binary64

Binary64 is the IEEE 754 64-bit double-precision floating-point format used to represent real numbers in binary. It encodes numbers with 64 bits: 1 sign bit, 11 exponent bits, and 52 fraction bits. The value is (-1)^sign × (1.fraction) × 2^(exponent − 1023) for normal numbers. If the exponent field is zero, the number is subnormal and has the form (-1)^sign × 0.fraction × 2^(−1022). When the exponent field is all ones (2047), the representation denotes either infinity (fraction zero) or NaN (fraction nonzero).

With 52 fraction bits plus the implicit leading 1, binary64 provides 53 bits of precision, roughly 15 to 17 decimal digits. The finite range spans about 5.0×10^−324 to 1.8×10^308; the largest finite value is around 1.7976931348623157×10^308. The smallest positive normal is ~2.2250738585072014×10^−308, and the smallest positive subnormal is ~4.9406564584124654×10^−324.

Usage and implications: Binary64 is the default floating-point format in many languages, including C, C++, Java, and Python. It supports standard IEEE rounding modes, typically round-to-nearest, ties-to-even. Because only 53 bits of precision are available, integers larger than 2^53 cannot be represented exactly. Arithmetic can produce rounding errors, overflow to infinity, underflow to subnormals, or NaN results in invalid operations. The format is a cornerstone of numeric computing, balancing range with precision for a wide range of applications.