bash: performance of let vs expr vs (( ))

A simple counting loop is used to test the performance of the bash utilities

# i=1; time while :;do [ $i -gt 1000000 ] && break; let i=$i+1; done; echo $i

real 0m17.590s
user 0m17.109s
sys 0m0.400s
1000001

# i=1; time while :;do [ $i -gt 1000000 ] && break; i=$(($i+1)); done; echo $i

real 0m16.043s
user 0m15.629s
sys 0m0.384s
1000001

# i=1; time while :;do [ $i -gt 1000 ] && break; i=`expr $i + 1`; done

real 0m1.411s
user 0m0.052s
sys 0m0.232s

Both let and (( )) perform calculations at almost the same speed. 'expr' being a process creation suffers heavily during calculations