I am home or I am at home When do we use at home when home without at?

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They mean the same actually. Generally,
I am at home== formal, grammatically exact. 
I am home = informal,casual, daily use.

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I'd use "I'm home" to announce that I have just arrived home.

I'd use "I'm at home" to explain where I am at that moment.

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They are both right c: 
But I would say 
I'm home not I am home
For some reason it just sounds weird

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I would use I am home or I'm home when announcing my arrival at home to family.
Example - Mom! I'm home. What's to eat?

I would use I am at my home/house if a friend called wanting to know where I was.
Example - Bob - "Hey Jon. Where are you?" Me - "I'm at my house. Why?" Bob - "Some of they guys are gonna watch a movie. Wanna come?" Me - "Sure."

Answer

i am at home is formal as Aseer said but i'm home is for casual speech :)

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