Short overview of the costs of 2-of-3 taproot inputs

Cost of 2-of-3 taproot inputs

Everything below is best effort. Corrections welcome! –Murch

==== Key path spending costs ====

* outpoint (txid:vout): 32+4vB
* scriptSig size: 1vB
* nSequence: 4vB
* num witness items: 1WU
* witness item size: 1WU
* signature: 64WU

32+4+1+4+(1+1+64)/4 = 57.5vB


==== Control Blocks ====

* Depth 0 control block:
  * Length of control block: 1WU
  * Header byte (script version, sign of output key): 1WU
  * Inner key of root key: 32WU
= 34WU

* Depth 1 control block:
  * CB size: 1WU
  * Header byte: 1WU
  * Inner key of root key: 32WU
  * Hashing partner in tree: 32WU
= 66WU


==== Script path spending cost added to key path spending cost ====

Script path spend assuming 2-of-2 MuSig leaf:
* script size: 1WU
* script "<pk> OP_CHECKSIG": 33+1WU
* Depth 1 Control block: 66WU

57.5+(1+34+66)/4 = 82.75vB

Leafs cannot do MuSig, construction with 2-of-2 OP_CHECKSIG:
* +2nd sig: 1+64WU
* script size: 1WU
* Script "<pk1> OP_CHECKSIGVERIFY <pk2> OP_CHECKSIG": 33+1+33+1=68WU
* Depth 1 Control block: 66WU

57.5+(1+64+1+68+1+1+32+32)/4 = 107.5vB 


==== Discarded other options ====

First less inefficient construction for 2-of-2 OP_CHECKSIG:
~Improved in second iteration below!~
* +2nd sig: 1+64WU
* Length of script: 1WU
* Script "<pk1> OP_CHECKSIGVERIFY <pk2> OP_CHECKSIG 2 OP_EQUAL": 33+1+33+1+1+1=70WU
* Depth 1 Control block: 66WU

57.5+(1+64+1+70+66)/4 = 108vB

Less private, more costly variant with a single 2-of-3 leaf in lieu of two 2-of-2 leaves:
* +2nd sig: 1+64WU
* +1 empty witness item: 2WU
* Length of script: 1WU
* Script "<pk1> OP_CHECKSIG <pk2> OP_CHECKSIGADD <pk3> OP_CHECKSIGADD 2 OP_EQUAL": 33+1+33+1+33+1+2=104WU
* Depth 0 Control block:

57.5+(1+64+2+1+104+1+1+32)/4 = 109vB