To a Muslim the only thing that is relevant is the teachings of authentic Islamic sources, the Quran, the Sunnah, and the Hadith in that order. The Holy Quran has clearly stated that there is "no compulsion in religion". Let me make this clear for those Muslims who have abandoned reasoning at the feet of their Scholars.
Premise 1: Islam is a religion.
Holy Quran: "Surely, the true religion with Allah is Islam." (3:20)
Premise 2: There is no compulsion in religion.
Holy Quran: "There should be no compulsion in religion." (2:257)
Conclusion: Since there is no compulsion in religion and Islam is a religion, there cannot be compulsion in Islam!
Definition of Compulsion: "the action or state of forcing or being forced to do something; constraint."
Since there is no compulsion in Religion one cannot be forced to stay within that religion. They cannot be forced to remain Muslim, Christian, Jewish, Sikh, Hindu or for that matter any religion if they choose to apostate.
This is a logically sound deductive argument, which means the truth of the premise guarantees the truth of the conclusion. Just to prove that I am not putting the verse out of context here is the complete verse:
[2:257] There should be no compulsion in religion. Surely, right has become distinct from wrong; so whosoever refuses to be led by those who transgress, and believes in Allah, has surely grasped a strong handle which knows no breaking. And Allah is All-Hearing, All-Knowing.
Islam is such a beautiful religion it has nothing to fear. It has a return policy, if you don't like it then feel free to leave. Islam has nothing to lose if you change your religion. I have spent extensive time researching this topic, I have established that the Holy Quran is crystal clear that there is no punishment for apostasy, in fact it clear states that one can apostate and remain alive:
[4:138] Those who believe, then disbelieve, then again believe, then disbelieve, and then increase in disbelief, Allah will never forgive them nor will He guide them to the way.
If the punishment for apostasy was death then once a person goes from belief to disbelief he/she would be put to death. They wouldn't be able to believe again. But this verse clearly shows it is possible! Moreover the Holy Quran has asked these people will they try to raise above God and enforce belief when Allah (SWT) does not:
[10:100] And if thy Lord had enforced His will, surely, all who are on the earth would have believed together.
Wilt thou, then, force men to become believers?
The arguments don't end here, I can keep going but since my conclusion is already reached it should sufficient for those whose hearts pursue truth. I know what comes after this, posting of Hadiths without having any context of them. There are all addressed in this conclusive book,
Apostasy in Islam, leaving a follower of truth no reason to believe in punishment of death for apostasy in Islam.