-19.981 999 999 999 999 317 878 973 670 303 821 563 715 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -19.981 999 999 999 999 317 878 973 670 303 821 563 715 8(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
-19.981 999 999 999 999 317 878 973 670 303 821 563 715 8(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-19.981 999 999 999 999 317 878 973 670 303 821 563 715 8| = 19.981 999 999 999 999 317 878 973 670 303 821 563 715 8


2. First, convert to binary (in base 2) the integer part: 19.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

19(10) =

1 0011(2)


4. Convert to binary (base 2) the fractional part: 0.981 999 999 999 999 317 878 973 670 303 821 563 715 8.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.981 999 999 999 999 317 878 973 670 303 821 563 715 8 × 2 = 1 + 0.963 999 999 999 998 635 757 947 340 607 643 127 431 6;
  • 2) 0.963 999 999 999 998 635 757 947 340 607 643 127 431 6 × 2 = 1 + 0.927 999 999 999 997 271 515 894 681 215 286 254 863 2;
  • 3) 0.927 999 999 999 997 271 515 894 681 215 286 254 863 2 × 2 = 1 + 0.855 999 999 999 994 543 031 789 362 430 572 509 726 4;
  • 4) 0.855 999 999 999 994 543 031 789 362 430 572 509 726 4 × 2 = 1 + 0.711 999 999 999 989 086 063 578 724 861 145 019 452 8;
  • 5) 0.711 999 999 999 989 086 063 578 724 861 145 019 452 8 × 2 = 1 + 0.423 999 999 999 978 172 127 157 449 722 290 038 905 6;
  • 6) 0.423 999 999 999 978 172 127 157 449 722 290 038 905 6 × 2 = 0 + 0.847 999 999 999 956 344 254 314 899 444 580 077 811 2;
  • 7) 0.847 999 999 999 956 344 254 314 899 444 580 077 811 2 × 2 = 1 + 0.695 999 999 999 912 688 508 629 798 889 160 155 622 4;
  • 8) 0.695 999 999 999 912 688 508 629 798 889 160 155 622 4 × 2 = 1 + 0.391 999 999 999 825 377 017 259 597 778 320 311 244 8;
  • 9) 0.391 999 999 999 825 377 017 259 597 778 320 311 244 8 × 2 = 0 + 0.783 999 999 999 650 754 034 519 195 556 640 622 489 6;
  • 10) 0.783 999 999 999 650 754 034 519 195 556 640 622 489 6 × 2 = 1 + 0.567 999 999 999 301 508 069 038 391 113 281 244 979 2;
  • 11) 0.567 999 999 999 301 508 069 038 391 113 281 244 979 2 × 2 = 1 + 0.135 999 999 998 603 016 138 076 782 226 562 489 958 4;
  • 12) 0.135 999 999 998 603 016 138 076 782 226 562 489 958 4 × 2 = 0 + 0.271 999 999 997 206 032 276 153 564 453 124 979 916 8;
  • 13) 0.271 999 999 997 206 032 276 153 564 453 124 979 916 8 × 2 = 0 + 0.543 999 999 994 412 064 552 307 128 906 249 959 833 6;
  • 14) 0.543 999 999 994 412 064 552 307 128 906 249 959 833 6 × 2 = 1 + 0.087 999 999 988 824 129 104 614 257 812 499 919 667 2;
  • 15) 0.087 999 999 988 824 129 104 614 257 812 499 919 667 2 × 2 = 0 + 0.175 999 999 977 648 258 209 228 515 624 999 839 334 4;
  • 16) 0.175 999 999 977 648 258 209 228 515 624 999 839 334 4 × 2 = 0 + 0.351 999 999 955 296 516 418 457 031 249 999 678 668 8;
  • 17) 0.351 999 999 955 296 516 418 457 031 249 999 678 668 8 × 2 = 0 + 0.703 999 999 910 593 032 836 914 062 499 999 357 337 6;
  • 18) 0.703 999 999 910 593 032 836 914 062 499 999 357 337 6 × 2 = 1 + 0.407 999 999 821 186 065 673 828 124 999 998 714 675 2;
  • 19) 0.407 999 999 821 186 065 673 828 124 999 998 714 675 2 × 2 = 0 + 0.815 999 999 642 372 131 347 656 249 999 997 429 350 4;
  • 20) 0.815 999 999 642 372 131 347 656 249 999 997 429 350 4 × 2 = 1 + 0.631 999 999 284 744 262 695 312 499 999 994 858 700 8;
  • 21) 0.631 999 999 284 744 262 695 312 499 999 994 858 700 8 × 2 = 1 + 0.263 999 998 569 488 525 390 624 999 999 989 717 401 6;
  • 22) 0.263 999 998 569 488 525 390 624 999 999 989 717 401 6 × 2 = 0 + 0.527 999 997 138 977 050 781 249 999 999 979 434 803 2;
  • 23) 0.527 999 997 138 977 050 781 249 999 999 979 434 803 2 × 2 = 1 + 0.055 999 994 277 954 101 562 499 999 999 958 869 606 4;
  • 24) 0.055 999 994 277 954 101 562 499 999 999 958 869 606 4 × 2 = 0 + 0.111 999 988 555 908 203 124 999 999 999 917 739 212 8;
  • 25) 0.111 999 988 555 908 203 124 999 999 999 917 739 212 8 × 2 = 0 + 0.223 999 977 111 816 406 249 999 999 999 835 478 425 6;
  • 26) 0.223 999 977 111 816 406 249 999 999 999 835 478 425 6 × 2 = 0 + 0.447 999 954 223 632 812 499 999 999 999 670 956 851 2;
  • 27) 0.447 999 954 223 632 812 499 999 999 999 670 956 851 2 × 2 = 0 + 0.895 999 908 447 265 624 999 999 999 999 341 913 702 4;
  • 28) 0.895 999 908 447 265 624 999 999 999 999 341 913 702 4 × 2 = 1 + 0.791 999 816 894 531 249 999 999 999 998 683 827 404 8;
  • 29) 0.791 999 816 894 531 249 999 999 999 998 683 827 404 8 × 2 = 1 + 0.583 999 633 789 062 499 999 999 999 997 367 654 809 6;
  • 30) 0.583 999 633 789 062 499 999 999 999 997 367 654 809 6 × 2 = 1 + 0.167 999 267 578 124 999 999 999 999 994 735 309 619 2;
  • 31) 0.167 999 267 578 124 999 999 999 999 994 735 309 619 2 × 2 = 0 + 0.335 998 535 156 249 999 999 999 999 989 470 619 238 4;
  • 32) 0.335 998 535 156 249 999 999 999 999 989 470 619 238 4 × 2 = 0 + 0.671 997 070 312 499 999 999 999 999 978 941 238 476 8;
  • 33) 0.671 997 070 312 499 999 999 999 999 978 941 238 476 8 × 2 = 1 + 0.343 994 140 624 999 999 999 999 999 957 882 476 953 6;
  • 34) 0.343 994 140 624 999 999 999 999 999 957 882 476 953 6 × 2 = 0 + 0.687 988 281 249 999 999 999 999 999 915 764 953 907 2;
  • 35) 0.687 988 281 249 999 999 999 999 999 915 764 953 907 2 × 2 = 1 + 0.375 976 562 499 999 999 999 999 999 831 529 907 814 4;
  • 36) 0.375 976 562 499 999 999 999 999 999 831 529 907 814 4 × 2 = 0 + 0.751 953 124 999 999 999 999 999 999 663 059 815 628 8;
  • 37) 0.751 953 124 999 999 999 999 999 999 663 059 815 628 8 × 2 = 1 + 0.503 906 249 999 999 999 999 999 999 326 119 631 257 6;
  • 38) 0.503 906 249 999 999 999 999 999 999 326 119 631 257 6 × 2 = 1 + 0.007 812 499 999 999 999 999 999 998 652 239 262 515 2;
  • 39) 0.007 812 499 999 999 999 999 999 998 652 239 262 515 2 × 2 = 0 + 0.015 624 999 999 999 999 999 999 997 304 478 525 030 4;
  • 40) 0.015 624 999 999 999 999 999 999 997 304 478 525 030 4 × 2 = 0 + 0.031 249 999 999 999 999 999 999 994 608 957 050 060 8;
  • 41) 0.031 249 999 999 999 999 999 999 994 608 957 050 060 8 × 2 = 0 + 0.062 499 999 999 999 999 999 999 989 217 914 100 121 6;
  • 42) 0.062 499 999 999 999 999 999 999 989 217 914 100 121 6 × 2 = 0 + 0.124 999 999 999 999 999 999 999 978 435 828 200 243 2;
  • 43) 0.124 999 999 999 999 999 999 999 978 435 828 200 243 2 × 2 = 0 + 0.249 999 999 999 999 999 999 999 956 871 656 400 486 4;
  • 44) 0.249 999 999 999 999 999 999 999 956 871 656 400 486 4 × 2 = 0 + 0.499 999 999 999 999 999 999 999 913 743 312 800 972 8;
  • 45) 0.499 999 999 999 999 999 999 999 913 743 312 800 972 8 × 2 = 0 + 0.999 999 999 999 999 999 999 999 827 486 625 601 945 6;
  • 46) 0.999 999 999 999 999 999 999 999 827 486 625 601 945 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 654 973 251 203 891 2;
  • 47) 0.999 999 999 999 999 999 999 999 654 973 251 203 891 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 309 946 502 407 782 4;
  • 48) 0.999 999 999 999 999 999 999 999 309 946 502 407 782 4 × 2 = 1 + 0.999 999 999 999 999 999 999 998 619 893 004 815 564 8;
  • 49) 0.999 999 999 999 999 999 999 998 619 893 004 815 564 8 × 2 = 1 + 0.999 999 999 999 999 999 999 997 239 786 009 631 129 6;
  • 50) 0.999 999 999 999 999 999 999 997 239 786 009 631 129 6 × 2 = 1 + 0.999 999 999 999 999 999 999 994 479 572 019 262 259 2;
  • 51) 0.999 999 999 999 999 999 999 994 479 572 019 262 259 2 × 2 = 1 + 0.999 999 999 999 999 999 999 988 959 144 038 524 518 4;
  • 52) 0.999 999 999 999 999 999 999 988 959 144 038 524 518 4 × 2 = 1 + 0.999 999 999 999 999 999 999 977 918 288 077 049 036 8;
  • 53) 0.999 999 999 999 999 999 999 977 918 288 077 049 036 8 × 2 = 1 + 0.999 999 999 999 999 999 999 955 836 576 154 098 073 6;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


5. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.981 999 999 999 999 317 878 973 670 303 821 563 715 8(10) =

0.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2)

6. Positive number before normalization:

19.981 999 999 999 999 317 878 973 670 303 821 563 715 8(10) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


19.981 999 999 999 999 317 878 973 670 303 821 563 715 8(10) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2) × 20 =

1.0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2) × 24


8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 1 (a negative number)

Exponent (unadjusted): 4

Mantissa (not normalized):
1.0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(10)


10. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1027(10) =

100 0000 0011(2)


12. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1 1111 =

0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
100 0000 0011

Mantissa (52 bits) =
0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111


Decimal number -19.981 999 999 999 999 317 878 973 670 303 821 563 715 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0011 - 0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100