Just say you're sorry. Never say you're sorry "if." Say you're sorry.
"I'm sorry I was rude" is good.
"I'm sorry if I was rude" is not. It weasels. It implies that maybe you weren't rude. It implies that the person being apologized to has a twisted little worldview if they think "Oh, shut up, frog-lips" is rude.
An apology should give the sense that you actually feel some form of regret. "Sorry if" is a conditional apology. Conditional apologies make things worse, not better.
"I'm sorry your frog is dead" is better than "I'm sorry if your frog's death causes you pain."
Similarly, "I'm sorry I taunted you about your frog's death" is better than "I'm sorry if my taunting you about your frog's death caused you pain."
When "I'm sorry" is an apology, it conveys remorse. "I'm sorry" can also be an expression of sympathy, a thing people occasionally forget.
"I'm sorry your frog died."
"Why are you sorry? You didn't kill my frog!"
But what if you did kill their frog? "I'm sorry I killed your frog" is better than "I'm sorry if my killing your frog caused you pain."
Making the if silent does not help. "I'm sorry my killing your frog caused you pain" contains a silent if, because it still implies that your regret is not for the action (killing the frog) but for the suffering it caused (oh, boo hoo), which by implication need not have followed from the action. It implies an argument about the value of the frog, and although you may differ on this subject, an apology is not the time to bring it up. Do you say, "Sorry about your whole family being killed, but, you know, I never liked them"? No.
What if there is genuine uncertainty? That's different. "I'm sorry your frog died" contains no silent if. "I'm sorry if your frog died" is a different sentence that implies that the frog may turn up at any moment, having just stepped out to catch a movie and forgotten to turn its pager back on.
"I'm sorry if my anvil fell on you -- it didn't? -- Oh, I'm so glad."
Even worse than "sorry if" is the poisoned apology: "I'm sorry my taunting you about your frog's death caused you pain. You should seek therapy."
Taking personal frogs out of the matter enables us to take an objective look at more general apologies.
"I'm sorry that hundreds of frog species around the world are going extinct and that woodlands that once throbbed with their gladsome croaking are now silent." This denotes simple regret.
"I'm sorry that my air-conditioned car has contributed to a situation that has caused hundreds of frog species around the world to go extinct and that woodlands that once throbbed with their gladsome croaking are now silent." This denotes regret and acknowledgement of personal responsibility.
"I'm sorry that you feel that my air-conditioned car has caused hundreds of frog species around the world to go extinct and that woodlands that once throbbed with their gladsome croaking are now silent, but would you please move your bicycle?" This includes a skillful substitution of "that" for "if," and denotes faux regret and implicit announcement of imminent homicidal attack. It does not count as an apology.
Let's try some examples without frogs. (An editor suggests that these lessons are too simplistic. But first of all, you might be surprised at how hard it apparently is for certain people to understand the most basic principles of a modest apology -- it's like it would kill them to say they're sorry! Second of all, those who require this information include not only individuals but nations. And when you are explaining things to nations, you cannot make it too simple.)
"I'm sorry for what I said" is better than "I'm sorry if you took what I said the wrong way."
"I'm sorry we bombed your embassy" is better than "I regret the suffering you experienced upon learning of the bombing of your embassy, and the suffering, though it was surely quite brief, of any who were in your bombed embassy, and indeed the suffering of anyone in any bombed building anywhere in the world today is something I regret, even if it was accidental, in fact I regret all suffering. It is sad that there is suffering. I regret it. OK?"
"I'm sorry your pilot is dead" is better than "I'm sorry if your pilot's death has caused irrational national mourning."
It is also better than "I'm sorry our airplane knocked your pilot out of the air," which is in turn better than "I'm sorry if our airplane knocking your pilot out of the air is something you insist on taking as a national insult."
Having established that "I'm sorry I hurt your feelings" is far better than "I'm sorry if I hurt your feelings" we might also note that specificity is also a crowd-pleaser in apologies:
"You mean for standing me up at the restaurant, making a pass at my best friend and telling my mother it was my fault for drunkenly cracking up your car when in fact it was you who totaled it when you swerved to run over a frog?"
"OK, OK, I'm sorry about standing you up, I'm sorry about hitting on Chris, I'm sorry about what I said to your mom, OK? I'm sorry!"
"What about the frog? Are you sorry about the frog?"
Here the dialogue can go two ways:
"Yes, I'm sorry about the frog."
"Thank you. I appreciate the apology."
"No, I'm not sorry about the frog. The frog was asking for it. The frog kept me up all night with his incessant croaking and this whole thing would never have happened if I was better rested."
"Oh? Well, I don't think we can be together until you've learned to accept responsibility."
"Oh? Well, maybe you should tell Chris that. Chris understands how I feel about frogs. I'm going to call Chris right now."
Because specificity is so valuable, "I'm sorry I hurt your feelings," while good, is not quite as good as "I'm sorry I called you flaming dog poop."
We now understand that the "if" in "I'm sorry if my calling you flaming dog poop hurt your feelings" is not good, implying as it does:
"I'm sorry if my calling you flaming dog poop wasn't something that you were able to accept in the lighthearted spirit of give-and-take in which it was intended and in which most well-balanced people certainly would have taken it. My previous significant other -- God, how I miss that merry jokester! -- wouldn't have minded a bit."
There are various wording subtleties that can confuse the distinction between "sorry" and "sorry if." "Sorry for" is one such variant. "I'm sorry for what I did" is OK.
"I'm sorry for you" has come to be a deadly insult and is often used in "sorry if" phrases:
"I'm sorry for you if you can't understand that 'space cadet' is a term of endearment" or "I'm sorry for you if you can't see that my setting your clothes alight was a friendly joke, a gesture of intimacy."
Fooling with the basic "I'm sorry" formula is not as good an idea as you think. "Regret" and "rue" just make people suspicious. "Remorse" can work, though, especially if you describe just how eaten through you are with it.
"I apologize" is good, although if you have a history of issuing weasel apologies it can be worrisome: What kind of apology did you make? Was it a "sorry" or a "sorry if"? Tip: If you secretly feel that your apology left your pride unbowed, it may have been a weasel apology.
To sum up:
"I'm sorry I hurt your feelings": good.
"I'm sorry I made fun of the way you walk": better.
"I'm sorry if my making fun of the way you walk hurt your feelings": bad.
These rules do not describe the entire world of apologies. Creative souls will always find their own way.
"I'm sorry if it seemed like I was making fun of the way you walk: I was bringing my sister her purse and I choked on a moth at the same time as I slipped on a dead frog and turned my ankle": good if you can carry it off.