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 0011 0000 - 1000 0011 1110 0110 1010 0011 1111 1100 0000 0101 0000 1010 0010 = -853 003 187 259 924.25 Sep 25 00:47 UTC (GMT)
0 - 100 0000 0111 - 1000 0100 1101 1110 1011 1000 0101 0001 1110 1011 1000 0101 0010 = 388.870 000 000 000 004 547 473 508 864 641 189 575 195 312 5 Sep 25 00:47 UTC (GMT)
0 - 100 0000 1111 - 1110 0010 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 123 468 Sep 25 00:45 UTC (GMT)
0 - 100 0001 1011 - 0001 1101 0010 0111 1111 0001 1100 0100 0011 0110 1100 0001 1110 = 299 007 772.263 368 487 358 093 261 718 75 Sep 25 00:44 UTC (GMT)
1 - 100 0000 1000 - 1000 0000 1000 1011 1000 0011 1101 0011 0000 0011 1111 1100 0000 = -769.089 960 457 749 839 406 460 523 605 346 679 687 5 Sep 25 00:44 UTC (GMT)
0 - 100 0001 1000 - 0011 0001 0000 1111 0100 1101 0100 0011 1001 1100 1100 0010 0110 = 39 984 794.528 221 413 493 156 433 105 468 75 Sep 25 00:43 UTC (GMT)
0 - 100 0100 1000 - 1111 0111 0001 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 18 560 306 841 913 326 043 136 Sep 25 00:37 UTC (GMT)
0 - 100 0000 0011 - 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 = 22.857 142 857 142 854 097 673 989 599 570 631 980 895 996 093 75 Sep 25 00:37 UTC (GMT)
0 - 110 0000 0000 - 1011 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 46 298 836 758 083 030 609 466 555 071 929 562 408 952 185 099 233 108 382 451 673 110 351 716 416 347 716 904 268 972 185 857 588 398 742 141 260 300 209 644 354 368 876 585 002 999 346 278 694 224 134 144 Sep 25 00:34 UTC (GMT)
0 - 100 0000 0100 - 0000 0111 0010 0011 1001 0010 0111 1000 1110 0001 1110 1100 0111 = 32.892 369 217 295 303 940 318 262 903 019 785 881 042 480 468 75 Sep 25 00:24 UTC (GMT)
0 - 110 0000 0000 - 1101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 49 022 297 743 852 620 645 317 528 899 690 124 903 596 431 281 540 938 287 301 771 528 607 699 734 956 406 133 931 852 902 672 740 657 491 678 981 494 339 623 434 037 634 031 179 646 366 648 029 178 494 976 Sep 25 00:19 UTC (GMT)
0 - 100 0001 1000 - 1000 0100 1011 1110 0001 1001 0111 0111 1010 0110 1011 0001 1011 = 50 953 266.934 774 599 969 387 054 443 359 375 Sep 25 00:18 UTC (GMT)
0 - 100 0000 0110 - 0001 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0011 = 140.153 267 198 019 051 420 487 812 720 239 162 445 068 359 375 Sep 25 00:17 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)