This page summarizes IEEE-754 binary floating-point formats as implemented on Intel x86 CPUs up to Core 2.

Intel x86 supports:

• Binary32 (single precision)
• Binary64 (double precision)
• x87 80-bit extended precision

Intel CPUs do not implement IEEE quad (128-bit) precision in hardware.

Binary32 (Single Precision, 32-bit)

Bit layout

+----+------------------+----------------------+
| S  |   Exponent (8)   |    Fraction (23)     |
+----+------------------+----------------------+
31  30                23 22                  0

• Sign: 1 bit (bit 31)
• Exponent: 8 bits (bits 30–23)
• Fraction: 23 bits (bits 22–0)
• Bias: 127 (0x7F)
• value = (−1)^S × 2^(E − 127) × (1.F)
• Minimum normal: 2⁻¹²⁶ ≈ 1.17549435e−38
• Maximum finite: (2 − 2⁻²³) × 2¹²⁷ ≈ 3.40282347e+38

Special values

• +0 Exponent = 0x00, Fraction = 0x000000 Hex: 0x00000000
• −0 Exponent = 0x00, Fraction = 0x000000 Hex: 0x80000000
• +∞ Exponent = 0xFF, Fraction = 0x000000 Hex: 0x7F800000
• −∞ Exponent = 0xFF, Fraction = 0x000000 Hex: 0xFF800000
• Quiet NaN Exponent = 0xFF, Fraction MSB = 1 Hex: 0x7FC00000
• Signaling NaN Exponent = 0xFF, Fraction MSB = 0, Fraction ≠ 0 Hex: 0x7FA00000

Binary64 (Double Precision, 64-bit)

Bit layout

+----+----------------------+----------------------------------------+
| S  |   Exponent (11)      |           Fraction (52)                 |
+----+----------------------+----------------------------------------+
63  62                    52 51                                      0

• Sign: 1 bit (bit 63)
• Exponent: 11 bits (bits 62–52)
• Fraction: 52 bits (bits 51–0)
• Bias: 1023 (0x3FF)
• value = (−1)^S × 2^(E − 1023) × (1.F)
• Minimum normal: 2⁻¹⁰²² ≈ 2.225074e−308
• Maximum finite: (2 − 2⁻⁵²) × 2¹⁰²³ ≈ 1.797693e+308

Special values

• +0 Exponent = 0x000, Fraction = 0x0000000000000 Hex: 0x0000000000000000
• −0 Exponent = 0x000, Fraction = 0x0000000000000 Hex: 0x8000000000000000
• +∞ Exponent = 0x7FF, Fraction = 0x0000000000000 Hex: 0x7FF0000000000000
• −∞ Exponent = 0x7FF, Fraction = 0x0000000000000 Hex: 0xFFF0000000000000
• Quiet NaN Exponent = 0x7FF, Fraction MSB = 1 Hex: 0x7FF8000000000000
• Signaling NaN Exponent = 0x7FF, Fraction MSB = 0, Fraction ≠ 0 Hex: 0x7FF4000000000000

x87 Extended Precision (80-bit)

Bit layout

+----+----------------------+--+--------------------------------------+
| S  |   Exponent (15)      |J |           Fraction (63)               |
+----+----------------------+--+--------------------------------------+
79  78                    64 63                                      0

• Sign: 1 bit (bit 79)
• Exponent: 15 bits (bits 78–64)
• Integer bit (J): 1 bit (bit 63, explicit)
• Fraction: 63 bits (bits 62–0)
• Bias: 16383 (0x3FFF)
• value = (−1)^S × 2^(E − 16383) × (J.F)
• Minimum normal: 2⁻¹⁶³⁸² ≈ 3.3621e−4932
• Maximum finite: (2 − 2⁻⁶³) × 2¹⁶³⁸³ ≈ 1.1897e+4932

Special values

• +0 Exponent = 0x0000, J = 0, Fraction = 0 Hex: 00 00 00 00 00 00 00 00 00 00
• −0 Exponent = 0x0000, J = 0, Fraction = 0 Hex: 00 00 00 00 00 00 00 00 00 80
• +∞ Exponent = 0x7FFF, J = 1, Fraction = 0 Hex: 00 00 00 00 00 00 00 80 FF 7F
• −∞ Exponent = 0x7FFF, J = 1, Fraction = 0 Hex: 00 00 00 00 00 00 00 80 FF FF
• Quiet NaN Exponent = 0x7FFF, J = 1, Fraction MSB = 1 Hex: 00 00 00 00 00 00 00 C0 FF 7F
• Signaling NaN Exponent = 0x7FFF, J = 1, Fraction MSB = 0 Hex: 00 00 00 00 00 00 00 A0 FF 7F
• Pseudo-NaN Exponent = 0x7FFF, J = 0 Hex: architecture-specific

Intel x86 Notes

• x87 always computes using 80-bit extended precision
• SSE instructions operate only on 32-bit and 64-bit formats
• Intel CPUs up to Core 2 never implement IEEE-754 quad precision
• The C type long double is ABI-defined
• Extra precision may exist only in registers, not memory

Summary

• Binary32: 32 bits total, 8 exponent bits, 23 fraction bits
• Binary64: 64 bits total, 11 exponent bits, 52 fraction bits
• x87 Extended: 80 bits total, 15 exponent bits, 63 fraction bits, explicit integer bit