64 bit double precision IEEE 754 binary floating point number 0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000.

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 0000 0011


The last 52 bits contain the mantissa:
1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000

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

100 0000 0011(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =


1,024 + 2 + 1 =


1,027(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 = 1,027 - 1023 = 4

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 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =


0.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 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)0 × (1 + 0.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5) × 24 =


1.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 × 24 =


30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125

0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125(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 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 = 30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 Nov 18 17:03 UTC (GMT)
1 - 101 1110 1110 - 0001 1000 1011 0001 0101 1011 1110 0101 1000 0011 1100 1000 1010 = -112 160 303 098 420 918 950 169 373 941 268 746 235 202 318 350 958 141 201 590 755 275 183 852 985 354 454 512 506 389 707 015 830 140 880 199 611 223 533 365 352 640 767 361 235 614 081 277 231 104 Nov 18 17:01 UTC (GMT)
0 - 000 0000 0000 - 0100 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Nov 18 16:59 UTC (GMT)
1 - 100 0000 0100 - 1011 0001 1100 0010 1000 1111 0100 1110 0001 0100 0111 1010 0111 = -54.219 999 895 095 774 888 886 808 184 906 840 324 401 855 468 75 Nov 18 16:58 UTC (GMT)
0 - 011 1110 0001 - 0001 0010 1110 0000 1011 1110 1000 0010 0110 1101 0110 1001 0101 = 0.000 000 001 000 000 000 000 000 062 281 591 457 779 856 418 897 068 692 785 978 782 922 029 495 239 257 812 5 Nov 18 16:58 UTC (GMT)
1 - 100 0000 1101 - 1111 1110 0010 1111 0011 1011 1000 1000 0010 0110 1010 1010 1000 = -32 651.808 136 562 496 656 551 957 130 432 128 906 25 Nov 18 16:57 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 Nov 18 16:55 UTC (GMT)
0 - 100 0010 0011 - 0001 0110 0110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 74 742 497 279.999 984 741 210 937 5 Nov 18 16:55 UTC (GMT)
0 - 111 1111 0010 - 1001 1100 0111 0111 1001 1010 0110 1011 0101 0000 1011 0000 1111 = 35 356 972 398 561 564 069 611 490 667 011 397 668 352 972 360 942 080 801 462 626 265 829 916 801 015 199 130 165 305 944 897 285 380 947 168 285 413 522 069 890 245 003 801 304 055 770 267 122 103 589 597 932 503 707 032 473 943 018 051 822 190 107 149 284 036 342 759 899 493 295 873 536 260 293 681 166 910 636 370 972 680 561 206 858 348 334 018 112 649 182 093 226 773 897 174 373 774 038 489 104 384 Nov 18 16:54 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 1100 1101 0001 0111 0000 0110 1010 0010 = 0 Nov 18 16:53 UTC (GMT)
0 - 000 1000 0110 - 0000 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 244 418 581 662 055 655 286 970 480 047 224 281 439 2 Nov 18 16:51 UTC (GMT)
0 - 111 1111 1111 - 0111 1000 0001 1011 0000 0100 0011 1101 1110 1111 0100 1110 1111 = SNaN, Signalling Not a Number Nov 18 16:45 UTC (GMT)
1 - 100 1100 0111 - 0111 0110 0110 0000 0110 0000 1001 0011 0101 1010 1010 0101 0110 = -2 349 999 212 290 342 694 039 218 688 336 252 332 275 839 940 449 760 595 410 944 Nov 18 16:44 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)