library forecast and back-transformation

[newsagent]

I was looking the source code of forecast

https://github.com/robjhyndman/forecast/blob/master/R/forecast.R

I would like to see in simple code terms how the forecast function obtains the mean forecasts at the original scale of the variable when we have these two transformations: A) Difference of the dependent variable and difference of logs of it

For example,

library(forecast)
h=5
  arima( goog200, order=c(0,1,1))|>  forecast(h=h)
  
Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
201       531.2263 523.2780 539.1746 519.0704 543.3822
202       531.2263 520.2113 542.2413 514.3803 548.0723
203       531.2263 517.8291 544.6234 510.7371 551.7155
204       531.2263 515.8108 546.6418 507.6503 554.8023
205       531.2263 514.0277 548.4249 504.9233 557.5293

 

  arima( log(goog200), order=c(0,1,1))|>  forecast(h=h) 

    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
201       6.275259 6.257671 6.292847 6.248360 6.302158
202       6.275259 6.250808 6.299709 6.237865 6.312652
203       6.275259 6.245488 6.305030 6.229728 6.320789
204       6.275259 6.240984 6.309534 6.222840 6.327678
205       6.275259 6.237006 6.313511 6.216757 6.333761

By the way the last code provides the forecasts that they seem to be in logs as well. Are these forecasts back-transformed?
But my main question is if I can see somewhere the back-transformation when I have first differences and differences of logs.

That is, I would like to understand how from the forecasts that we obtain at the transformed scale we eventually obtain the forecasts reported above that are back-transformed. If you could refer me to a line of code from the above link that would be helpful

[robjhyndman]

To back transform to obtain means rather than medians, you need to do some bias adjustment as explained in https://otexts.com/fpp2/transformations.html#bias-adjustments. The bias adjustment code can be seen here: https://github.com/robjhyndman/forecast/blob/a9db8d4e7169dac3a8d5c1bbf783e3e9bbad5533/R/forecast2.R#L81

Backtransforming differences is handled within the stats::arima() function and uses the diffinv() function.

[newsagent]

thanks a lot