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)

1 - 111 1111 0000 - 1110 0001 1111 1111 1111 1111 1110 0000 0000 0011 1111 1111 1111 = -10 329 342 932 420 764 619 602 851 503 332 582 973 775 058 453 042 077 815 115 580 669 957 101 221 666 349 642 658 151 599 361 955 710 513 876 610 375 090 045 331 069 173 983 874 578 825 476 037 396 291 823 577 854 012 121 898 365 103 227 479 626 406 999 888 955 530 547 709 916 545 223 409 682 335 993 362 700 011 626 829 104 053 086 219 717 566 698 360 082 410 577 797 259 358 126 317 350 631 805 812 736 Aug 10 09:20 UTC (GMT)
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 Aug 10 09:19 UTC (GMT)
1 - 100 1100 1100 - 1100 1100 1100 1100 1110 1100 1100 1100 0001 0000 1100 0110 0000 = -92 559 729 420 238 819 161 529 508 229 666 862 439 337 676 554 375 889 342 693 376 Aug 10 09:13 UTC (GMT)
0 - 100 0000 0100 - 1111 0101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 62.625 Aug 10 09:10 UTC (GMT)
1 - 100 0001 1011 - 0001 1110 0001 1010 0011 0000 0000 0000 0000 0000 0000 0000 0000 = -300 000 000 Aug 10 09:09 UTC (GMT)
0 - 011 1110 0101 - 0011 0010 1100 1000 1000 1011 0111 1010 1011 0101 1000 1010 0001 = 0.000 000 017 857 142 857 142 855 862 113 075 431 953 090 888 015 367 454 499 937 593 936 920 166 015 625 Aug 10 09:07 UTC (GMT)
0 - 000 0000 0010 - 0001 0001 0100 0011 0001 0011 0010 0011 0011 0101 0011 0100 0011 = 0 Aug 10 09:06 UTC (GMT)
1 - 100 0000 0000 - 1111 1111 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3.998 046 875 Aug 10 09:05 UTC (GMT)
0 - 100 0000 0111 - 0101 0101 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 341.5 Aug 10 09:04 UTC (GMT)
0 - 110 0011 1100 - 0011 1000 1000 0011 0010 1110 0010 1000 0000 1001 0100 0000 0000 = 37 741 124 539 049 449 170 023 199 796 648 604 675 577 532 922 783 804 582 289 632 384 814 524 734 880 572 579 694 451 045 456 944 042 836 881 924 672 022 861 178 178 528 343 989 364 757 217 017 772 326 655 845 845 868 713 345 024 Aug 10 09:03 UTC (GMT)
0 - 101 1111 0000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 613 760 738 980 526 600 627 473 068 155 174 841 040 620 383 633 074 648 028 117 131 518 025 086 825 876 479 040 613 142 311 107 029 465 791 253 273 917 513 314 228 235 602 353 511 197 499 098 923 008 Aug 10 09:02 UTC (GMT)
0 - 100 0001 0111 - 1011 1100 1010 1110 0000 0011 1010 1011 1011 1000 0010 0000 1110 = 29 142 531.670 778 326 690 196 990 966 796 875 Aug 10 09:02 UTC (GMT)
0 - 100 0001 1001 - 0110 1100 1110 1001 0010 1100 0000 0111 0110 0001 1001 0111 1010 = 95 659 184.115 331 560 373 306 274 414 062 5 Aug 10 08:59 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)