It depends on what the bridge is for and the amount of risk you're assuming. Generally, just look at using a note with a cap and / or discount. Warrants are a pain.
If you're bridging to the close of a round and the term sheet is signed then you may just apply a small (5%, maybe even zero) discount and no cap.
If you're early to mid-process on a fundraise then 10-20% discount seems fair.
If you're bridging to an M&A event and there’s still some uncertainty then you’d also add a multiplier on exit (as a cap / discount is less relevant in these scenario is).
If it’s more of an extension to the next round then find a cap that reflects where the company is on the continuum from, say, seed to series A and price it as so.
Depending on what the need is, there are continuations of the previous round or a discount to the next as a note if a new financing is coming up. If a financing happens in short order, that 'discount' can be minimal or zero.
It really depends on a company by company basis. In most cases if a company needs to be genuinely bridged, then it's recommended that they do it by extending the previous round by adding some additional cash or if it's a true bridge to an impending follow-on round, then putting in a small amount of cash on a note with a 20% discount to the future round is also okay.
A bridge with a discount to the next round of 20 to 25% along with a multiple (2x or 3x) return on the bridge if there is an acquisition rather than another round of financing is typical.
The bridge terms depend on how likely another financing is. 20-30% discount on next round and 2x or higher on exit.