Converter of 64 bit double precision IEEE 754 binary floating point standard system numbers: converting to base ten decimal (double)

Convert 64 bit double precision IEEE 754 floating point standard binary numbers to base ten decimal system (double)

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest 64 bit double precision IEEE 754 floating point binary standard numbers converted to decimal base ten (double)

0 - 011 1110 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 465 661 287 307 739 257 812 5 Jun 18 09:35 UTC (GMT)
0 - 100 0011 0001 - 0001 1000 1011 0101 0100 1111 0010 0110 1110 1011 1100 0001 1100 = 1 234 567 891 234 567 Jun 18 09:35 UTC (GMT)
0 - 100 0000 0011 - 0100 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 20.015 625 Jun 18 09:34 UTC (GMT)
0 - 110 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 26 815 615 859 885 194 199 148 049 996 411 692 254 958 731 641 184 786 755 447 122 887 443 528 060 147 093 953 603 748 596 333 806 855 380 063 716 372 972 101 707 507 765 623 893 139 892 867 298 012 168 192 Jun 18 09:34 UTC (GMT)
0 - 000 0111 1010 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 118 305 218 616 677 457 962 757 395 596 998 655 1 Jun 18 09:34 UTC (GMT)
0 - 011 1010 1111 - 0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0000 0001 = 0.000 000 000 000 000 000 000 001 034 178 108 905 430 803 126 524 836 103 967 232 711 645 855 864 447 099 629 801 685 356 268 736 715 179 556 995 281 018 316 745 758 056 640 625 Jun 18 09:34 UTC (GMT)
0 - 000 0000 0000 - 0100 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Jun 18 09:34 UTC (GMT)
0 - 011 0000 0010 - 0001 0011 0100 0011 0001 0011 0010 0011 0011 0101 0011 0100 0011 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 074 287 785 055 795 137 939 741 304 349 450 029 658 975 820 371 247 588 273 594 372 502 120 995 371 217 134 220 996 593 516 232 878 722 278 755 088 962 770 349 511 847 022 397 422 111 826 897 398 923 102 371 259 953 841 734 843 340 168 183 922 848 357 467 194 624 348 394 427 215 680 480 003 356 933 593 75 Jun 18 09:32 UTC (GMT)
0 - 100 0001 0000 - 1001 1010 1110 0010 1011 0010 1011 1000 0110 0000 1000 0101 0011 = 210 373.396 251 740 487 059 578 299 522 399 902 343 75 Jun 18 09:32 UTC (GMT)
1 - 011 1111 1110 - 0111 1111 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.749 511 718 75 Jun 18 09:31 UTC (GMT)
0 - 111 1111 1100 - 1111 1111 1111 1111 1111 1111 1111 1111 1110 0000 0000 0000 0000 = 44 942 328 370 903 900 740 604 292 781 842 058 129 841 513 212 228 681 889 338 029 562 233 614 270 973 415 190 449 497 185 525 468 979 147 401 603 434 892 606 432 792 445 362 059 267 143 175 044 889 117 519 279 579 999 796 405 911 860 162 943 145 830 256 769 300 381 874 381 239 590 271 485 517 571 481 080 398 544 270 826 175 709 196 472 572 561 614 641 181 221 763 744 894 561 044 491 759 957 232 886 743 040 Jun 18 09:31 UTC (GMT)
0 - 011 1010 0110 - 0110 0001 1010 1111 0001 1101 0110 1010 1000 0011 0000 0111 1010 = 0.000 000 000 000 000 000 000 000 002 232 060 112 083 283 453 478 333 432 534 178 338 406 333 632 699 827 663 113 095 009 573 134 648 930 836 654 301 401 722 477 748 990 058 898 925 781 25 Jun 18 09:31 UTC (GMT)
1 - 011 1111 0011 - 0100 1001 0100 0101 1110 0110 1011 0000 0100 0111 1101 1111 0000 = -0.000 314 019 252 089 152 954 565 687 281 274 222 186 766 564 846 038 818 359 375 Jun 18 09:30 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

How to convert numbers from 64 bit double precision IEEE 754 binary floating point standard to decimal system in base 10

Follow the steps below to convert a number from 64 bit double precision IEEE 754 binary floating point representation to base 10 decimal system:

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent.
    The last 52 bits contain the mantissa.
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10).
  • 3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1,023, that is due to the 11 bit excess/bias notation.
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10).
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted)

Example: convert the number 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 from 64 bit double precision IEEE 754 binary floating point system to base ten decimal (double):

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent: 100 0011 1101
    The last 52 bits contain the mantissa:
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):
    100 0011 1101(2) =
    1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
    1,024 + 0 + 0 + 0 + 0 + 32 + 16 + 8 + 4 + 0 + 1 =
    1,024 + 32 + 16 + 8 + 4 + 1 =
    1,085(10)
  • 3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1,023, that is due to the 11 bit excess/bias notation:
    Exponent adjusted = 1,085 - 1,023 = 62
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10):
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000(2) =
    1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 1 × 2-47 + 0 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
    0.5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =
    0.5 + 0.000 488 281 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =
    0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5(10)
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =
    (-1)1 × (1 + 0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5) × 262 =
    -1.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5 × 262 =
    -6 919 868 872 153 800 704(10)
  • 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 converted from 64 bit double precision IEEE 754 binary floating point representation to a decimal number (float) in decimal system (in base 10) = -6 919 868 872 153 800 704(10)