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

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Mantissa: empty

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

64 bit double precision IEEE 754 binary floating point standard representation of numbers requires three building blocks: 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 0001 0101 - 0001 1111 1110 0001 0010 1101 1001 1110 0001 0100 0111 0010 0001 = 4 716 619.404 374 868 609 011 173 248 291 015 625 Jun 26 12:46 UTC (GMT)
0 - 100 0001 1111 - 0010 0010 1010 0101 0110 1111 0000 1101 0000 0000 0000 0000 0000 = 4 876 234 509 Jun 26 12:39 UTC (GMT)
0 - 100 0001 1011 - 0010 0110 1111 1101 1110 0001 1011 1011 0010 1001 1010 0011 1011 = 309 321 243.697 665 870 189 666 748 046 875 Jun 26 12:39 UTC (GMT)
0 - 010 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 149 166 814 624 004 134 865 819 306 309 258 676 747 529 430 692 008 137 885 430 366 664 125 567 701 402 366 098 723 497 808 008 556 067 230 232 065 116 722 029 068 254 561 904 506 053 209 723 296 591 841 6 Jun 26 12:34 UTC (GMT)
0 - 100 0001 0010 - 1010 0100 0110 0011 1001 0100 1110 0101 1001 0111 0100 0111 0000 = 860 956.653 026 236 221 194 267 272 949 218 75 Jun 26 12:32 UTC (GMT)
1 - 100 1111 1111 - 1000 1111 0011 0111 0001 1111 1000 1111 0110 0001 1110 0000 0011 = -180 570 220 993 002 741 689 188 481 472 832 370 404 368 915 030 297 950 134 427 552 840 043 433 295 872 Jun 26 12:13 UTC (GMT)
1 - 000 0000 0100 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1 Jun 26 12:06 UTC (GMT)
0 - 011 0111 0111 - 0100 1001 0100 0111 1011 1101 1011 0101 1110 0011 1111 1001 1100 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 014 765 436 561 480 878 676 891 108 523 061 052 917 694 300 423 277 484 568 837 503 572 922 127 941 354 390 218 840 310 480 130 975 566 924 987 629 757 737 295 221 886 597 573 757 171 630 859 375 Jun 26 12:06 UTC (GMT)
1 - 111 0000 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -6 016 007 607 665 247 123 702 295 517 792 197 401 620 433 316 327 199 368 163 826 647 350 947 441 659 710 434 532 153 834 211 005 102 058 139 956 451 382 908 673 914 872 670 944 905 766 846 613 190 784 576 914 417 263 032 080 793 384 007 277 981 822 443 973 697 894 121 875 651 316 409 709 781 558 537 224 192 Jun 26 12:02 UTC (GMT)
0 - 011 1111 0000 - 1101 0011 0100 1000 0010 1011 1110 1000 1011 1100 0001 0110 0000 = 0.000 055 704 345 703 124 933 872 341 095 764 113 561 017 438 769 340 515 136 718 75 Jun 26 12:01 UTC (GMT)
0 - 010 0001 0101 - 0111 0010 1101 1000 1101 1010 0111 1110 1111 1010 1111 1010 0100 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 453 165 696 864 982 712 902 616 777 766 009 519 625 102 296 349 291 037 920 051 097 910 297 051 060 435 015 760 166 726 497 731 790 664 877 195 040 898 941 584 453 028 416 679 854 269 536 211 170 308 037 892 075 8 Jun 26 12:00 UTC (GMT)
0 - 101 1000 0000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 0000 0000 = 118 206 018 589 237 191 510 649 387 601 484 220 029 972 860 474 071 259 495 778 511 801 574 765 052 914 762 839 151 122 627 343 887 982 258 402 541 699 072 Jun 26 11:59 UTC (GMT)
0 - 100 0001 1010 - 0111 1111 0011 0110 1101 1111 1110 0011 1010 0011 1111 0110 0001 = 200 914 687.113 764 792 680 740 356 445 312 5 Jun 26 11:54 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)