64 bit double precision IEEE 754 binary floating point number 1 - 100 0100 1111 - 1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 100 0100 1111 - 1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101.

1. Identify the elements that make up the binary representation of the number:

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.


The next 11 bits contain the exponent:
100 0100 1111


The last 52 bits contain the mantissa:
1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101

2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):

100 0100 1111(2) =


1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =


1,024 + 0 + 0 + 0 + 64 + 0 + 0 + 8 + 4 + 2 + 1 =


1,024 + 64 + 8 + 4 + 2 + 1 =


1,103(10)

3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1023, that is due to the 11 bit excess/bias notation:

Exponent adjusted = 1,103 - 1023 = 80

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):

1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101(2) =

1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 1 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 0 × 2-30 + 1 × 2-31 + 0 × 2-32 + 1 × 2-33 + 0 × 2-34 + 1 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 1 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 1 × 2-44 + 1 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 1 × 2-52 =


0.5 + 0 + 0 + 0.062 5 + 0 + 0 + 0 + 0.003 906 25 + 0 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0 + 0 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 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 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.062 5 + 0.003 906 25 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 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 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.567 769 129 449 613 219 051 684 609 439 689 666 032 791 137 695 312 5(10)

Conclusion:

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 + 0.567 769 129 449 613 219 051 684 609 439 689 666 032 791 137 695 312 5) × 280 =


-1.567 769 129 449 613 219 051 684 609 439 689 666 032 791 137 695 312 5 × 280 =


-1 895 316 579 786 387 326 238 720

1 - 100 0100 1111 - 1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


-1 895 316 579 786 387 326 238 720(10)

Convert 64 bit double precision IEEE 754 floating point standard binary numbers to base ten decimal system (double)

64 bit double precision IEEE 754 binary floating point standard representation of numbers requires three building blocks: 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 0100 1111 - 1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101 = -1 895 316 579 786 387 326 238 720 Apr 18 14:21 UTC (GMT)
0 - 011 1111 1100 - 1111 1111 1001 1010 1111 0010 1011 1111 1011 1100 0000 0000 0000 = 0.249 807 259 060 162 323 294 207 453 727 722 167 968 75 Apr 18 14:21 UTC (GMT)
0 - 100 0000 0100 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 60 Apr 18 14:21 UTC (GMT)
1 - 011 1111 1110 - 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.609 375 Apr 18 14:14 UTC (GMT)
0 - 011 1111 1111 - 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 = 1.099 999 999 999 999 866 773 237 044 981 215 149 164 199 829 101 562 5 Apr 18 14:09 UTC (GMT)
0 - 100 0000 1111 - 1100 0000 0000 0000 0000 1100 0000 0000 0000 0100 0000 0000 0000 = 114 688.046 875 238 418 579 101 562 5 Apr 18 14:08 UTC (GMT)
1 - 111 1101 1001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -653 996 952 628 336 987 883 560 210 607 911 261 328 982 429 019 490 727 199 554 680 401 825 592 727 622 145 076 415 026 132 626 866 532 955 732 981 904 996 841 544 888 480 036 812 770 751 011 814 861 973 559 810 459 458 912 611 754 481 266 760 562 888 863 640 011 851 938 052 153 014 134 639 969 934 006 809 031 100 094 365 055 109 531 933 378 765 047 739 725 368 031 717 079 125 173 169 291 264 Apr 18 14:05 UTC (GMT)
0 - 010 0000 0011 - 0100 0000 0110 1111 1110 1000 0010 1000 1111 0101 1100 0010 1000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 493 705 840 585 533 843 639 099 403 500 740 599 816 313 020 776 182 558 382 325 059 098 827 707 778 537 008 005 585 892 859 959 621 235 849 912 573 802 556 937 918 280 680 732 758 355 682 697 736 611 923 Apr 18 14:04 UTC (GMT)
0 - 101 1111 0000 - 0000 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 412 370 496 502 541 309 796 583 467 666 758 096 324 166 820 253 472 029 143 891 197 738 673 105 211 135 759 355 411 954 990 275 035 422 328 498 293 413 329 257 997 095 795 331 265 335 819 707 088 896 Apr 18 14:04 UTC (GMT)
0 - 100 0000 0000 - 0101 0010 0000 1011 1101 1001 0000 0111 1100 1110 1001 1011 1000 = 2.640 986 565 410 653 241 769 978 194 497 525 691 986 083 984 375 Apr 18 14:01 UTC (GMT)
1 - 011 1110 1111 - 0100 1111 1000 1011 0111 0000 0011 1111 0001 1001 1010 1000 0000 = -0.000 020 000 021 546 807 359 579 256 413 439 907 191 786 915 063 858 032 226 562 5 Apr 18 14:00 UTC (GMT)
0 - 111 1111 1000 - 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 0000 0000 = 2 340 746 269 351 920 615 780 945 207 978 231 785 292 726 542 302 699 884 547 509 236 206 870 814 957 662 131 806 691 792 760 427 193 062 588 345 858 796 385 212 599 646 648 231 210 507 029 260 423 953 866 283 254 231 850 131 727 472 717 239 788 773 184 716 174 473 501 258 948 641 858 900 232 505 027 663 280 912 773 736 956 650 947 757 301 160 028 017 235 108 463 077 996 177 509 826 004 226 368 384 008 192 Apr 18 13:57 UTC (GMT)
0 - 010 0000 0001 - 0100 0000 0010 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 373 106 408 492 638 427 667 865 932 604 502 807 781 842 560 194 390 484 906 864 297 700 011 715 253 415 007 355 614 355 973 307 861 842 912 133 585 730 167 918 280 172 330 040 095 798 095 423 405 862 654 3 Apr 18 13:54 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)