1: Software Calculation of IEEE Floating-Point Values, Intel Format

series crbx x value signexp sign exponent exp_biased exp_use
0 00000000000014C0 -5 5 C014 1 (-) 0401 1025 2
00000000000010C0 -4 4 C010 1 (-) 0401 1025 2
00000000000008C0 -3 3 C008 1 (-) 0400 1024 1
00000000000000C0 -2 2 C000 1 (-) 0400 1024 1
000000000000F0BF -1 1 BFF0 1 (-) 03FF 1023 0
0000000000000000 0 0 0000 0 (+) 0000 0 -1023
000000000000F03F 1 1 3FF0 0 (+) 03FF 1023 0
0000000000000040 2 2 4000 0 (+) 0400 1024 1
0000000000000840 3 3 4008 0 (+) 0400 1024 1
0000000000001040 4 4 4010 0 (+) 0401 1025 2
0000000000001440 5 5 4014 0 (+) 0401 1025 2

series crbx x value signexp sign exponent exp_biased exp_use
8 0000000000001840 6 6 4018 0 (+) 0401 1025 2
0000000000001C40 7 7 401C 0 (+) 0401 1025 2
0000000000002040 8 8 4020 0 (+) 0402 1026 3
0000000000002240 9 9 4022 0 (+) 0402 1026 3
0000000000002440 10 10 4024 0 (+) 0402 1026 3

series crbx x value signexp sign exponent exp_biased exp_use
16 0000000000002C40 14 14 402C 0 (+) 0402 1026 3
0000000000002E40 15 15 402E 0 (+) 0402 1026 3
0000000000003040 16 16 4030 0 (+) 0403 1027 4
0000000000003140 17 17 4031 0 (+) 0403 1027 4
0000000000003240 18 18 4032 0 (+) 0403 1027 4

series crbx x value signexp sign exponent exp_biased exp_use
32 0000000000003E40 30 30 403E 0 (+) 0403 1027 4
0000000000003F40 31 31 403F 0 (+) 0403 1027 4
0000000000004040 32 32 4040 0 (+) 0404 1028 5
0000000000804040 33 33 4040 0 (+) 0404 1028 5
0000000000004140 34 34 4041 0 (+) 0404 1028 5

series crbx x value signexp sign exponent exp_biased exp_use
64 0000000000004F40 62 62 404F 0 (+) 0404 1028 5
0000000000804F40 63 63 404F 0 (+) 0404 1028 5
0000000000005040 64 64 4050 0 (+) 0405 1029 6
0000000000405040 65 65 4050 0 (+) 0405 1029 6
0000000000805040 66 66 4050 0 (+) 0405 1029 6

series crbx x value signexp sign exponent exp_biased exp_use
128 0000000000805F40 126 126 405F 0 (+) 0405 1029 6
0000000000C05F40 127 127 405F 0 (+) 0405 1029 6
0000000000006040 128 128 4060 0 (+) 0406 1030 7
0000000000206040 129 129 4060 0 (+) 0406 1030 7
0000000000406040 130 130 4060 0 (+) 0406 1030 7

series crbx x value signexp sign exponent exp_biased exp_use
256 0000000000C06F40 254 254 406F 0 (+) 0406 1030 7
0000000000E06F40 255 255 406F 0 (+) 0406 1030 7
0000000000007040 256 256 4070 0 (+) 0407 1031 8
0000000000107040 257 257 4070 0 (+) 0407 1031 8
0000000000207040 258 258 4070 0 (+) 0407 1031 8

series crbx x value signexp sign exponent exp_biased exp_use
512 0000000000F07F40 511 511 407F 0 (+) 0407 1031 8
0000000000008040 512 512 4080 0 (+) 0408 1032 9
0000000000088040 513 513 4080 0 (+) 0408 1032 9

series crbx x value signexp sign exponent exp_biased exp_use
1024 0000000000F88F40 1,023 1,023 408F 0 (+) 0408 1032 9
0000000000009040 1,024 1,024 4090 0 (+) 0409 1033 10
0000000000049040 1,025 1,025 4090 0 (+) 0409 1033 10

series crbx x value signexp sign exponent exp_biased exp_use
2048 0000000000FC9F40 2,047 2,047 409F 0 (+) 0409 1033 10
000000000000A040 2,048 2,048 40A0 0 (+) 040A 1034 11
000000000002A040 2,049 2,049 40A0 0 (+) 040A 1034 11

series crbx x value signexp sign exponent exp_biased exp_use
4096 0000000000FEAF40 4,095 4,095 40AF 0 (+) 040A 1034 11
000000000000B040 4,096 4,096 40B0 0 (+) 040B 1035 12
000000000001B040 4,097 4,097 40B0 0 (+) 040B 1035 12

series crbx x value signexp sign exponent exp_biased exp_use
8192 0000000000FFBF40 8,191 8,191 40BF 0 (+) 040B 1035 12
000000000000C040 8,192 8,192 40C0 0 (+) 040C 1036 13
000000008000C040 8,193 8,193 40C0 0 (+) 040C 1036 13

series crbx x value signexp sign exponent exp_biased exp_use
16384 0000000080FFCF40 16,383 16,383 40CF 0 (+) 040C 1036 13
000000000000D040 16,384 16,384 40D0 0 (+) 040D 1037 14
000000004000D040 16,385 16,385 40D0 0 (+) 040D 1037 14

series crbx x value signexp sign exponent exp_biased exp_use
32768 00000000C0FFDF40 32,767 32,767 40DF 0 (+) 040D 1037 14
000000000000E040 32,768 32,768 40E0 0 (+) 040E 1038 15
000000002000E040 32,769 32,769 40E0 0 (+) 040E 1038 15

series crbx x value signexp sign exponent exp_biased exp_use
65536 00000000E0FFEF40 65,535 65,535 40EF 0 (+) 040E 1038 15
000000000000F040 65,536 65,536 40F0 0 (+) 040F 1039 16
000000001000F040 65,537 65,537 40F0 0 (+) 040F 1039 16

series crbx x value signexp sign exponent exp_biased exp_use
131072 00000000F0FFFF40 131,071 131,071 40FF 0 (+) 040F 1039 16
0000000000000041 131,072 131,072 4100 0 (+) 0410 1040 17
0000000008000041 131,073 131,073 4100 0 (+) 0410 1040 17

series crbx x value signexp sign exponent exp_biased exp_use
262144 00000000F8FF0F41 262,143 262,143 410F 0 (+) 0410 1040 17
0000000000001041 262,144 262,144 4110 0 (+) 0411 1041 18
0000000004001041 262,145 262,145 4110 0 (+) 0411 1041 18

series crbx x value signexp sign exponent exp_biased exp_use
524288 00000000FCFF1F41 524,287 524,287 411F 0 (+) 0411 1041 18
0000000000002041 524,288 524,288 4120 0 (+) 0412 1042 19
0000000002002041 524,289 524,289 4120 0 (+) 0412 1042 19

series crbx x value signexp sign exponent exp_biased exp_use
1048576 00000000FEFF2F41 1,048,575 1,048,575 412F 0 (+) 0412 1042 19
0000000000003041 1,048,576 1,048,576 4130 0 (+) 0413 1043 20
0000000001003041 1,048,577 1,048,577 4130 0 (+) 0413 1043 20

series crbx x value signexp sign exponent exp_biased exp_use
2097152 00000000FFFF3F41 2,097,151 2,097,151 413F 0 (+) 0413 1043 20
0000000000004041 2,097,152 2,097,152 4140 0 (+) 0414 1044 21
0000008000004041 2,097,153 2,097,153 4140 0 (+) 0414 1044 21

series crbx x value signexp sign exponent exp_biased exp_use
4194304 00000080FFFF4F41 4,194,303 4,194,303 414F 0 (+) 0414 1044 21
0000000000005041 4,194,304 4,194,304 4150 0 (+) 0415 1045 22
0000004000005041 4,194,305 4,194,305 4150 0 (+) 0415 1045 22

series crbx x value signexp sign exponent exp_biased exp_use
8388608 000000C0FFFF5F41 8,388,607 8,388,607 415F 0 (+) 0415 1045 22
0000000000006041 8,388,608 8,388,608 4160 0 (+) 0416 1046 23
0000002000006041 8,388,609 8,388,609 4160 0 (+) 0416 1046 23

2: The Two Original Values and Their Difference

Obs x crbx y crby d crbd signexp exponent exp_biased exp_use
1 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 0.01562500 000000000000903F 3F90 03F9 1017 -6

3: Values Immediately Adjacent to the Customer's Integer

Obs x crbx y crby lsb d lagd
1 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B165E7E555BDDB42 10110001 -0.23437500 .
2 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B265E7E555BDDB42 10110010 -0.21875000 0.015625
3 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B365E7E555BDDB42 10110011 -0.20312500 0.015625
4 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B465E7E555BDDB42 10110100 -0.18750000 0.015625
5 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B565E7E555BDDB42 10110101 -0.17187500 0.015625
6 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B665E7E555BDDB42 10110110 -0.15625000 0.015625
7 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B765E7E555BDDB42 10110111 -0.14062500 0.015625
8 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B865E7E555BDDB42 10111000 -0.12500000 0.015625
9 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 B965E7E555BDDB42 10111001 -0.10937500 0.015625
10 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 BA65E7E555BDDB42 10111010 -0.09375000 0.015625
11 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 BB65E7E555BDDB42 10111011 -0.07812500 0.015625
12 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 BC65E7E555BDDB42 10111100 -0.06250000 0.015625
13 122000015596951.000000 C065E7E555BDDB42 122000015596950.000000 BD65E7E555BDDB42 10111101 -0.04687500 0.015625
14 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 BE65E7E555BDDB42 10111110 -0.03125000 0.015625
15 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 BF65E7E555BDDB42 10111111 -0.01562500 0.015625
16 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C065E7E555BDDB42 11000000 0.00000000 0.015625
17 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 11000001 0.01562500 0.015625
18 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C265E7E555BDDB42 11000010 0.03125000 0.015625
19 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C365E7E555BDDB42 11000011 0.04687500 0.015625
20 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C465E7E555BDDB42 11000100 0.06250000 0.015625
21 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C565E7E555BDDB42 11000101 0.07812500 0.015625
22 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C665E7E555BDDB42 11000110 0.09375000 0.015625
23 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C765E7E555BDDB42 11000111 0.10937500 0.015625
24 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C865E7E555BDDB42 11001000 0.12500000 0.015625
25 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C965E7E555BDDB42 11001001 0.14062500 0.015625
26 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CA65E7E555BDDB42 11001010 0.15625000 0.015625
27 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CB65E7E555BDDB42 11001011 0.17187500 0.015625
28 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CC65E7E555BDDB42 11001100 0.18750000 0.015625
29 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CD65E7E555BDDB42 11001101 0.20312500 0.015625
30 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CE65E7E555BDDB42 11001110 0.21875000 0.015625
31 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 CF65E7E555BDDB42 11001111 0.23437500 0.015625

4: Incrementing the Least-Significant of 15 Significant Decimal Digits

Obs y crby ls_12bits_hex ls_12bits_bin d_10 exp_use Lowest 12 Bits
Decoded
1 122000015596946.000000 8064E7E555BDDB42 480 010010000000 . 46 18.000
2 122000015596947.000000 C064E7E555BDDB42 4C0 010011000000 64 46 19.000
3 122000015596948.000000 0065E7E555BDDB42 500 010100000000 64 46 20.000
4 122000015596949.000000 4065E7E555BDDB42 540 010101000000 64 46 21.000
5 122000015596950.000000 8065E7E555BDDB42 580 010110000000 64 46 22.000
6 122000015596951.000000 C065E7E555BDDB42 5C0 010111000000 64 46 23.000
7 122000015596952.000000 0066E7E555BDDB42 600 011000000000 64 46 24.000
8 122000015596953.000000 4066E7E555BDDB42 640 011001000000 64 46 25.000
9 122000015596954.000000 8066E7E555BDDB42 680 011010000000 64 46 26.000
10 122000015596955.000000 C066E7E555BDDB42 6C0 011011000000 64 46 27.000
11 122000015596956.000000 0067E7E555BDDB42 700 011100000000 64 46 28.000

5: Results of Miscellaneous Math Operations

operation x crbx y crby w crbw d
Divide 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 1.0000000 010000000000F03F 2.22045E-16
Multiply 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 1.4884003805656E28 E8CC8833E10BC845 2.19902E+12
Add 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 244000031193902.000000 C065E7E555BDEB42 0.00000E+00
Fuzz 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 1.56250E-02
Round 122000015596951.000000 C065E7E555BDDB42 122000015596951.000000 C165E7E555BDDB42 122000015596951.000000 C065E7E555BDDB42 0.00000E+00