Converter of 64 bit double precision IEEE 754 binary floating point standard system numbers: converting to base ten decimal (double)

Convert 64 bit double precision IEEE 754 floating point standard binary numbers to base ten decimal system (double)

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: 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 0000 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -6 016 007 607 665 247 123 702 295 517 792 197 401 620 433 316 327 199 368 163 826 647 350 947 441 659 710 434 532 153 834 211 005 102 058 139 956 451 382 908 673 914 872 670 944 905 766 846 613 190 784 576 914 417 263 032 080 793 384 007 277 981 822 443 973 697 894 121 875 651 316 409 709 781 558 537 224 192 May 24 15:02 UTC (GMT)
0 - 100 0001 0000 - 0000 0000 0100 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 131 208 May 24 14:59 UTC (GMT)
1 - 110 1011 1110 - 1011 1110 1011 1110 1011 1110 1011 1110 1011 1110 1011 1110 1011 = -73 435 622 782 400 813 686 143 151 113 110 365 575 435 840 418 095 255 275 297 506 064 069 659 137 818 951 812 398 040 764 533 185 524 715 064 856 092 106 547 425 013 440 122 109 769 664 608 675 607 646 651 085 489 240 444 353 012 735 852 259 223 372 053 716 078 873 457 780 064 256 May 24 14:58 UTC (GMT)
1 - 100 0001 0100 - 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3 211 264 May 24 14:56 UTC (GMT)
0 - 100 0000 0101 - 1110 0001 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 = 120.349 999 999 999 994 315 658 113 919 198 513 031 005 859 375 May 24 14:56 UTC (GMT)
1 - 001 1101 1011 - 0110 1010 0100 1100 1010 0001 0000 1011 0000 0100 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 535 995 504 094 096 888 706 114 211 351 265 688 284 156 805 335 130 987 673 622 212 286 245 723 919 702 014 238 036 962 118 360 220 573 071 253 387 972 783 597 904 279 074 487 061 100 083 6 May 24 14:56 UTC (GMT)
0 - 000 0001 1011 - 0110 1001 1011 0100 1011 1010 1100 1101 0000 0101 1111 0001 0101 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 109 793 7 May 24 14:55 UTC (GMT)
0 - 011 1111 0011 - 0001 0000 0001 0110 1110 0101 0011 0011 0001 0011 1001 0010 1101 = 0.000 259 484 705 741 014 007 058 381 496 406 695 987 388 957 291 841 506 958 007 812 5 May 24 14:52 UTC (GMT)
0 - 010 0000 0000 - 1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 269 199 485 766 757 462 140 658 279 354 990 268 192 807 019 451 983 436 340 112 614 839 164 110 461 124 582 568 790 062 450 390 441 027 579 559 430 015 334 286 834 115 654 687 038 267 901 922 511 818 089 1 May 24 14:52 UTC (GMT)
1 - 100 0000 0000 - 0110 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1101 = -2.850 000 000 000 000 088 817 841 970 012 523 233 890 533 447 265 625 May 24 14:48 UTC (GMT)
0 - 010 1001 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 326 531 125 006 367 749 929 729 465 818 210 037 692 653 698 680 220 000 722 695 384 255 006 897 457 462 606 472 685 065 411 997 579 646 945 972 791 215 254 204 058 266 949 967 710 391 868 727 437 133 322 359 195 194 490 533 570 238 064 438 569 426 280 522 502 5 May 24 14:48 UTC (GMT)
1 - 100 0000 0011 - 0111 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -23.375 May 24 14:47 UTC (GMT)
1 - 000 0111 1100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 236 610 437 233 354 942 194 550 319 503 605 306 3 May 24 14:47 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)