs: Self) -> Option<Self>);
			func!(checked_rem_euclid(self, rhs: Self) -> Option<Self>);
			func!(checked_neg(self) -> Option<Self>);
			func!(checked_shl(self, rhs: u32) -> Option<Self>);
			func!(checked_shr(self, rhs: u32) -> Option<Self>);
			func!(checked_pow(self, rhs: u32) -> Option<Self>);
			func!(saturating_add(self, rhs: Self) -> Self);
			func!(saturating_sub(self, rhs: Self) -> Self);
			func!(saturating_mul(self, rhs: Self) -> Self);
			func!(saturating_pow(self, rhs: u32) -> Self);
			func!(wrapping_add(self, rhs: Self) -> Self);
			func!(wrapping_sub(self, rhs: Self) -> Self);
			func!(wrapping_mul(self, rhs: Self) -> Self);
			func!(wrapping_div(self, rhs: Self) -> Self);
			func!(wrapping_div_euclid(self, rhs: Self) -> Self);
			func!(wrapping_rem(self, rhs: Self) -> Self);
			func!(wrapping_rem_euclid(self, rhs: Self) -> Self);
			func!(wrapping_neg(self) -> Self);
			func!(wrapping_shl(self, rhs: u32) -> Self);
			func!(wrapping_shr(self, rhs: u32) -> Self);
			func!(wrapping_pow(self, rhs: u32) -> Self);
			func!(overflowing_add(self, rhs: Self) -> (Self, bool));
			func!(overflowing_sub(self, rhs: Self) -> (Self, bool));
			func!(overflowing_mul(self, rhs: Self) -> (Self, bool));
			func!(overflowing_div(self, rhs: Self) -> (Self, bool));
			func!(overflowing_div_euclid(self, rhs: Self) -> (Self, bool));
			func!(overflowing_rem(self, rhs: Self) -> (Self, bool));
			func!(overflowing_rem_euclid(self, rhs: Self) -> (Self, bool));
			func!(overflowing_neg(self) -> (Self, bool));
			func!(overflowing_shl(self, rhs: u32) -> (Self, bool));
			func!(overflowing_shr(self, rhs: u32) -> (Self, bool));
			func!(overflowing_pow(self, rhs: u32) -> (Self, bool));
			func!(pow(self, rhs: u32) -> Self);
			func!(div_euclid(self, rhs: Self) -> Self);
			func!(rem_euclid(self, rhs: Self) -> Self);
		}
	)+ };
	( IsSigned => $($t:ty),+ $(,)? ) => { $(
		impl IsSigned for $t {
			func!(checked_abs(self) -> Option<Self>);
			func!(wrapping_abs(self) -> Self);
			func!(overflowing_abs(self) -> (Self, bool));
			func!(abs(self) -> Self);
			func!(signum(self) -> Self);
			func!(is_positive(self) -> bool);
			func!(is_negative(self) -> bool);
		}
	)+ };
	( IsUnsigned => $($t:ty),+ $(,)? ) => { $(
		impl IsUnsigned for $t {
			func!(is_power_of_two(self) -> bool);
			func!(next_power_of_two(self) -> Self);
			func!(checked_next_power_of_two(self) -> Option<Self>);
		}
	)+ };
	( IsFloat => $($t:ident | $u:ty),+ $(,)? ) => { $(
		impl IsFloat for $t {
			type Raw = $u;

			const RADIX: u32 = core::$t::RADIX;
			const MANTISSA_DIGITS: u32 = core::$t::MANTISSA_DIGITS;
			const DIGITS: u32 = core::$t::DIGITS;
			const EPSILON: Self = core::$t::EPSILON;
			const MIN: Self = core::$t::MIN;
			const MIN_POSITIVE: Self = core::$t::MIN_POSITIVE;
			const MAX: Self = core::$t::MAX;
			const MIN_EXP: i32 = core::$t::MIN_EXP;
			const MAX_EXP: i32 = core::$t::MAX_EXP;
			const MIN_10_EXP: i32 = core::$t::MIN_10_EXP;
			const MAX_10_EXP: i32 = core::$t::MAX_10_EXP;
			const NAN: Self = core::$t::NAN;
			const INFINITY: Self = core::$t::INFINITY;
			const NEG_INFINITY: Self = core::$t::NEG_INFINITY;

			const PI: Self = core::$t::consts::PI;
			const FRAC_PI_2: Self = core::$t::consts::FRAC_PI_2;
			const FRAC_PI_3: Self = core::$t::consts::FRAC_PI_3;
			const FRAC_PI_4: Self = core::$t::consts::FRAC_PI_4;
			const FRAC_PI_6: Self = core::$t::consts::FRAC_PI_6;
			const FRAC_PI_8: Self = core::$t::consts::FRAC_PI_8;
			const FRAC_1_PI: Self = core::$t::consts::FRAC_1_PI;
			const FRAC_2_PI: Self = core::$t::consts::FRAC_2_PI;
			const FRAC_2_SQRT_PI: Self = core::$t::consts::FRAC_2_SQRT_PI;
			const SQRT_2: Self = core::$t::consts::SQRT_2;
			const FRAC_1_SQRT_2: Self = core::$t::consts::FRAC_1_SQRT_2;
			const E: Self = core::$t::consts::E;
			const LOG2_E: Self = core::$t::consts::LOG2_E;
			const LOG10_E: Self = core::$t::consts::LOG10_E;
			const LN_2: Self = core::$t::consts::LN_2;
			const LN_10: Self = core::$t::consts::LN_10;

			stdfunc!(floor(self) -> Self);
			stdfunc!(ceil(self) -> Self);
			stdfunc!(round(self) -> Self);
			stdfunc!(trunc(self) -> Self);
			stdfunc!(fract(self) -> Self);
			stdfunc!(abs(self) -> Self);
			stdfunc!(signum(self) -> Self);
			stdfunc!(copysign(self, sign: Self) -> Self);
			stdfunc!(mul_add(self, a: Self, b: Self) -> Self);
			stdfunc!(div_euclid(self, rhs: Self) -> Self);
			stdfunc!(rem_euclid(self, rhs: Self) -> Self);
			stdfunc!(powi(self, n: i32) -> Self);
			stdfunc!(powf(self, n: Self) -> Self);
			stdfunc!(sqrt(self) -> Self);
			stdfunc!(exp(self) -> Self);
			stdfunc!(exp2(self) -> Self);
			stdfunc!(ln(self) -> Self);
			stdfunc!(log(self, base: Self) -> Self);
			stdfunc!(log2(self) -> Self);
			stdfunc!(log10(self) -> Self);
			stdfunc!(cbrt(self) -> Self);
			stdfunc!(hypot(self, other: Self) -> Self);
			stdfunc!(sin(self) -> Self);
			stdfunc!(cos(self) -> Self);
			stdfunc!(tan(self) -> Self);
			stdfunc!(asin(self) -> Self);
			stdfunc!(acos(self) -> Self);
			stdfunc!(atan(self) -> Self);
			stdfunc!(atan2(self, other: Self) -> Self);
			stdfunc!(sin_cos(self) -> (Self, Self));
			stdfunc!(exp_m1(self) -> Self);
			stdfunc!(ln_1p(self) -> Self);
			stdfunc!(sinh(self) -> Self);
			stdfunc!(cosh(self) -> Self);
			stdfunc!(tanh(self) -> Self);
			stdfunc!(asinh(self) -> Self);
			stdfunc!(acosh(self) -> Self);
			stdfunc!(atanh(self) -> Self);

			func!(is_nan(self) -> bool);
			func!(is_infinite(self) -> bool);
			func!(is_finite(self) -> bool);
			func!(is_normal(self) -> bool);
			func!(classify(self) -> FpCategory);
			func!(is_sign_positive(self) -> bool);
			func!(is_sign_negative(self) -> bool);
			func!(recip(self) -> Self);
			func!(to_degrees(self) -> Self);
			func!(to_radians(self) -> Self);
			func!(max(self, other: Self) -> Self);
			func!(min(self, other: Self) -> Self);
			func!(to_bits(self) -> Self::Raw);
			func!(from_bits(bits: Self::Raw) -> Self);
		}
	)+ };
	( $which:ty => $($t:ty),+ $(,)? ) => { $(
		impl $which for $t {}
	)+ };
}

impl_for!(IsNumber => i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize, f32, f64);
impl_for!(IsInteger => i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize);
impl_for!(IsSigned => i8, i16, i32, i64, i128, isize);
impl_for!(IsUnsigned => u8, u16, u32, u64, u128, usize);
impl_for!(IsFloat => f32 | u32, f64 | u64);

impl_for!(Is8 => i8, u8);
impl_for!(Is16 => i16, u16);
impl_for!(Is32 => i32, u32, f32);
impl_for!(Is64 => i64, u64, f64);
impl_for!(Is128 => i128, u128);

#[cfg(target_pointer_width = "16")]
impl_for!(Is16 => isize, usize);

#[cfg(target_pointer_width = "32")]
impl_for!(Is32 => isize, usize);

#[cfg(target_pointer_width = "64")]
impl_for!(Is64 => isize, usize);

impl_for!(AtLeast8 => i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize, f32, f64);
impl_for!(AtLeast16 => i16, i32, i64, i128, u16, u32, u64, u128, f32, f64);
impl_for!(AtLeast32 => i32, i64, i128, u32, u64, u128, f32, f64);
impl_for!(AtLeast64 => i64, i128, u64, u128, f64);
impl_for!(AtLeast128 => i128, u128);

#[cfg(any(
	target_pointer_width = "16",
	target_pointer_width = "32",
	target_pointer_width = "64"
))]
impl_for!(AtLeast16 => isize, usize);

#[cfg(any(target_pointer_width = "32", target_pointer_width = "64"))]
impl_for!(AtLeast32 => isize, usize);

#[cfg(target_pointer_width = "64")]
impl_for!(AtLeast64 => isize, usize);

impl_for!(AtMost8 => i8, u8);
impl_for!(AtMost16 => i8, i16, u8, u16);
impl_for!(AtMost32 => i8, i16, i32, u8, u16, u32, f32);
impl_for!(AtMost64 => i8, i16, i32, i64, isize, u8, u16, u32, u64, usize, f32, f64);
impl_for!(AtMost128 => i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize, f32, f64);

#[cfg(target_pointer_width = "16")]
impl_for!(AtMost16 => isize, usize);

#[cfg(any(target_pointer_width = "16", target_pointer_width = "32"))]
impl_for!(AtMost32 => isize, usize);

#[cfg(test)]
mod tests {
	use super::*;
	use static_assertions::*;

	assert_impl_all!(i8: IsInteger, IsSigned, Is8);
	assert_impl_all!(i16: IsInteger, IsSigned, Is16);
	assert_impl_all!(i32: IsInteger, IsSigned, Is32);
	assert_impl_all!(i64: IsInteger, IsSigned, Is64);
	assert_impl_all!(i128: IsInteger, IsSigned, Is128);
	assert_impl_all!(isize: IsInteger, IsSigned);

	assert_impl_all!(u8: IsInteger, IsUnsigned, Is8);
	assert_impl_all!(u16: IsInteger, IsUnsigned, Is16);
	assert_impl_all!(u32: IsInteger, IsUnsigned, Is32);
	assert_impl_all!(u64: IsInteger, IsUnsigned, Is64);
	assert_impl_all!(u128: IsInteger, IsUnsigned, Is128);
	assert_impl_all!(usize: IsInteger, IsUnsigned);

	assert_impl_all!(f32: IsFloat, Is32);
	assert_impl_all!(f64: IsFloat, Is64);
}