Game Boy: What constitutes a "half-carry"?

For 16-bit operations, the carry from bit 3 to bit 4 in the register's high byte sets the flag. In other words, bit 11 to bit 12.

(Note the above bits are labeled 0-15, from least to most significant)

See here: http://www.z80.info/z80code.htm

16 bit arithmetic

If  you want to add numbers that are more than the 0-255 that can
be stored in the A register,  then the HL, IX or IY registers can
be used. Thus LD HL,1000H:LD BC,2000H:ADD HL,BC will give

A  CZPSNH  BC   DE   HL   IX   IY  A' CZPSNH' BC'  DE'  HL'  SP
00 000000 2000 0000 3000 0000 0000 00 000000 0000 0000 0000 0000

The flags are set as follows.

C or carry flag          1 if answer >65535 else 0
Z or zero flag           not changed
P flag                   not changed
S or sign flag           not changed
N flag                   0
H or half carry flag     1 if carry from bit 11 to bit 12 else 0

It's the carry from bit 3 to bit 4, just like the normal carry flag records carry from bit 7. So, e.g. to get the half carry bit in an add:

((a&0xf) + (value&0xf))&0x10

Which gives 0x10 if half carry should be set, 0 otherwise. Getting half carry from the other relevant ops follows naturally - the questions is whether there was carry from the low nibble to the high.

To put things in perspective, the z80 has a 4bit ALU and performs 8bit ops by doing two 4bit ops. So it gets half carry very naturally, as an intermediate result.

DAA is interested in the flag because if half carry is set then two digits that add up to more than 16 were added in the low nibble; that will have correctly produced carry into the upper nibble but will have left the low nibble 6 lower than it should be, since there were six more values between 10, when it should have generated carry, and 16, when it did.