IEEE 754 Double Precision Format

IEEE 754 double precision, known as double in C/C++/Java and the default number type in JavaScript and Python, uses 64 bits to represent floating-point values with much greater range and precision than single precision.

The 64-bit layout:

Field	Bits	Position
Sign	1 bit	Bit 63
Exponent	11 bits	Bits 62-52
Mantissa	52 bits	Bits 51-0

How Pi (3.14159...) is stored:

The mathematical constant Pi cannot be represented exactly in any finite binary format. The closest double-precision representation is:

1. Sign = 0 (positive)
2. Biased exponent = 1024 (actual exponent = 1024 - 1023 = 1)
3. Mantissa = the 52-bit fraction closest to Pi/2 - 1

The stored value is 3.141592653589793, which differs from true Pi by about 1.22e-16. The hex representation is 0x400921FB54442D18.

Why double precision matters:

• 15-16 decimal digits of precision (vs. 7 for float32)
• Exponent range from -1022 to +1023 (vs. -126 to +127)
• Largest value approximately 1.80e+308 (vs. 3.40e+38)
• Smallest normalized approximately 2.23e-308

JavaScript and double precision:

All numbers in JavaScript are IEEE 754 double precision. There is no native 32-bit float type (though Float32Array provides typed array access). This means expressions like 0.1 + 0.2 use 64-bit arithmetic, and the maximum safe integer (Number.MAX_SAFE_INTEGER) is 2^53 - 1 = 9007199254740991.

Memory trade-off:

Double precision uses 8 bytes per value compared to 4 bytes for single precision. In applications processing millions of values (scientific simulation, financial modeling, data analytics), this doubled memory footprint can be significant.