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 - 011 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.75 Sep 23 10:48 UTC (GMT)
1 - 100 0000 1110 - 0101 0000 0111 0011 0001 0110 0011 0111 0001 1000 0000 0000 0000 = -43 065.543 389 081 954 956 054 687 5 Sep 23 10:45 UTC (GMT)
1 - 011 1111 1110 - 1111 1011 1001 1000 0100 1101 1110 0101 0110 1101 1000 0111 0011 = -0.991 396 364 456 490 153 393 986 020 091 688 260 436 058 044 433 593 75 Sep 23 10:36 UTC (GMT)
0 - 101 1111 0000 - 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 498 680 600 421 677 863 009 821 867 876 079 558 345 504 061 701 873 151 522 845 169 358 395 383 046 024 639 220 498 178 127 774 461 440 955 393 285 057 979 567 810 441 426 912 227 847 968 017 874 944 Sep 23 10:33 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 689 964 596 767 805 383 442 867 854 275 239 384 948 099 050 998 034 608 370 797 314 674 312 438 223 714 234 551 696 550 951 604 564 015 091 723 428 328 493 871 551 256 955 877 586 285 902 168 064 Sep 23 10:31 UTC (GMT)
0 - 011 1111 1110 - 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 = 0.833 333 333 333 333 259 318 465 024 989 563 971 757 888 793 945 312 5 Sep 23 10:23 UTC (GMT)
0 - 011 1111 1010 - 0110 1111 0000 0000 0110 1000 0110 1100 0010 0010 0110 1000 0000 = 0.044 799 999 189 376 826 791 431 085 439 398 884 773 254 394 531 25 Sep 23 10:22 UTC (GMT)
0 - 101 1100 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 1 453 677 448 591 213 781 098 647 615 776 009 068 707 282 721 374 636 120 562 980 398 361 278 576 226 795 846 652 382 101 427 527 131 121 525 043 212 532 355 867 069 203 257 229 312 Sep 23 10:13 UTC (GMT)
0 - 000 0000 0000 - 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 = 0 Sep 23 10:06 UTC (GMT)
0 - 100 0100 0110 - 0100 0011 1101 1100 0100 0110 0111 0010 0100 0110 1010 1100 0100 = 2 987 085 417 696 381 108 224 Sep 23 10:03 UTC (GMT)
0 - 100 0000 1010 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 3 328 Sep 23 10:03 UTC (GMT)
0 - 110 0110 0101 - 1100 1010 0000 1110 1010 1001 0101 0101 1100 0100 1010 1110 1010 = 121 645 809 849 328 817 838 453 027 721 477 665 430 371 818 032 316 022 618 992 546 564 605 743 267 254 531 046 833 805 967 248 094 130 521 624 124 275 271 003 622 589 874 418 714 440 228 137 384 492 468 931 423 112 757 283 680 325 852 425 355 264 Sep 23 10:00 UTC (GMT)
1 - 100 0001 1111 - 1000 0000 0101 0011 1010 1010 0111 1101 0101 0111 1101 1010 1011 = -6 447 934 077.343 180 656 433 105 468 75 Sep 23 09:56 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)