Decimal to binary conversion

[Fred Tydeman]

Subject: Decimal to binary conversion for IEEE 754/854 double format

What is a 'hard' decimal number to convert to IEEE double format with
its 53-bits of fraction? How many extra bits are required to get a
correctly (directed) rounded result?

One such 'hard' 17 digit number is: 26153245263757307e49

It requires 66 bits beyond the 53 bits of IEEE double to do all
directed rounding modes.  Its fraction value in hexadecimal is:
     (1)3DE005BD620DE-FFFFFFFFFFFFFFFFB4C6953FA1 
with (1) being the hidden bit. 

Tim Peters in March, 1991, posted to numeric-interest several articles
on finding decimal numbers that are 'hard' to convert to IEEE-754
double (a sign bit, 11 exponent bits, a hidden bit, and 52 bits of the
fraction).  There are hard in that they require many additional bits
beyond the 53 kept to determine the correctly rounded value and the
setting (or not) of the inexact flag. 

I do not recall if anyone else posted any other such tables or their
results in trying to duplicate finding 'hard' decimal numbers. 

Tim also posted hints on how to write a program to generate 'hard'
numbers to convert.  I wrote such a program based upon those hints and
the following tables are its output.  These results mostly match the
results posted by Tim.  Some of these (indicated by a '!' character)
require more extra bits than what Tim posted.  In one case, Tim found
a number with more extra bits than found by this program.  Tim's hints
referred to Knuth Volume 2 (2nd edition).  My copy of Knuth is in
Texas and I am temporarily living in Germany, so I was unable to refer
to it.  Tim said that Knuth gave hints on how to solve the modulo
equation when the continued fraction approach fails.  I know that my
program found many cases where the continued fraction approach failed
(so did not find a decimal value in a given binade), so I am not
surprised that Tim's list and my list do not agree. 

I kept track of 8 cases of numbers.  The eight came from looking at
the last bit kept and the first two extra bits (bits 53, 54, and 55 if
the implicit bit is bit 1).  These allow for all cases of odd/even,
just above/below exact or 1/2 way point.  All 8 cases would be needed
to do correct directed rounding (nearest, +infinity, -infinity). 

Each of the following groups is made up of 16 lines: 2 lines for each
of the 8 cases.  The 2 lines are two numbers with the largest extra
bits required for correct conversion found by the program. 
 
Each line has the following parts:

  Number of bits, beyond the 53 kept, that are needed to determine the
  correct (directed) rounding. 

  The decimal number being converted. 

  The 53-bits, with the hidden bit made explicit, followed by a '-',
  followed by enough extra bits to show the run.  These are hexadecimal
  digits.

================ 1 decimal digit numbers follow
 11 9e0306 19A2028368022E-003382B4FC287700CF95357446
  8 4e-079 17B6D71D20B96C-01EA801D30F77839241054A25F
 11 7e-261 1B8D7E32BE6396-7FC04E77F5640ACC6A6C4AC739
 10 6e-025 17361CB863DE62-7F9D7C5CC9C45EFB306D21D4E9
 11 7e-161 1F7E0DB3799AA2-802FBCB2A54B58C9158C61CC5B
 11 7e0289 1CBB547777A284-8033CAEF6A534FE3B571EAB93E
  9 5e0079 1AFCEF51F0FB5E-FF7B866E8BD92F7D20C6531D11
  9 1e0080 1AFCEF51F0FB5E-FF7B866E8BD92F7D20C6531D11
 11 7e-303 13339131C46F8B-0024DDEA531260F883CE3B6834
 10 5e0152 1317E5EF3AB327-005DE5EEE7FF7B2F4C3C76D96E
 13 5e0125 17A2ECC414A03F-7FF6CA1CB527787B130A97D51E
 13 2e0126 17A2ECC414A03F-7FF6CA1CB527787B130A97D51E
 11 7e-141 15570F59BD178B-8027EC1004BABE9522BFC7E97B
  9 4e-192 123FF06EEA8479-80CF6C60D468C4FBB9E38B4812
 11 9e0043 102498EA6DF0C3-FFC9B5066A68A0000000000000
 10 1e0303 1754E31CD072D9-FFBA60AC04B1EA9CD757CC87E0
 
================ 2 decimal digit numbers follow
 14 95e-089 1E3CBC9907FDC8-0007C06B97C8BB51F5912738BB
 12 85e0194 1D1C26DB7D0DAE-0014E06256CB9186970DA0525F
 17 69e0267 10B7CB60C994DA-7FFF6E455DD11B7956F4E42065
 12 97e-019 165DDE8E688BA6-7FE5577A67D47E3FD2341B2145
 15 37e0046 1033D7ECA0ADEE-8002654DD40602D00000000000
 15 74e0046 1033D7ECA0ADEE-8002654DD40602D00000000000
 14 61e-099 10AD836F269A16-FFF97988E51EB03639C93B5397
 12 53e-208 1B39AE1909C31A-FFEDB8DD27C8A0294CA9FE0673
 16 93e-234 127B2E4F210075-000166B54D4518313B205BCB2F
 13 79e-095 1A5DB812948281-000891C447878E75DB6FC2B324
 15 87e-274 12D36CF48E7ABD-7FFC3577544385C3E3F65554B8
 13 83e0025 15747AB143E353-7FF00000000000000000000000
 16 17e-036 1698CCDC600159-8001786B91DE65B8B11135BD43
 13 53e0033 146A3418629EF5-800FB224000000000000000000
 16 51e-074 1CD5BEE57763E5-FFFE6AA23EA3D02B7EC56960E4
 11 63e-022 1DC03B8FD70169-FFD1BB066E1CD23DB571E0FD06
 
================ 3 decimal digit numbers follow
 15 839e0143 1AE03F245703E2-0003A1AD0365D37714C9F86789
 15 749e-182 1F14727744C63E-00020D99183B429E891AD205BA
 20 999e-026 182782AFE1869E-7FFFE236F529F2D6FE64C4018F
 17 345e0266 10B7CB60C994DA-7FFF6E455DD11B7956F4E42065
 18 914e-102 1FFC81BC29F02A-800060A5DBD2C74C198CF47399
 16 829e0102 17221A79CDD1D8-8001B8193E42F3B62DD12E40C9
 17 307e0090 12D6B183FE4B54-FFFF29968182D3A0DD00D2E267
 17 859e0182 14374D8B87AC62-FFFF088A8857AA402A2DF72C49
 17 283e0085 16C309024BAB4B-0000D51A3CE10CBD543022A5D3
 17 589e0187 1526BE9C22EB17-0000E3765D06DC54C15FD5361A
 19 302e0176 1DCF7DF8F573B7-7FFFC0F867814FE6D6935667C7
 19 604e0176 1DCF7DF8F573B7-7FFFC0F867814FE6D6935667C7
 18 761e-244 103CEA3586452D-80006C0F0C04B3847A36DD444E
 17 647e0230 14D64D4079150B-8000F425D15F6DDF49B8AA8D22
 19 755e0174 17D93193F78FC5-FFFFCD9386010CB8ABA911EC9F
 16 255e-075 1CD5BEE57763E5-FFFE6AA23EA3D02B7EC56960E4
 
================ 4 decimal digit numbers follow
 19 3391e0055 159BD3AD46E346-000032D4D02F50617529A60000
 18 4147e-015 123D1B5EB1D778-00006FDF841403054A0D77C8F0
 20 3996e-026 182782AFE1869E-7FFFE236F529F2D6FE64C4018F
 20 1998e-026 182782AFE1869E-7FFFE236F529F2D6FE64C4018F
 22 3338e-296 15519AC5142AAA-8000070F02115D6BE09D8A395B
 22 1669e-296 15519AC5142AAA-8000070F02115D6BE09D8A395B
 22 8699e-276 12D2DF246ECD2C-FFFFF9C9169BAC447C637182B4
 20 5311e0243 184E91F4AA0FDA-FFFFEB2A65BD57B3AA940E0872
 19 7903e-096 107C2D27A5B989-00002C504015BF546C4216EB1A
 19 7611e-226 119B8744033457-000023A370FADFF65AB410CDD1
 20 3257e0058 1444B34A6FB3EB-7FFFE25F5EFFC58F5CA374C400
 20 6514e0058 1444B34A6FB3EB-7FFFE25F5EFFC58F5CA374C400
 24 3571e0263 162644C61D41A9-800001716F10BC2D7AFD9675AE
 24 7142e0263 162644C61D41A9-800001716F10BC2D7AFD9675AE
 20 5311e0242 13720E5D54D97B-FFFFEF551E3112F622100B39F5
 19 1617e-063 14C98FCE16152D-FFFFD30F3A06FB744A8026626F
 
================ 5 decimal digit numbers follow
 21 51881e0037 17D2950DC76DA4-00000951493E80000000000000
 21 31441e-118 1EF890F5DE4C86-0000084147B42C6D79EFF7F0DE
 24 30179e0079 13E272A77478E8-7FFFFE5AC31A06DBACB11FBFF9
 24 60358e0079 13E272A77478E8-7FFFFE5AC31A06DBACB11FBFF9
 22 63876e-020 1703856844BDBE-8000049505EB1E788782724827
 22 31938e-020 1703856844BDBE-8000049505EB1E788782724827
 23 46073e-032 12405B773FBDF2-FFFFFDEA4E8D3BEC45246BB7E7
 22 32941e0051 157EB8AD52A5C8-FFFFFB26AB424A577CDC800000
 27 82081e0041 170105DF3D47CB-00000027ABDB40A00000000000
 25 38701e-215 10492A4A8A37FD-000000D6B0BB845975831C3209
 24 62745e0047 10C52FE6DC6A1B-7FFFFE410E649A76A280000000
 24 12549e0048 10C52FE6DC6A1B-7FFFFE410E649A76A280000000
 25 64009e-183 1099B393B84831-800000BD34BA9A27DF71468016
 24 89275e0261 162644C61D41A9-800001716F10BC2D7AFD9675AE
 23 75859e0025 132645E1BA93EF-FFFFFC00000000000000000000
 22 57533e0287 1272ED2307F569-FFFFFA01B0DA4165D57CCE9E44
 
================ 6 decimal digit numbers follow
 29 584169e0229 1D657059DC79AA-0000000AA1BEC504C0C965CD00
 26 940189e-112 1B99D6240C1A28-0000007B902E20D88FF17FFDB9
 28 416121e0197 10FD07ED297F80-7FFFFFE440EAFA83E3A5DA9CE4
 28 832242e0197 10FD07ED297F80-7FFFFFE440EAFA83E3A5DA9CE4
 27 584738e0076 18A85EB277E644-8000002EE9B7876D6EE6FDF044
 27 933587e-140 1B248728B9C116-800000280B8CD1179005207818
 28 252601e0121 1DDA592E398DD6-FFFFFFEE82BC3B2AB61E2A94C9
 26 358423e0274 19463B59B8F2BC-FFFFFFB7BA154A4F173DA8B7F2
 28 892771e-213 125818C7294F27-0000001911FF9B80B78FA49CD6
 27 410405e0040 170105DF3D47CB-00000027ABDB40A00000000000
 30 928609e-261 1BE2DD66200BEF-7FFFFFF8CEA72B6BDFF2542D7E
 28 302276e-254 15A462D91C6AB3-7FFFFFE93298612B6DB60E4DC2
 30 920657e-023 153A9985DBDE6B-8000000638E097FE0941EECD53
 26 609019e-025 11F99E11EA0A23-800000629219A926DEAD2F45A9
 29 252601e0120 17E1E0F1C7A4AB-FFFFFFF202302F555E7E887707
 27 654839e-060 100E7DB3B3F241-FFFFFFCCDD52680319AD306D3D
 
================ 7 decimal digit numbers follow
 32 8823691e0130 1E597C0B94B7AE-00000001A66459030F6383FB12
 29 2920845e0228 1D657059DC79AA-0000000AA1BEC504C0C965CD00
 31 9210917e0080 1DA232347E6032-7FFFFFFDA6BA75E9946EEF4C06
 29 5800419e-303 1E58FFA48F4FCE-7FFFFFF29428F374F93E80FE49
 29 6119898e-243 13ECF22EA07862-8000000EB3412D84B5DFE132B6
 29 3059949e-243 13ECF22EA07862-8000000EB3412D84B5DFE132B6
 30 2572231e0223 10F73BE1DFF9AC-FFFFFFF87D805A44F1C14EF45B
 29 5444097e-021 18849DD33C95AE-FFFFFFF4F68F85A3D0BB3A0F7D
!33 5783893e-127 17E5902CE0E151-00000000DFBB53C98FB0925372
 29 3865421e-225 15D4FE53AFEC65-0000000F24C7A54FBE1BCACA3D
 30 4590831e0156 14689B4A5FA201-7FFFFFF846CBE66ADEEB6FB1C1
 30 9181662e0156 14689B4A5FA201-7FFFFFF846CBE66ADEEB6FB1C1
 29 5906361e-027 1BE45A312D089F-8000000BE83F39F4CD13A43290
 28 7315057e0235 15F0D408362A71-80000018D1C85D2DD5E127AAC9
 29 9088115e0106 12E51A84A3C69F-FFFFFFF68C805750AF70E070BD
 29 1817623e0107 12E51A84A3C69F-FFFFFFF68C805750AF70E070BD
 
================ 8 decimal digit numbers follow
 32 44118455e0129 1E597C0B94B7AE-00000001A66459030F6383FB12
 32 35282041e0293 1512D5273E62E8-000000016F61DC440953C68E59
 33 31279898e-291 129B01B6885D36-7FFFFFFF31062D23C880D01D7B
 33 15639949e-291 129B01B6885D36-7FFFFFFF31062D23C880D01D7B
 34 27966061e0145 155BCF72FD10F8-80000000665E4AFC2573450D41
 34 55932122e0145 155BCF72FD10F8-80000000665E4AFC2573450D41
 34 70176353e-053 100683A21DE854-FFFFFFFFACA567A0A60C76A6E7
 33 40277543e-032 1F29CA0FF893B0-FFFFFFFF1BF6BFF47B9237A08F
 33 50609263e0157 1193AFF1F1C8E3-00000000A83DFAFCDE29EA5C1D
 32 66094077e0077 1B37C4B7928317-0000000192B53E4B2B85F96105
 34 84863171e0114 106E98F5EC8F37-7FFFFFFF9F7823B0A617613BE1
 34 89396333e0264 1526F061CA9053-7FFFFFFF845874089815ABEAEA
 37 87575437e-309 1E07320602056B-800000000F66BD57EA0F28581D
 31 78693511e-044 10BC7B7603A2C9-80000003FAE3B73DFF946E1160
 32 90285923e-206 11470083F89D47-FFFFFFFE0573B23C37E8132CA2
 31 30155207e-030 123A4AD20748A1-FFFFFFFC258B818C61DB9DCE79
 
================ 9 decimal digit numbers follow
 41 245540327e0121 1C569E968E0944-0000000000A0E8E36D022848A8
 37 263125459e0287 14997A298B2F2E-000000000C14D92837E0BF4D79
 37 566446538e-257 14472BA9550E86-7FFFFFFFF2B23A422A94621062
 37 283223269e-257 14472BA9550E86-7FFFFFFFF2B23A422A94621062
 42 245540327e0122 11B6231E18C5CA-800000000064918E2421592D69
 42 491080654e0122 11B6231E18C5CA-800000000064918E2421592D69
 39 971212611e-126 1397D3C9745D2E-FFFFFFFFFDCBAADFA9DDF044B4
 37 229058583e0052 16CE94FEBDC7A4-FFFFFFFFF352FAE6901142E700
 36 325270231e0039 1C7CCF90C9F8AB-0000000012839A52C000000000
 35 989648089e-035 1880A3D515E849-0000000027173658732FDE9CD1
 40 653777767e0273 120223F2B3A881-7FFFFFFFFE599A5E48EE62B98F
 36 923091487e0209 10BC60E6896717-7FFFFFFFECDD8AF7A44F10AFB5
 37 526250918e0288 19BFD8B3EDFAF9-800000000F1A0F7245D8EF20D7
 37 350301748e-309 1E07320602056B-800000000F66BD57EA0F28581D
 36 741111169e-203 114FE7DAF8F3AD-FFFFFFFFE023B8F95BD69E9E61
 36 667284113e-240 109355F8050C01-FFFFFFFFE9E9D22838EB2F5511
 
================ 10 decimal digit numbers follow
 41 1227701635e0120 1C569E968E0944-0000000000A0E8E36D022848A8
 40 9981396317e-182 18AFE10A2A66AA-0000000001A0B11F8941A5559F
 41 5232604057e-298 1465B896C24520-7FFFFFFFFF67C0F1F395AAACA9
 40 5572170023e-088 10847822F765B2-7FFFFFFFFE16F8297BF496FBD5
 42 1964322616e0122 11B6231E18C5CA-800000000064918E2421592D69
 42 3928645232e0122 11B6231E18C5CA-800000000064918E2421592D69
 41 8715380633e-058 14614C3219891E-FFFFFFFFFF48B91C2C93B04164
 39 4856063055e-127 1397D3C9745D2E-FFFFFFFFFDCBAADFA9DDF044B4
 39 8336960483e-153 1A06A1024B95E1-00000000031EACECE86024ADDC
 38 1007046393e-155 101891FC4717FD-00000000073470E60FEAC84D1E
 41 5378822089e-176 195FD4C88D4B1B-7FFFFFFFFF39C7BE53FED26575
 41 5981342308e-190 13DB11AC608107-7FFFFFFFFF118B3E9A37A36984
 40 7214782613e-086 10B552D2EDCDE9-8000000001269B72BBCD857FAB
 40 5458466829e0142 10ACDE6A98EB4B-80000000014A36A3189A4F3161
!45 9078555839e-109 1FC575867314ED-FFFFFFFFFFF542E995DAB0415A
 41 6418488827e0079 1021F14ED7B3F9-FFFFFFFFFF11EBF78695DBE32E
 
================ 11 decimal digit numbers follow
 42 65325840981e0069 11A151DDBD3C4A-0000000000795A9803498E1A30
 42 49573485983e0089 1221BD871D2CF4-000000000064CD7E1B61FCDA9A
 41 46275205733e0074 130E6C7D4E3480-7FFFFFFFFF25EC70559AF4AF96
 41 92550411466e0074 130E6C7D4E3480-7FFFFFFFFF25EC70559AF4AF96
 43 41129842097e-202 12C3E72D179606-80000000003CB551015D795288
 42 93227267727e-049 1960FE08D5847E-80000000004AD264B3137202BC
 43 41297294357e0185 1C49437FCCFADA-FFFFFFFFFFC743C79E4ABDB56E
 42 41534892987e-067 1A12666477886C-FFFFFFFFFF8DC4AD10FE4A0B59
 45 42333842451e0201 10189A26DF575F-00000000000926D47ED2954FB0
 44 78564021519e-227 1155515FD37265-00000000001BABBB3285F46D4C
 45 53587107423e-061 100A19A3FFD981-7FFFFFFFFFF0F1837B84C2F664
 43 53827010643e-200 132FA69A69BD6D-7FFFFFFFFFD3239D18933DFC1E
 45 83356057653e0193 1544E6DAEE2A17-80000000000808878EDE80A57C
 43 45256834646e-118 1541ECDFD48693-800000000023C3453802336B07
 45 45392779195e-110 1FC575867314ED-FFFFFFFFFFF542E995DAB0415A
 44 23934638219e0291 11DEAF40EAD99F-FFFFFFFFFFE569C565C300423F
 
================ 12 decimal digit numbers follow
 44 995779191233e0113 12D44EDCC51304-000000000018BEE0D3D939A578
 44 997422852243e-265 176688FAEE99BC-00000000001C8F45D10A2860E4
 47 653532977297e-123 1925A0AABCDC68-7FFFFFFFFFFC87DCA09275F00A
 45 938885684947e0147 111894B202E9F4-7FFFFFFFFFF4D789A5F47C97FB
 49 619534293513e0124 110C20303FE0F0-800000000000979AF6FEF0B57C
 45 539879452414e-042 15E66DC3D6BDB4-80000000000895CB9EB4173259
 47 742522891517e0259 1C1C352FC3C308-FFFFFFFFFFFCC3CF9BBDEF3E9B
 44 254901016865e-022 1C06D366394440-FFFFFFFFFFE0B6DDCF57CEC457
 48 685763015669e0280 15FD7AA44D9477-000000000001B75ECC89500807
 47 384865004907e-285 1AA65B58639E69-0000000000020B630FA5EA4CD6
 47 286556458711e0081 11958B36C5102B-7FFFFFFFFFFD74B0EAE567573B
 47 573112917422e0081 11958B36C5102B-7FFFFFFFFFFD74B0EAE567573B
 45 769525178383e-150 1253EC0E16141F-8000000000081D3D7AD13224DA
 45 416780288265e0192 1544E6DAEE2A17-80000000000808878EDE80A57C
 45 226963895975e-111 1FC575867314ED-FFFFFFFFFFF542E995DAB0415A
 44 665592809339e0063 178B6516C2B477-FFFFFFFFFFEAF548643B25230A
 
================ 13 decimal digit numbers follow
 52 3891901811465e0217 19AB8261990292-000000000000133D9FA00B6A67
 47 4764593340755e0069 14177A9915FBF8-000000000003A5E64E8FC1FA19
 50 6336156586177e0269 1173F76C63B792-7FFFFFFFFFFFA38A618DC2490C
 49 8233559360849e0095 166FEE05649A7A-7FFFFFFFFFFF2ABDF169D37593
 49 3662265515198e-107 138E6EDD48F2A2-800000000000F59CD029FF04F3
 49 1831132757599e-107 138E6EDD48F2A2-800000000000F59CD029FF04F3
 53 7812878489261e-179 126DAE7BBEDA74-FFFFFFFFFFFFF2CE2CF8B733BC
 52 6363857920591e0145 128A61CF9483B6-FFFFFFFFFFFFE4197156AEE3BC
 49 8811915538555e0082 151F508B287AE7-000000000000C7E7F64F8C9938
 49 9997878507563e-195 153DB2FEA1EA31-000000000000E8596918548671
 51 9224786422069e-291 14EE5D56B32957-7FFFFFFFFFFFCDBE8BB2B10983
 51 6284426329974e-294 1D3409DFBCA26F-7FFFFFFFFFFFD920C691192634
 50 9199302046091e-062 1135972630774B-80000000000042815C908B8788
 49 6070482281213e-122 123FA9E6FCF47D-800000000000D8E3D1C3E21529
 50 2780161250963e-301 15ACC2053064C1-FFFFFFFFFFFF989794AD4C4ECD
 49 8233559360849e0094 11F324D11D4861-FFFFFFFFFFFF5564C12175F7A9
 
================ 14 decimal digit numbers follow !58
 52 72027097041701e0206 14677812D3A606-0000000000001F58FC812AA097
 52 97297545286625e0215 19AB8261990292-000000000000133D9FA00B6A67
 55 99021992302453e-025 15C6714DEF374C-7FFFFFFFFFFFFC7E7E54263FBE
 53 54104687080198e-022 173CDF8DB7A7BC-7FFFFFFFFFFFF0E36FF7F4508D
 53 33519685743233e0089 17F203339C9628-8000000000000BDAA0A1ED76F8
 53 67039371486466e0089 17F203339C9628-8000000000000BDAA0A1ED76F8
 53 39064392446305e-180 126DAE7BBEDA74-FFFFFFFFFFFFF2CE2CF8B733BC
 53 17796979903653e0261 1072F3819C1320-FFFFFFFFFFFFF5394A8A7E13A7
 57 28921916763211e0038 1EEBABE0957AF3-00000000000000D06C23000000
 53 87605699161665e0155 12920F96E7F9EF-0000000000000B6064733EE0B1
 52 41921560615349e-067 19B2A5C4041E4B-7FFFFFFFFFFFE9BDE74AE0F2F9
 52 80527976643809e0061 1C7C5AEA080A49-7FFFFFFFFFFFEA6D58CD0139E4
 57 72335858886654e-159 1CE77C2B3328FB-800000000000008B6C8A7774CD
 56 52656615219377e0102 1561DEF4A9EE31-800000000000015C50BFBCDBFF
 57 15400733123779e-072 18BF7E7FA6F029-FFFFFFFFFFFFFF77468AEAC200
 57 77003665618895e-073 18BF7E7FA6F029-FFFFFFFFFFFFFF77468AEAC200
 
================ 15 decimal digit numbers follow
 57 475603213226859e-042 12D73088F4050A-00000000000000CD5720AEEF52
 55 972708181182949e0116 118A7F36172332-00000000000002F36FA4C3AA22
 57 246411729980464e-071 1EEF5E1F90AC34-7FFFFFFFFFFFFF55182DA57280
 57 123205864990232e-071 1EEF5E1F90AC34-7FFFFFFFFFFFFF55182DA57280
 57 609610927149051e-255 104273B18918B0-80000000000000ECD1A3F300A3
 56 475603213226859e-041 178CFCAB31064C-8000000000000100ACE8DAAB26
 55 672574798934795e0065 1226C684C87260-FFFFFFFFFFFFFCCB530D966E86
 55 134514959786959e0066 1226C684C87260-FFFFFFFFFFFFFCCB530D966E86
 58 294897574603217e-151 15F2DF5E675A0F-0000000000000056D7AC9E5D7D
 57 723047919080275e0036 1EEBABE0957AF3-00000000000000D06C23000000
 56 660191429952702e-088 1DDC7E975C5045-7FFFFFFFFFFFFE94C241C22B31
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 57 144671717773308e-159 1CE77C2B3328FB-800000000000008B6C8A7774CD
 57 385018328094475e-074 18BF7E7FA6F029-FFFFFFFFFFFFFF77468AEAC200
 56 330095714976351e-089 17E3987916A69D-FFFFFFFFFFFFFEDD68349B55C1
 
================ 16 decimal digit numbers follow
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 61 3729901848043846e0213 13AE60753AF6CA-7FFFFFFFFFFFFFF0A7AB586553
 63 7487252720986826e-165 18823A57ADBEF8-8000000000000003F982068899
 63 3743626360493413e-165 18823A57ADBEF8-8000000000000003F982068899
 60 4988915232824583e0119 15F6DE9D5D6B5A-FFFFFFFFFFFFFFEEE74C49E3F2
 60 3771476185376383e0277 1E3C14D6916CE8-FFFFFFFFFFFFFFE8AAE7152501
 61 6182410494241627e-119 11B96458445D07-000000000000000DCBCB0E5384
 59 2572981889477453e0142 1DFC11FBF46087-000000000000002B6A898E1D43
 58 7793560217139653e0051 1280461B856EC5-7FFFFFFFFFFFFFA78F4BBD1865
 58 9163942927285259e-202 1FE601457DCE4D-7FFFFFFFFFFFFF9C6110D2D70C
 61 6353227084707473e0155 150AFF653FFE89-800000000000000A0CB6BDACF7
 61 4431803091515554e-211 1090684F5FE997-800000000000000CEA2B24301E
 60 9324754620109615e0211 1F7D6721F7F143-FFFFFFFFFFFFFFE772ABC0A21F
 60 8870461176410409e0263 190529A37B7E21-FFFFFFFFFFFFFFEB326C71395A
 
================ 17 decimal digit numbers follow
 64 90372559027740405e0143 1491DAAD0BA280-0000000000000001C7455ED4E4
 64 18074511805548081e0146 1011F2D73116F4-000000000000000163AE321652
 64 54897030182071313e0029 1EC55666D8F9EC-7FFFFFFFFFFFFFFE9400000000
 63 76232626624829156e-032 1B7738011E75FE-7FFFFFFFFFFFFFFD0270C14C91
 63 59898021767894608e-165 18823A57ADBEF8-8000000000000003F982068899
 63 29949010883947304e-165 18823A57ADBEF8-8000000000000003F982068899
 66 26153245263757307e0049 13DE005BD620DE-FFFFFFFFFFFFFFFFB4C6953FA1
 65 27176258005319167e-261 17C0747BD76FA0-FFFFFFFFFFFFFFFF48F1A171DC
 64 18074511805548081e0147 14166F8CFD5CB1-0000000000000001BC99BE9BE7
 63 24691002732654881e-115 159A2783CE70AB-0000000000000003B3D1FB5A2E
 64 58483921078398283e0057 108CE499519CE3-7FFFFFFFFFFFFFFE53C0C0B6A3
 63 64409240769861689e-159 192238F7987779-7FFFFFFFFFFFFFFCB78CABF429
 64 94080055902682397e-242 164981E39E66C9-80000000000000014E3A82E654
 61 31766135423537365e0154 150AFF653FFE89-800000000000000A0CB6BDACF7
 62 68985865317742005e0164 1A999DDEC72AC9-FFFFFFFFFFFFFFFBAFAAAC5601
 62 13797173063548401e0165 1A999DDEC72AC9-FFFFFFFFFFFFFFFBAFAAAC5601
 
================ 18 decimal digit numbers follow
 67 902042358290366539e-281 122DC01CA1CB8C-000000000000000027FD092417
 65 238296178309629163e0272 138FD93F1F5342-00000000000000008E3314DFA3
 67 783308178698887621e0226 125AE62CB346D8-7FFFFFFFFFFFFFFFC3A965B086
 64 439176241456570504e0029 1EC55666D8F9EC-7FFFFFFFFFFFFFFE9400000000
 69 899810892172646163e0283 1ADF51FA055E02-80000000000000000D0F6861FE
 68 926145344610700019e-225 1307A67F1F69FE-80000000000000001BB057C8A3
 66 653831131593932675e0047 13DE005BD620DE-FFFFFFFFFFFFFFFFB4C6953FA1
 66 130766226318786535e0048 13DE005BD620DE-FFFFFFFFFFFFFFFFB4C6953FA1
 68 557035730189854663e-294 13BFAC6BC4767B-0000000000000000125BF1DAD0
 67 902042358290366539e-280 16B93023CA3E6F-000000000000000031FC4B6D1C
 72 272104041512242479e0200 13BBB4BF05F087-7FFFFFFFFFFFFFFFFE14128914
 72 544208083024484958e0200 13BBB4BF05F087-7FFFFFFFFFFFFFFFFE14128914
 66 680429695511221511e0192 108EBC116F8A1F-8000000000000000623DCC1A25
 66 308975121073410857e0236 10DB75CC001071-80000000000000007893BF4E10
 69 792644927852378159e0078 17BFF336D8FF05-FFFFFFFFFFFFFFFFF7817774E0
 67 783308178698887621e0223 12CBAC35F7113F-FFFFFFFFFFFFFFFFC236ADC319
 
================ 19 decimal digit numbers follow
 73 8396094300569779681e-252 1AB223EFCEE35A-000000000000000000D0084B60
 71 3507665085003296281e-074 1B85C026A264E4-0000000000000000029493108D
 71 7322325862592278999e0074 16775B6CAA5AE0-7FFFFFFFFFFFFFFFFCBB2235D2
 71 6014546754280072926e0209 196397B06732A4-7FFFFFFFFFFFFFFFFCB8201B2F
 73 7120190517612959703e0120 13220DCD5899FC-8000000000000000009228C1E1
 72 3507665085003296281e-073 11339818257F0E-8000000000000000019CDBEA58
 70 4345544743100783551e-218 1A9C42E5B6D89E-FFFFFFFFFFFFFFFFF81ABCF8D2
 69 9778613303868468131e-090 1146FE1075E1EE-FFFFFFFFFFFFFFFFF0E695E484
 68 7539204280836061195e-082 13D969E3DBE723-00000000000000001D2CB834F9
 68 7862637540082247119e-202 1ABA3262EE707B-000000000000000013354A27A6
 72 2176832332097939832e0200 13BBB4BF05F087-7FFFFFFFFFFFFFFFFE14128914
 69 8643988913946659879e0115 1E71EC1ED0A4F9-7FFFFFFFFFFFFFFFF776C925EE
 69 5529436763613147623e0138 19C677BE6F236D-80000000000000000A51495C99
 68 6764958008109694533e-173 1D06692E6F5EF5-80000000000000001F29730F20
 71 6802601037806061975e0197 1F92BACB3CB40B-FFFFFFFFFFFFFFFFFCECEA74ED
 71 1360520207561212395e0198 1F92BACB3CB40B-FFFFFFFFFFFFFFFFFCECEA74ED
 
================ 20 decimal digit numbers follow
 73 62259110684423957791e0047 1D8F2CFC20D6E8-000000000000000000EE680C48
 72 88800290202542652011e-226 16CEC51A43F41A-00000000000000000151343CED
 74 41010852717673354694e-221 1012954B6AABBA-7FFFFFFFFFFFFFFFFF93F5D154
 74 20505426358836677347e-221 1012954B6AABBA-7FFFFFFFFFFFFFFFFF93F5D154
 73 66102447903809911604e0055 1762068A24FD54-8000000000000000008E5EDC01
 73 35600952588064798515e0119 13220DCD5899FC-8000000000000000009228C1E1
 73 14371240869903838702e0205 18D92D2BCC7A80-FFFFFFFFFFFFFFFFFF4D3666BD
 71 57500690832492901689e0043 165D3E2ACD616A-FFFFFFFFFFFFFFFFFC36D9C412
 74 23432630639573022093e-107 1DD54C40A2F25F-000000000000000000506016E7
 73 62259110684423957791e0048 12797C1D948651-000000000000000000950107AD
 73 35620497849450218807e-306 15B22082529425-7FFFFFFFFFFFFFFFFF5C9E448C
 72 69658634627134074624e0200 13BBB4BF05F087-7FFFFFFFFFFFFFFFFE14128914
 72 99440755792436956989e-062 162D10462A26F3-80000000000000000111AA4908
 72 55277197169490210673e0081 1945BFA911E329-800000000000000001248F8511
 72 36992084760177624177e-318 18C5F9551C2F99-FFFFFFFFFFFFFFFFFE874C95EA
 71 30888265282878466443e-111 101B8EF28251FB-FFFFFFFFFFFFFFFFFC4CD6B6FF

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