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 1000 - 1111 0100 0000 0010 1111 0001 1010 1001 1111 1011 1110 0111 0100 = 1 000.022 999 999 999 683 495 843 783 020 973 205 566 406 25 Nov 24 21:36 UTC (GMT)
0 - 000 0000 0011 - 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 = 0 Nov 24 19:16 UTC (GMT)
0 - 011 1111 1000 - 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000 = 0.015 374 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 5 Nov 24 19:15 UTC (GMT)
1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 = -3 875 739 974 753 834 706 075 163 486 913 256 793 856 570 234 510 836 280 412 770 513 148 715 349 080 425 257 484 493 363 182 218 688 710 603 984 309 181 221 006 228 382 042 997 444 461 286 993 983 555 596 632 805 592 952 453 024 263 932 572 099 783 602 225 529 645 252 313 717 991 116 420 093 794 541 292 813 610 327 893 705 685 088 582 916 171 632 511 843 259 383 808 Nov 24 18:43 UTC (GMT)
0 - 000 1100 0100 - 0100 0101 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 422 616 857 683 481 690 287 229 092 469 781 029 414 825 797 336 916 325 319 6 Nov 24 18:43 UTC (GMT)
0 - 011 1111 1100 - 1010 0100 0101 1010 0001 1100 1010 1100 0000 1000 0011 0000 0000 = 0.205 249 999 999 999 488 409 230 252 727 866 172 790 527 343 75 Nov 24 17:16 UTC (GMT)
0 - 100 0000 0010 - 1000 0111 0101 1000 0101 0001 1110 1011 1000 0101 0001 1110 1100 = 12.229 531 250 000 000 852 651 282 912 120 223 045 349 121 093 75 Nov 24 17:16 UTC (GMT)
1 - 100 1111 0110 - 1001 1110 1101 1110 1110 1001 0111 0111 1010 1110 0001 0100 0111 = -366 506 583 389 293 531 152 712 530 761 625 941 712 628 926 189 466 468 387 534 772 000 629 194 752 Nov 24 16:56 UTC (GMT)
1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 = -3 875 739 974 753 834 706 075 163 486 913 256 793 856 570 234 510 836 280 412 770 513 148 715 349 080 425 257 484 493 363 182 218 688 710 603 984 309 181 221 006 228 382 042 997 444 461 286 993 983 555 596 632 805 592 952 453 024 263 932 572 099 783 602 225 529 645 252 313 717 991 116 420 093 794 541 292 813 610 327 893 705 685 088 582 916 171 632 511 843 259 383 808 Nov 24 16:56 UTC (GMT)
0 - 011 1111 0011 - 0001 0000 0001 0110 1110 0101 0011 0011 0001 0011 1001 0010 1101 = 0.000 259 484 705 741 014 007 058 381 496 406 695 987 388 957 291 841 506 958 007 812 5 Nov 24 16:50 UTC (GMT)
0 - 111 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = +∞ (Infinity, positive) Nov 24 16:49 UTC (GMT)
1 - 011 0111 1111 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 003 673 419 846 319 648 462 402 301 678 819 517 743 183 329 864 912 773 504 714 849 082 120 053 935 796 022 415 161 132 812 5 Nov 24 16:46 UTC (GMT)
0 - 100 1010 1000 - 0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000 = 1 064 608 701 888 006 151 252 602 686 964 999 266 829 445 191 696 384 Nov 24 16:27 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)