64 bit double precision IEEE 754 binary floating point number 0 - 011 1111 1110 - 1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 011 1111 1110 - 1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000.

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 1111 1110


The last 52 bits contain the mantissa:
1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000

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

011 1111 1110(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =


512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =


1,022(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 = 1,022 - 1023 = -1

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):

1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 0 + 0 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 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 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 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 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 =


0.904 517 034 162 978 461 608 872 748 911 380 767 822 265 625(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)0 × (1 + 0.904 517 034 162 978 461 608 872 748 911 380 767 822 265 625) × 2-1 =


1.904 517 034 162 978 461 608 872 748 911 380 767 822 265 625 × 2-1 =


0.952 258 517 081 489 230 804 436 374 455 690 383 911 132 812 5

0 - 011 1111 1110 - 1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0.952 258 517 081 489 230 804 436 374 455 690 383 911 132 812 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)

0 - 011 1111 1110 - 1110 0111 1000 1110 0110 1101 1010 1000 0110 0111 1010 1000 0000 = 0.952 258 517 081 489 230 804 436 374 455 690 383 911 132 812 5 Jun 24 16:13 UTC (GMT)
0 - 100 0000 0110 - 0011 0111 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 155.625 Jun 24 16:13 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Jun 24 16:10 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 Jun 24 16:09 UTC (GMT)
0 - 111 0010 0110 - 0101 0111 0011 0111 0011 0110 0001 0100 1110 0110 1000 0100 0001 = 1 144 283 517 142 482 188 112 020 266 720 057 419 136 298 954 335 226 472 190 306 906 372 871 084 264 114 568 379 010 961 296 102 317 094 940 241 499 474 946 985 053 109 511 587 646 001 110 865 029 627 005 804 607 780 259 462 420 924 572 488 588 619 024 838 453 814 155 236 534 054 031 332 999 004 074 641 232 794 316 963 840 Jun 24 16:08 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 789 860 550 377 917 134 847 078 965 628 620 706 732 054 191 563 313 685 198 290 337 382 324 333 865 946 734 395 546 346 249 636 307 092 206 185 224 213 505 804 205 592 177 614 811 415 703 453 696 Jun 24 16:08 UTC (GMT)
0 - 100 0000 1000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 768 Jun 24 16:07 UTC (GMT)
1 - 100 0100 1001 - 1001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -29 680 811 214 598 668 550 144 Jun 24 16:06 UTC (GMT)
0 - 011 1110 1000 - 0001 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 127 591 192 722 320 556 640 625 Jun 24 16:05 UTC (GMT)
1 - 000 0000 0000 - 1100 0010 0011 0000 0000 0111 1111 1100 0000 0000 0010 0000 0000 = -0 Jun 24 16:04 UTC (GMT)
0 - 000 0000 0101 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 3 Jun 24 16:03 UTC (GMT)
0 - 100 0001 1001 - 0110 1100 1110 1001 0010 1100 0000 0111 0110 0001 1001 0111 1010 = 95 659 184.115 331 560 373 306 274 414 062 5 Jun 24 16:02 UTC (GMT)
1 - 011 1111 1111 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1.5 Jun 24 16:01 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)