64 Bit IEEE 754 Binary to Double: Convert 0 - 111 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001, Number Written in 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation, to a Base Ten Decimal System Double
0 - 111 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001: 64 bit double precision IEEE 754 binary floating point standard representation number converted to a base ten decimal system double
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 11 bits contain the exponent:
111 1111 1110
The last 52 bits contain the mantissa:
1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001
1. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1111 1110(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =
2,046(10)
2. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 2,046 - 1023 = 1023
2. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(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 + 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 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =
0.5 + 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 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.500 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5(10)
3. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.500 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 21023 =
1.500 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 21023 = ...
= 134 826 985 114 673 713 038 100 984 656 374 971 585 075 403 789 058 653 629 585 165 222 874 921 879 598 146 721 650 276 681 446 309 865 419 740 336 260 555 911 668 560 768 535 764 906 594 015 456 915 542 200 675 043 902 916 963 317 157 283 394 211 791 335 764 967 026 657 693 626 153 162 180 457 196 087 094 413 539 384 114 324 316 761 065 567 493 881 298 751 169 503 211 420 203 652 211 469 550 658 267 381 760
0 - 111 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = 134 826 985 114 673 713 038 100 984 656 374 971 585 075 403 789 058 653 629 585 165 222 874 921 879 598 146 721 650 276 681 446 309 865 419 740 336 260 555 911 668 560 768 535 764 906 594 015 456 915 542 200 675 043 902 916 963 317 157 283 394 211 791 335 764 967 026 657 693 626 153 162 180 457 196 087 094 413 539 384 114 324 316 761 065 567 493 881 298 751 169 503 211 420 203 652 211 469 550 658 267 381 760(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.