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₎