Below is my (initial) solution to an issue I was having with *TaylorSeries.jl* , which leads to another question >> And more generally I’m interested in using *TaylorSeries.jl* as an implementation of high-order automatic differentiation

One partial SOLN : Using *TaylorSeries.jl* to get the inverse function *exp* by defining *Taylor1* function *log* works now

```
julia> tBig = Taylor1(BigFloat, 50) # Independent variable with BigFloats, With order 50 precision 1.0 t + ?(t⁵¹)
julia> p = log(tBig + 1.0)
julia> TaylorSeries.evaluate(inverse(p), TaylorSeries.evaluate(p, 0.9))
8.999082e-01
julia> TaylorSeries.evaluate(inverse(p), 0.9)
1.459603
julia> exp(0.9)
2.45960311115695
julia> TaylorSeries.evaluate(inverse(p), 0.9) + 1.0
2.459603111156949
```

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Using *TaylorSeries.jl* to get the inverse function *exp* by defining *Taylor1* function *log* works now ( see above )

Next Question: but How do we do vice-versa aka for an **arbitrary** function *f* and *f^-1* ?

tBig = Taylor1(BigFloat, 50) # Independent variable with BigFloats, With order 50 precision 1.0 t + ?(t⁵¹)

TaylorSeries.evaluate(inverse(

exp(tBig)), TaylorSeries.evaluate(exp(tBig), 0.9))DEBUG ERROR: ArgumentError: Evaluation of Taylor1 series at 0 is non-zero. For high accuracy, revert a Taylor1 series with first coefficient 0 and re-expand about f(0).

More generally I’m interested in Introduction using TaylorSeries.jl as an implementation of high-order automatic differentiation, as presented in the book by W. Tucker per description here >> https://www.juliadiff.org/TaylorSeries.jl/stable/background/ and would appreciated any tips or advice there also.