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 - 100 0000 0011 - 1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 27.566 406 250 000 003 552 713 678 800 500 929 355 621 337 890 625 Mar 26 22:14 UTC (GMT)
0 - 100 0011 1101 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 1111 1111 = 9 223 372 036 854 250 496 Mar 26 22:11 UTC (GMT)
0 - 000 0011 0000 - 1101 0000 0000 0000 0000 0000 0100 0001 0011 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 005 675 867 473 335 6 Mar 26 22:08 UTC (GMT)
1 - 011 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = -0.999 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 5 Mar 26 22:01 UTC (GMT)
0 - 100 0001 0101 - 1110 1101 1110 1011 1011 1110 1110 0010 0011 1110 1010 0001 1111 = 8 092 399.720 942 049 287 259 578 704 833 984 375 Mar 26 21:57 UTC (GMT)
0 - 110 0011 0000 - 0011 0111 0101 0111 0011 0110 0100 0110 1111 0110 1111 0100 0000 = 9 179 593 728 708 743 198 721 133 024 012 996 334 309 533 151 343 662 555 319 109 647 047 605 443 193 895 018 320 781 051 598 986 080 402 073 250 472 916 956 439 799 689 412 990 557 070 577 071 619 124 335 790 488 461 246 464 Mar 26 21:53 UTC (GMT)
1 - 100 0000 0111 - 1000 0000 1110 0111 1010 0011 1111 1100 0010 0101 0000 1010 1110 = -384.904 845 961 612 295 468 512 456 864 118 576 049 804 687 5 Mar 26 21:52 UTC (GMT)
0 - 100 0000 0100 - 0000 0001 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 32.140 625 Mar 26 21:51 UTC (GMT)
0 - 111 0000 0000 - 1101 0101 0111 0111 0011 0001 1000 1001 0001 1001 0100 1011 0000 = 5 694 170 602 308 671 965 166 068 618 511 705 724 345 668 289 993 741 912 369 720 926 797 396 743 789 792 690 160 534 977 501 014 985 738 433 070 291 802 462 858 806 368 563 648 035 786 183 756 550 289 646 628 515 260 855 508 516 689 615 445 335 357 684 971 246 091 701 945 166 203 857 183 303 857 149 575 168 Mar 26 21:49 UTC (GMT)
1 - 011 1010 0110 - 1000 1001 0001 0100 0000 1000 0000 1000 0000 0000 1000 0000 0000 = -0.000 000 000 000 000 000 000 000 002 480 672 497 462 602 225 561 518 112 086 004 677 081 747 040 009 791 243 840 543 088 078 343 103 575 207 351 241 260 766 983 032 226 562 5 Mar 26 21:49 UTC (GMT)
0 - 011 1111 1110 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.625 Mar 26 21:43 UTC (GMT)
0 - 100 0001 1011 - 0010 1001 1000 1111 0111 1011 1000 1011 1001 1011 1011 0011 0001 = 312 014 776.725 512 564 182 281 494 140 625 Mar 26 21:41 UTC (GMT)
1 - 001 1100 1010 - 0001 0010 0110 0011 1000 0010 1010 1111 0101 0010 1010 1000 0010 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 008 875 211 820 572 695 813 069 047 300 613 450 510 924 939 408 779 443 573 984 089 832 837 231 020 510 661 803 816 909 126 079 835 794 628 468 908 489 218 271 729 262 170 985 456 921 5 Mar 26 21:41 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)