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 - 111 1111 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 89 884 656 743 115 815 344 868 354 886 649 353 244 625 979 315 500 989 311 227 644 933 441 752 928 222 905 938 473 157 350 844 425 860 139 711 866 292 707 572 253 863 326 332 160 750 970 803 599 255 673 669 580 601 929 560 746 945 838 207 839 311 479 812 235 103 755 919 672 374 710 952 642 786 610 471 374 045 061 106 479 117 507 470 227 939 822 734 724 191 182 691 726 800 274 576 002 630 468 252 211 347 456 Jul 23 11:46 UTC (GMT)
0 - 100 0001 1000 - 0110 1100 1100 1111 0110 0101 1110 1001 0111 0011 0001 1100 1000 = 47 816 395.823 825 418 949 127 197 265 625 Jul 23 11:34 UTC (GMT)
0 - 011 1111 1110 - 1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000 = 0.952 258 517 081 489 230 804 436 374 455 690 383 911 132 812 5 Jul 23 11:28 UTC (GMT)
1 - 100 0000 1001 - 0100 1110 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1 337.5 Jul 23 11:23 UTC (GMT)
0 - 100 0000 1100 - 1110 0011 1000 0011 1000 0011 1000 1101 1001 0000 0000 0000 0000 = 15 472.439 234 852 790 832 519 531 25 Jul 23 11:23 UTC (GMT)
0 - 100 0001 0101 - 1110 1101 1110 1011 1011 1110 1110 0010 0011 1110 1010 0001 1111 = 8 092 399.720 942 049 287 259 578 704 833 984 375 Jul 23 11:19 UTC (GMT)
0 - 011 1111 1011 - 1010 0111 1101 1110 0001 0110 1101 0100 1011 1111 0011 1010 1101 = 0.103 483 285 125 504 934 076 623 442 251 730 011 776 089 668 273 925 781 25 Jul 23 11:10 UTC (GMT)
1 - 011 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.75 Jul 23 11:06 UTC (GMT)
0 - 100 0100 0011 - 0110 1011 1100 1100 0011 1110 0001 0110 1010 0100 1001 1011 1110 = 419 430 332 767 819 137 024 Jul 23 11:02 UTC (GMT)
0 - 100 0011 0011 - 0101 1111 0111 0001 0001 1110 1011 1000 0101 1001 1110 1011 1001 = 6 182 630 848 831 161 Jul 23 10:56 UTC (GMT)
0 - 100 0010 1000 - 0000 1110 0110 1110 1011 1110 0010 1000 1010 1111 0000 0000 0000 = 2 322 998 251 870 Jul 23 10:48 UTC (GMT)
0 - 100 0001 0111 - 1100 0100 1100 1101 1010 0010 1010 0101 0110 0011 1111 0110 1011 = 29 674 914.646 056 573 837 995 529 174 804 687 5 Jul 23 10:41 UTC (GMT)
0 - 100 0111 0000 - 0100 0110 0000 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 13 225 081 799 531 060 288 464 730 365 886 464 Jul 23 10:34 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)