When calculating a Playing Handicap™, the difference in results can be attributed to whether machine precision has been used in the calculation.
Here is an example calculation that explains the difference in Playing Handicap between your calculation and the computer. The computer uses machine precision which is to the first decimal place that has a value of 0.
HI = 13.6
CH = 13.6*(117/113)+71.2-70
CH = 13.6*1.03539823+71.2-70
CH = 15.281415928
PH = 0.95*15.281415928
PH = 14.51734 (rounded up = 15)
On the contrary, the 0.95*15 will result in 14.25 rounded to 14, providing a different playing handicap.
It is advised therefore that competition software is used to calculate the Playing Handicap™ as calculating this manually without machine precision can lead to golfers gaining / losing shots.