64 bit double precision IEEE 754 binary floating point number 1 - 111 0001 0101 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 111 0001 0101 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 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:
111 0001 0101


The last 52 bits contain the mantissa:
1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

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

111 0001 0101(2) =


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


1,024 + 512 + 256 + 0 + 0 + 0 + 16 + 0 + 4 + 0 + 1 =


1,024 + 512 + 256 + 16 + 4 + 1 =


1,813(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 = 1,813 - 1023 = 790

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):

1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 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 + 0 × 2-52 =


0.5 + 0.25 + 0.125 + 0.062 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 =


0.5 + 0.25 + 0.125 + 0.062 5 =


0.937 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)1 × (1 + 0.937 5) × 2790 =


-1.937 5 × 2790 =


-12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184

1 - 111 0001 0101 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


-12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184(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 - 111 0001 0101 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184 Aug 25 05:57 UTC (GMT)
1 - 011 1111 1101 - 1000 0101 1011 1100 1101 0001 1111 0111 1000 1001 0101 1101 0101 = -0.380 603 104 337 152 331 471 116 895 045 270 211 994 647 979 736 328 125 Aug 25 05:54 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 689 964 596 767 805 383 442 867 854 275 239 384 948 099 050 998 034 608 370 797 314 674 312 438 223 714 234 551 696 550 951 604 564 015 091 723 428 328 493 871 551 256 955 877 586 285 902 168 064 Aug 25 05:54 UTC (GMT)
1 - 010 0101 1011 - 0000 1100 0000 1001 0111 0001 1100 0111 0001 1001 0010 0101 0010 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 386 684 202 490 747 323 007 464 959 363 989 953 917 493 762 247 826 225 709 329 402 503 154 670 078 002 036 388 392 071 991 415 256 393 607 209 150 547 077 596 063 305 112 762 915 535 581 737 178 340 065 706 577 612 235 067 692 280 281 679 8 Aug 25 05:54 UTC (GMT)
0 - 011 1110 0101 - 0011 0010 1100 1000 1000 1011 0111 1010 1011 0101 1000 1010 0001 = 0.000 000 017 857 142 857 142 855 862 113 075 431 953 090 888 015 367 454 499 937 593 936 920 166 015 625 Aug 25 05:53 UTC (GMT)
0 - 100 0000 0010 - 0010 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1101 = 9.400 000 000 000 000 355 271 367 880 050 092 935 562 133 789 062 5 Aug 25 05:53 UTC (GMT)
0 - 111 1111 1111 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = QNaN, Quiet Not a Number Aug 25 05:50 UTC (GMT)
0 - 100 0000 1001 - 0100 0101 1000 0111 1110 0110 1011 0111 0100 0100 0010 0100 0111 = 1 302.123 456 779 999 969 512 573 443 353 176 116 943 359 375 Aug 25 05:49 UTC (GMT)
1 - 100 0000 0010 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -15 Aug 25 05:47 UTC (GMT)
0 - 100 0000 0100 - 0001 1110 0000 1010 0011 1101 0111 0000 0101 0001 1110 1011 1000 = 35.754 999 997 615 811 935 247 620 567 679 405 212 402 343 75 Aug 25 05:47 UTC (GMT)
1 - 011 1111 0000 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 038 146 972 656 25 Aug 25 05:46 UTC (GMT)
1 - 111 1100 0011 - 1111 1111 1111 1000 0000 0000 0000 0001 1111 1111 1111 1111 0000 = -311 831 014 620 979 148 475 404 892 213 894 993 487 302 883 978 800 993 037 993 467 169 936 537 533 746 718 649 714 626 435 482 700 853 631 332 545 859 624 875 785 259 429 637 622 242 153 583 666 775 801 040 220 521 346 203 578 736 249 726 868 306 238 435 279 628 088 036 214 268 641 183 784 275 801 622 685 803 861 613 246 514 419 272 870 386 437 519 211 884 880 488 000 104 997 847 040 Aug 25 05:44 UTC (GMT)
0 - 011 1111 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 0.500 000 000 000 000 111 022 302 462 515 654 042 363 166 809 082 031 25 Aug 25 05:43 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)