# A faster pow(x,1/12) function available in Fast12thRoot.jl

**URL:** <https://discourse.julialang.org/t/a-faster-pow-x-1-12-function-available-in-fast12throot-jl/10893>\
**Category:** Numerics\
**Created:** [May 14, 2018, 9:43pm UTC](https://discourse.julialang.org/t/a-faster-pow-x-1-12-function-available-in-fast12throot-jl/10893 "2018-05-14T21:43:25Z")\
**Posts on this page:** 1\
**Showing post:** 11

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**Author:** ![JonT](https://avatars.discourse-cdn.com/v4/letter/j/5daacb/32.png) [@JonT](https://discourse.julialang.org/u/JonT)\
**Post date:** [May 15, 2019, 6:42am UTC](https://discourse.julialang.org/t/a-faster-pow-x-1-12-function-available-in-fast12throot-jl/10893/11 "2019-05-15T06:42:15Z")

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@MrBelette ,

Further analysis suggests my earlier proposal is seriously flawed.

In part due to issues that Jeffrey raised about instability with certain sets of coefficients, and also because I didn’t pay attention to being clear about what was required.

- Is the desired interval [\frac{1}{2},1] or [\frac{1}{2},1) ?  
The semi-open interval does not need the special case of t=0, so its perhaps not sensible to use transform t=x-1 for the best approximation.
- The interval [\frac{1}{2},1) is best for simple range/strength reduction, but is it best for simple polynomial approximation ? (I conjecture [\frac{3}{4},\frac{3}{2}) might be numerically more stable, but will be more expensive for range reduction.)
- What is the actual required precision ?  
I was targeting _faithful_ rounding (which is basically \pm1 ulp) but you seem to be looking for _correct_ rounding (100% \pm\frac{1}{2} ulp). However, without resolving the [Table Makers Dilemma](https://en.wikipedia.org/wiki/Rounding#Table-maker's_dilemma) for this function (which I am not volunteering to do) I don’t know how much intermediate precision is required to guarantee correctness.

But If I go back and look at the text of your question, your use case suggests you code will end up a line that looks like `fast12throot(x)-1`, (see for example [converting annual interest rate to monthly when compounding frequency known](https://money.stackexchange.com/questions/51728/)) so I wonder whether what you really want is the function `fast12throotm1(x)`\equiv`fast12throot(x)-1`. It is often undesirable to calculate `(x+1)-1` for small `x` because it loses precision on the intermediate step, which is exactly why standard maths libraries include functions for log1p(x)\equiv\log(x)+1 and expm1(x)\equiv\exp(x)-1 specifically to avoid problems with precision near special cases (since the implementations include approximations then add 1 or subtract 1 respectively).

So I suppose there are two questions I should have asked earlier:

1. Do you really want both functions ?
2. What is actually the acceptable error for the desired function(s) ?

regards,  
JonT.

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