Rough gas cost calculation for me would be:
1/2 hour recirc time per day.
100,000 BTU per therm
Gas cost per therm = $1.80
Tankless modulated gas water heater uses up to 125,000 BTU per hour. Estimate use at 80,000 BTU per hour. Hot water pipes fully insulated.
80,000 BTU/hr X 1/2 hr/day X $1.80 per therm X 1/100,000 therm/BTU X 365 days per year = $263 per year.
Running a recirc pump for 1/2 hour per day would cost me around $263 per year just for gas!