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 0011 0000 - 0001 1100 0011 0111 1001 0011 0111 1110 0000 1000 0000 0000 0000 = 625 000 000 000 000 Oct 29 08:33 UTC (GMT)
0 - 100 0010 1000 - 0111 1001 0011 0010 0001 0100 0010 0000 0000 0000 0000 0000 0010 = 3 240 085 700 608.000 976 562 5 Oct 29 08:32 UTC (GMT)
0 - 111 1110 0100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 1 339 385 758 982 834 448 588 913 006 881 614 822 814 255 644 612 029 035 556 728 317 341 830 625 060 542 016 304 629 405 600 494 531 693 156 240 378 211 551 491 228 689 645 710 002 645 415 121 305 818 463 408 657 937 192 331 954 029 771 802 415 124 771 181 272 026 239 628 678 183 420 786 899 128 712 287 158 445 434 368 835 650 495 741 187 629 442 434 373 366 574 819 785 360 613 107 720 471 445 504 Oct 29 08:31 UTC (GMT)
1 - 010 1100 0100 - 0101 1100 0000 0101 1110 0100 0001 1100 1100 0110 0100 1111 0101 = -0 Oct 29 08:31 UTC (GMT)
0 - 000 0000 1000 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 005 3 Oct 29 08:31 UTC (GMT)
0 - 100 0000 0100 - 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = 37.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 Oct 29 08:30 UTC (GMT)
0 - 100 0000 1010 - 0001 1111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 2 300 Oct 29 08:29 UTC (GMT)
1 - 011 1111 1001 - 0111 0110 1111 1011 0000 1001 0010 0000 0011 1010 0001 0010 0011 = -0.022 886 999 999 998 224 736 286 189 795 464 451 890 438 795 089 721 679 687 5 Oct 29 08:29 UTC (GMT)
0 - 100 0000 0100 - 0001 1110 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 35.765 625 Oct 29 08:29 UTC (GMT)
0 - 000 1111 1111 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 288 229 753 919 426 375 600 940 538 964 375 127 192 154 826 724 343 021 784 075 301 458 295 130 763 9 Oct 29 08:28 UTC (GMT)
1 - 000 1000 0000 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0 Oct 29 08:28 UTC (GMT)
0 - 111 1111 1110 - 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 123 591 403 021 784 218 656 389 731 866 745 450 436 235 917 302 283 576 875 483 180 795 941 214 616 281 912 153 737 078 159 155 181 014 520 078 292 411 582 933 390 417 966 059 911 427 963 832 608 564 638 460 509 715 426 729 595 021 127 457 775 975 856 689 707 123 994 529 558 441 466 076 228 594 110 443 231 013 315 263 496 818 725 549 803 476 095 653 080 039 963 731 582 704 804 959 574 307 476 616 654 094 336 Oct 29 08:28 UTC (GMT)
0 - 110 0000 0001 - 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 93 854 655 509 598 167 788 492 516 128 217 628 132 234 292 307 080 462 794 004 876 605 033 248 558 579 405 462 019 023 637 256 748 217 516 048 113 003 144 208 443 123 182 618 566 726 594 122 459 820 064 768 Oct 29 08:27 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)