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 0001 0101 - 0000 0011 0011 0011 1110 0101 1110 1010 1000 0100 0100 1000 1111 = -4 246 777.479 020 251 892 507 076 263 427 734 375 Jan 24 04:09 UTC (GMT)
1 - 100 0001 0001 - 1111 1110 1001 0001 1100 1000 1011 0110 1010 0111 1101 1100 1001 = -522 823.136 148 419 755 045 324 563 980 102 539 062 5 Jan 24 04:09 UTC (GMT)
0 - 100 0001 0010 - 0000 0100 0110 1010 1010 1010 1000 1111 0100 0000 0000 0000 0000 = 533 333.329 986 572 265 625 Jan 24 03:57 UTC (GMT)
0 - 100 0000 0101 - 0101 1110 0000 0000 0000 0001 0000 1100 0110 1111 0111 1010 0001 = 87.500 004 000 000 004 111 825 546 715 408 563 613 891 601 562 5 Jan 24 03:55 UTC (GMT)
0 - 111 1111 1110 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 134 826 985 114 673 673 121 294 793 961 978 738 457 621 143 052 287 332 280 559 956 513 724 091 828 653 297 977 412 439 302 164 994 166 260 430 483 546 534 124 819 623 884 685 860 027 145 255 689 043 668 985 891 607 937 220 334 910 757 169 935 190 077 805 414 212 597 770 433 882 500 095 045 566 317 607 227 797 330 281 696 916 538 983 960 198 532 998 148 608 751 365 696 299 370 805 041 564 943 778 068 824 064 Jan 24 03:51 UTC (GMT)
0 - 011 1111 0001 - 1010 0011 0110 1110 0010 1110 1011 0001 1100 0100 0011 0010 1100 = 0.000 099 999 999 999 999 991 239 646 446 317 124 173 219 781 368 970 870 971 679 687 5 Jan 24 03:45 UTC (GMT)
1 - 111 1111 1111 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = QNaN, Quiet Not a Number Jan 24 03:44 UTC (GMT)
1 - 111 1111 1111 - 1100 0011 0001 1001 1001 1000 0000 0110 0100 0000 0000 0000 0000 = QNaN, Quiet Not a Number Jan 24 03:43 UTC (GMT)
1 - 111 1111 1111 - 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = SNaN, Signalling Not a Number Jan 24 03:43 UTC (GMT)
1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 = -3 875 739 974 753 834 706 075 163 486 913 256 793 856 570 234 510 836 280 412 770 513 148 715 349 080 425 257 484 493 363 182 218 688 710 603 984 309 181 221 006 228 382 042 997 444 461 286 993 983 555 596 632 805 592 952 453 024 263 932 572 099 783 602 225 529 645 252 313 717 991 116 420 093 794 541 292 813 610 327 893 705 685 088 582 916 171 632 511 843 259 383 808 Jan 24 03:43 UTC (GMT)
1 - 101 1100 1010 - 0001 0010 0110 0011 1000 0010 1010 1111 0101 0010 1010 1000 0010 = -1 595 490 736 029 241 154 837 177 729 981 154 941 948 596 263 229 714 338 765 839 157 038 697 237 824 497 339 153 227 557 953 318 823 221 288 655 574 310 275 200 659 640 852 120 862 720 Jan 24 03:43 UTC (GMT)
1 - 100 1111 1111 - 1000 1111 0011 0111 0001 1111 1000 1111 0110 0001 1110 0000 0011 = -180 570 220 993 002 741 689 188 481 472 832 370 404 368 915 030 297 950 134 427 552 840 043 433 295 872 Jan 24 03:43 UTC (GMT)
1 - 100 1111 0110 - 1001 1110 1101 1110 1110 1001 0111 0111 1010 1110 0001 0100 0111 = -366 506 583 389 293 531 152 712 530 761 625 941 712 628 926 189 466 468 387 534 772 000 629 194 752 Jan 24 03:43 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)