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 - 011 1111 1110 - 1111 1111 0111 1100 1110 1101 1001 0000 1000 1100 1000 1011 1011 = -0.998 999 999 899 999 990 837 784 480 390 837 416 052 818 298 339 843 75 Nov 19 08:37 UTC (GMT)
1 - 100 0000 1000 - 1000 0000 0000 1000 0000 1000 0000 1000 0000 0000 1000 0000 1000 = -768.062 745 094 533 056 544 605 642 557 144 165 039 062 5 Nov 19 08:34 UTC (GMT)
0 - 100 0001 1000 - 1110 0000 1000 1111 0010 1010 0000 1011 0110 1011 1101 1100 0010 = 62 987 860.089 229 121 804 237 365 722 656 25 Nov 19 08:33 UTC (GMT)
0 - 100 0000 0100 - 0101 0010 0001 1100 1010 1100 0000 1000 0011 0001 0010 0110 1111 = 42.264 000 000 000 002 899 014 361 901 208 758 354 187 011 718 75 Nov 19 08:33 UTC (GMT)
0 - 100 0000 1001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 1 024 Nov 19 08:32 UTC (GMT)
0 - 100 0000 0010 - 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1101 = 9.400 000 000 000 000 355 271 367 880 050 092 935 562 133 789 062 5 Nov 19 08:31 UTC (GMT)
0 - 100 0001 1011 - 0010 0010 1000 1110 0001 0001 1011 1111 1111 1100 0000 0010 0011 = 304 668 955.999 025 523 662 567 138 671 875 Nov 19 08:31 UTC (GMT)
1 - 011 1000 1001 - 0101 0010 1111 0101 1100 0111 1000 1101 0000 0110 1011 0011 1101 = -0.000 000 000 000 000 000 000 000 000 000 000 003 984 456 546 743 054 569 387 010 875 833 830 383 706 549 163 696 379 972 449 251 072 927 971 524 617 152 522 451 880 614 128 352 786 629 250 203 986 885 026 097 297 668 457 031 25 Nov 19 08:30 UTC (GMT)
1 - 011 1111 1111 - 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1.375 Nov 19 08:29 UTC (GMT)
0 - 100 0000 0000 - 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 2.062 5 Nov 19 08:27 UTC (GMT)
0 - 011 1000 0000 - 0011 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 007 209 086 448 402 310 107 464 517 044 683 303 570 997 284 859 891 318 003 002 891 323 660 605 848 999 693 989 753 723 144 531 25 Nov 19 08:26 UTC (GMT)
1 - 010 1010 0100 - 1000 1100 1010 0000 1011 0100 0001 0010 1011 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 005 404 240 067 590 393 402 494 609 241 296 241 688 924 155 027 546 467 961 581 633 284 898 775 675 283 521 821 205 302 484 649 543 690 678 877 960 488 849 468 510 772 315 906 694 316 842 135 325 950 720 596 073 791 988 305 654 608 574 158 408 078 256 474 899 377 117 273 1 Nov 19 08:26 UTC (GMT)
0 - 000 0111 1110 - 1111 1111 1111 1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 892 883 385 042 191 046 004 554 051 699 962 377 2 Nov 19 08:26 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)