IEEE 754 Floating Point
X-Ray: Special Cases
How edge values behave in real computations
+Infinity (+∞)
Result of overflow or division by zero.
0 |11111111 |00000000000000000000000
-Infinity (-∞)
Negative overflow or division by -0.
1 |11111111 |00000000000000000000000
NaN
Undefined results
Quiet NaN (qNaN)
1 |11111111 |1MMMMMMMMMMMMMMMMMMMMMM
Propagates silently
Signaling NaN (sNaN)
1 |11111111 |0MMMMMMMMMMMMMMMMMMMMMM
Raises exception; M ≠ 0
0 / 0 → NaN ; NaN ≠ NaN → true ; NaN < x → false ; NaN > x → false ; x ≠ NaN → true
±0
Two distinct zeros
± |00000000 |00000000000000000000000
1 / +0 → +∞ ; 1 / -0 → -∞ +0 == -0
Subnormal
Near-zero numbers
± |00000000 |MMMMMMMMMMMMMMMMMMMMMMM
Smooth underflow; M ≠ 0
IEEE 754 favors propagation over interruption: computations continue instead of crashing.
xeland314NaN → Breaks ordering, not execution
±0 → Preserves directional information
±0 → Preserves directional information