Half-precision floating-point format

The half-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 15; also known as exponent bias in the IEEE 754 standard.

• E_min = 00001₂ − 01111₂ = −14
• E_max = 11110₂ − 01111₂ = 15
• Exponent bias = 01111₂ = 15

Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 15 has to be subtracted from the stored exponent.

The stored exponents 00000₂ and 11111₂ are interpreted specially.

The minimum strictly positive (subnormal) value is 2⁻²⁴ ≈ 5.96 × 10⁻⁸. The minimum positive normal value is 2⁻¹⁴ ≈ 6.10 × 10⁻⁵. The maximum representable value is (2−2⁻¹⁰) × 2¹⁵ = 65504.

These examples are given in bit representation of the floating-point value. This includes the sign bit, (biased) exponent, and significand.

0 00000 00000000012 = 000116 = ${\displaystyle 2^{-14}\times (0+{\frac {1}{1024}})}$ ≈ 0.000000059604645
                              (smallest positive subnormal number)

0 00000 11111111112 = 03ff16 = ${\displaystyle 2^{-14}\times (0+{\frac {1023}{1024}})}$ ≈ 0.000060975552
                              (largest subnormal number)

0 00001 00000000002 = 040016 = ${\displaystyle 2^{-14}\times (1+{\frac {0}{1024}})}$ ≈ 0.00006103515625
                              (smallest positive normal number)

0 11110 11111111112 = 7bff16 = ${\displaystyle 2^{15}\times (1+{\frac {1023}{1024}})}$ = 65504
                              (largest normal number)

0 01110 11111111112 = 3bff16 = ${\displaystyle 2^{-1}\times (1+{\frac {1023}{1024}})}$ ≈ 0.99951172
                              (largest number less than one)

0 01111 00000000002 = 3c0016 = ${\displaystyle 2^{0}\times (1+{\frac {0}{1024}})}$ = 1
                              (one)

0 01111 00000000012 = 3c0116 = ${\displaystyle 2^{0}\times (1+{\frac {1}{1024}})}$ ≈ 1.00097656
                              (smallest number larger than one)

0 01101 01010101012 = 355516 = ${\displaystyle 2^{-2}\times (1+{\frac {341}{1024}})}$ = 0.33325195
                              (the rounding of 1/3 to nearest)

1 10000 00000000002 = c00016 = −2

0 00000 00000000002 = 000016 = 0
1 00000 00000000002 = 800016 = −0

0 11111 00000000002 = 7c0016 = infinity
1 11111 00000000002 = fc0016 = −infinity

By default, 1/3 rounds down like for double precision, because of the odd number of bits in the significand. The bits beyond the rounding point are 0101... which is less than 1/2 of a unit in the last place.

• Decimals between 2⁻²⁴ (minimum positive subnormal) and 2⁻¹⁴ (maximum subnormal): fixed interval 2⁻²⁴
• Decimals between 2⁻¹⁴ (minimum positive normal) and 2⁻¹³: fixed interval 2⁻²⁴
• Decimals between 2⁻¹³ and 2⁻¹²: fixed interval 2⁻²³
• Decimals between 2⁻¹² and 2⁻¹¹: fixed interval 2⁻²²
• Decimals between 2⁻¹¹ and 2⁻¹⁰: fixed interval 2⁻²¹
• Decimals between 2⁻¹⁰ and 2⁻⁹: fixed interval 2⁻²⁰
• Decimals between 2⁻⁹ and 2⁻⁸: fixed interval 2⁻¹⁹
• Decimals between 2⁻⁸ and 2⁻⁷: fixed interval 2⁻¹⁸
• Decimals between 2⁻⁷ and 2⁻⁶: fixed interval 2⁻¹⁷
• Decimals between 2⁻⁶ and 2⁻⁵: fixed interval 2⁻¹⁶
• Decimals between 2⁻⁵ and 2⁻⁴: fixed interval 2⁻¹⁵
• Decimals between 2⁻⁴ and 2⁻³: fixed interval 2⁻¹⁴
• Decimals between 2⁻³ and 2⁻²: fixed interval 2⁻¹³
• Decimals between 2⁻² and 2⁻¹: fixed interval 2⁻¹²
• Decimals between 2⁻¹ and 2⁻⁰: fixed interval 2⁻¹¹

• Decimals between 1 and 2: fixed interval 2⁻¹⁰ (1+2⁻¹⁰ is the next largest float after 1)
• Decimals between 2 and 4: fixed interval 2⁻⁹
• Decimals between 4 and 8: fixed interval 2⁻⁸
• Decimals between 8 and 16: fixed interval 2⁻⁷
• Decimals between 16 and 32: fixed interval 2⁻⁶
• Decimals between 32 and 64: fixed interval 2⁻⁵
• Decimals between 64 and 128: fixed interval 2⁻⁴
• Decimals between 128 and 256: fixed interval 2⁻³
• Decimals between 256 and 512: fixed interval 2⁻²
• Decimals between 512 and 1024: fixed interval 2⁻¹
• Decimals between 1024 and 2048: fixed interval 2⁰

• Integers between 0 and 2048 can be exactly represented (and also between −2048 and 0)
• Integers between 2048 and 4096 round to a multiple of 2 (even number)
• Integers between 4096 and 8192 round to a multiple of 4
• Integers between 8192 and 16384 round to a multiple of 8
• Integers between 16384 and 32768 round to a multiple of 16
• Integers between 32768 and 65519 round to a multiple of 32
• Integers above 65519 are rounded to "infinity" if using round-to-even, or above 65535 if using round-to-zero, or above 65504 (the largest representable finite value) if using round-to-infinity.