64 bit double precision IEEE 754 binary floating point number 1 - 011 1110 0000 - 0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 011 1110 0000 - 0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010.

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:
011 1110 0000


The last 52 bits contain the mantissa:
0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010

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

011 1110 0000(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 0 + 0 + 0 + 0 + 0 =


512 + 256 + 128 + 64 + 32 =


992(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 = 992 - 1023 = -31

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):

0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010(2) =

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


0 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 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 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =


0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 003 725 290 298 461 914 062 5 + 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 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 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =


0.469 397 820 073 935 267 970 455 242 902 971 804 141 998 291 015 625(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 + 0.469 397 820 073 935 267 970 455 242 902 971 804 141 998 291 015 625) × 2-31 =


-1.469 397 820 073 935 267 970 455 242 902 971 804 141 998 291 015 625 × 2-31 =


-0.000 000 000 684 241 680 462 814 526 618 670 319 627 491 666 070 184 805 903 409 142 047 166 824 340 820 312 5

1 - 011 1110 0000 - 0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


-0.000 000 000 684 241 680 462 814 526 618 670 319 627 491 666 070 184 805 903 409 142 047 166 824 340 820 312 5(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 - 011 1110 0000 - 0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010 = -0.000 000 000 684 241 680 462 814 526 618 670 319 627 491 666 070 184 805 903 409 142 047 166 824 340 820 312 5 May 20 17:40 UTC (GMT)
0 - 100 0000 0001 - 0011 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 1101 1011 = 4.857 142 857 142 856 762 209 248 699 946 328 997 611 999 511 718 75 May 20 17:39 UTC (GMT)
0 - 110 0000 0000 - 1110 1101 1100 1100 1100 1100 0000 0000 0000 0000 0000 0000 0000 = 51 724 807 751 063 469 163 770 000 339 488 765 440 385 572 120 128 293 341 926 993 490 197 177 134 715 841 225 867 482 689 619 627 314 829 139 640 844 942 230 731 094 663 059 971 788 634 994 400 099 827 712 May 20 17:36 UTC (GMT)
0 - 100 0010 0011 - 0100 0100 0000 1111 1010 0100 0010 1110 0101 0101 0101 1101 0101 = 86 989 488 869.335 281 372 070 312 5 May 20 17:35 UTC (GMT)
0 - 100 1111 0000 - 1000 0000 0010 0100 0110 1000 1010 1100 0000 0000 0000 0000 0000 = 5 302 504 356 965 246 824 333 661 319 592 128 864 273 019 323 720 407 554 286 467 208 479 506 432 May 20 17:35 UTC (GMT)
0 - 100 0001 0110 - 0000 0010 1111 1111 0000 1010 0001 0111 0000 0101 1001 1101 0110 = 8 486 789.044 964 712 113 142 013 549 804 687 5 May 20 17:34 UTC (GMT)
0 - 011 1111 1011 - 1010 0111 1101 1110 0001 0110 1101 0100 1011 1111 0011 1010 1101 = 0.103 483 285 125 504 934 076 623 442 251 730 011 776 089 668 273 925 781 25 May 20 17:29 UTC (GMT)
0 - 100 0001 1111 - 0000 0010 1110 1100 0010 1110 0000 0000 0000 0000 0000 0000 0000 = 4 344 000 000 May 20 17:28 UTC (GMT)
0 - 011 1011 1110 - 0111 0110 1100 1000 1011 1111 1111 1110 1011 1100 0110 1010 0110 = 0.000 000 000 000 000 000 039 681 818 530 146 577 878 838 259 635 139 474 006 523 138 770 412 810 092 877 431 422 664 358 251 495 286 822 319 030 761 718 75 May 20 17:26 UTC (GMT)
0 - 100 0001 1101 - 0111 0001 1011 0111 1010 0011 0101 0000 0000 0000 0000 0000 0000 = 1 550 706 900 May 20 17:26 UTC (GMT)
1 - 100 0000 1000 - 0100 0011 1101 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -647.703 125 May 20 17:18 UTC (GMT)
1 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -30 May 20 17:16 UTC (GMT)
0 - 010 0100 0000 - 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 783 507 823 781 784 385 008 671 819 696 767 919 374 061 782 681 257 739 146 242 645 026 470 426 212 709 840 062 720 496 782 514 395 838 304 552 162 787 354 679 547 675 607 846 089 238 139 475 213 279 437 274 496 323 759 185 386 6 May 20 17:16 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)