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 - 100 0100 0000 - 1000 1010 1110 1000 0101 1000 0001 0101 0111 1111 1110 0000 0001 = -56 912 182 189 806 854 144 Apr 04 19:17 UTC (GMT)
0 - 100 0000 0010 - 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 8.25 Apr 04 19:16 UTC (GMT)
1 - 100 0000 0010 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0111 = -8.000 000 000 000 012 434 497 875 801 753 252 744 674 682 617 187 5 Apr 04 19:15 UTC (GMT)
1 - 100 0000 1000 - 1010 0011 0010 1111 0011 1011 1000 1000 0010 0110 1010 1010 1000 = -838.369 004 267 578 020 517 248 660 326 004 028 320 312 5 Apr 04 19:13 UTC (GMT)
1 - 100 0110 0001 - 1000 0000 1111 1000 0101 1000 0011 1010 0000 0000 0000 0000 0000 = -476 569 896 051 290 332 918 303 424 512 Apr 04 19:10 UTC (GMT)
0 - 100 0000 0100 - 1111 0111 1011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 = 62.962 499 999 999 998 578 914 528 479 799 628 257 751 464 843 75 Apr 04 19:07 UTC (GMT)
0 - 010 0000 0100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 386 669 033 984 070 397 428 970 803 332 865 243 048 007 265 278 858 020 888 110 584 544 011 180 525 032 936 382 846 216 905 747 743 727 222 108 520 593 405 041 511 233 765 988 061 105 142 105 241 024 253 6 Apr 04 19:07 UTC (GMT)
0 - 000 0011 1111 - 1111 1000 0011 1110 0000 1111 1000 0011 1110 0000 1111 1000 0011 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 202 117 342 488 177 598 2 Apr 04 19:07 UTC (GMT)
1 - 111 1111 1101 - 0000 1111 0010 1011 0000 0100 0001 1000 1001 0011 0111 0100 1011 = -47 605 166 813 696 764 181 384 945 344 805 243 187 604 196 365 074 955 519 841 382 989 008 811 638 058 304 902 871 170 364 535 169 923 318 757 658 671 648 464 748 436 255 145 470 017 975 475 668 483 176 056 139 471 382 546 924 981 188 115 035 013 532 754 202 895 334 966 633 161 583 825 223 800 331 342 286 584 935 040 008 874 204 244 017 527 792 703 274 550 546 501 229 290 292 107 766 937 364 953 158 385 664 Apr 04 19:06 UTC (GMT)
0 - 000 0000 0000 - 0010 0000 0010 0000 0010 0000 0010 0000 0010 0000 0101 1000 0000 = 0 Apr 04 19:02 UTC (GMT)
0 - 010 0001 0000 - 0100 0010 0010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 012 302 383 869 300 117 018 923 581 468 549 800 106 075 742 266 892 679 163 962 984 060 257 092 070 605 458 741 626 121 758 217 697 653 088 746 159 338 436 532 625 375 226 738 512 232 232 418 719 163 115 551 856 8 Apr 04 18:59 UTC (GMT)
0 - 100 0000 0100 - 1001 0010 1010 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 50.333 984 375 Apr 04 18:54 UTC (GMT)
0 - 011 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Apr 04 18:53 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)