The mantissa, also called the significand, supplies the significant value in scientific notation. In 1.898 × 10²⁷, the mantissa is 1.898; the exponent sets the scale. An f32 uses the same idea in binary: its bits encode a sign, an exponent, and a mantissa. The radix is fixed at 2, so it needs no field of its own. section 5.4
For an ordinary f32 value, 23 stored mantissa bits supply the fractional part. Start at 1.0 for an implicit leading bit, then add 0.5 for the first fractional bit if it is set, 0.25 for the next, and so on. The resulting significand combines with the sign and exponent to give the value. For example, the bits shown for 42.42 decode to a significand of 1.325625 and a scale factor of 2⁵. section 5.4.1 section 5.4.4
Listing 5.9 Isolating and decoding the mantissa from an f32
let n: f32 = 42.42;
let n_bits: u32 = n.to_bits();
let mut mantissa: f32 = 1.0;
for i in 0..23 {
let mask = 1 << i;
let one_at_bit_i = n_bits & mask;
if one_at_bit_i != 0 {
let i_ = i as f32;
let weight = 2_f32.powf( i_ - 23.0 );
mantissa += weight;
}
}
Try decoding the bits yourself: `to_bits()` exposes the f32 pattern, and the loop checks each of its 23 fractional mantissa bits. Each set bit adds its binary weight to the initial 1.0. This procedure is for nonspecial values. section 5.4.4
The exponent determines when that initial 1.0 rule changes. An all-zero exponent calls for subnormal-number handling, which allows values closer to zero. An all-one exponent instead marks infinities or NaN. Do not apply the ordinary mantissa-decoding loop to those cases unchanged. section 5.4.4
Listing 5.16 Generating f32 values in interval [0,1] from a u8
fn mock_rand(n: u8) -> f32 {
let base: u32 = 0b0_01111110_00000000000000000000000;
let large_n = (n as u32) << 15; #1
let f32_bits = base | large_n; #2
let m = f32::from_bits(f32_bits); #3
2.0 * ( m - 0.5 ) #4
}
The reverse trick is useful too: place a `u8` into selected mantissa bits while holding the sign and exponent fixed. This Rust example produces a value from 0.5 to 0.998, then scales it to a result from 0 to 0.996 without division. section 5.6
