Do you really use the formula for speaker placement?

There are lots of good discussions on here about improving the room your speakers are in to get the best listening experience. This includes speaker placement. I've moved my speakers all over creation and found where they sound good to me. But then I saw several "speaker placement calculators" and I thought, great, something else for me to tinker with.  :-)    Besides enforcing the standard guidelines of using an equilateral triangle between the 2 speakers and the sweet spot, and the speakers being equally distant from side walls, the formula specifies the optimum distance from the front and side walls. My room is 15.5 ft x 23 ft. The algorithm says the front of my speakers should be a little over 7 feet from the wall and a little over 4 feet from the side walls. The distance between the speakers then is about 80 inches, which is the measure for the sides of the equilateral triangle. Basically, I was about the same distance from the speakers as they were from the wall. I felt like I might as well be sitting between the speakers. For me, this algorithm did not result in a great listening experience. There are about 5 different algorithms, each arriving at slightly different results. Has anyone found success using one of these algorithms, for example, the Cardas algorithm? Or any other algorithm? Obviously, I'll let my ears be my guide. But I'm always open to the proposition that there's something I don't know that I could benefit from.
There are two main aspects to speaker placement with just two speakers- bass response and imaging. As usual its a trade off.  

Closer to walls improves bass. But closer to walls brings more early reflections which harms imaging. So its a trade off between finding the location with the smoothest bass and good imaging.  

An algorithm is simply a fancy word for a set of steps you go through. With speakers that means first trying them different distances from the front and side walls, and moving the listening position, listening primarily for smooth response. In this step you don't pay a lot of attention to imaging, although you do always keep them equidistant its not critical to be precise about it.  

Once you've found the smoothest frequency response the next step is to get them precisely equidistant, and level, and experiment with toe. Listen for your preferred balance between wide and deep. When you're done with that you'd done with speaker positioning. Time for acoustic embedding, electrical embedding, vibration.....
As you say, there are several calculators/algorithms/methods out there, and if you try two or three of them you'll find they give different recommendations for the same room.  Because the creator/designer prioritized different factors.  They *are* good for getting people to think differently and creatively about speaker placement, and they are often good for getting speakers somewhere in the general region of where an individual likes them best.  Most are free, so there's very little downside.
to add to millercarbon's apt comments

there is also the issue of toe in / out... trading off direct radiation vs side wall reflections... the importance of this depends on the dispersion pattern of the speakers in question
The one algorithm that is absolute garbage is the hokey Master Set where the goal is supposed to be to get your speakers cockeyed just the right amount to sound equally crappy from everywhere in the room. Of course they spend 5 pages detailing exactly how to achieve this exalted state of crap. But crap is crap no matter how many pages or hours you put into it. Avoid.
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thanks for all of your suggestions. As I imagined, I learned some new tricks, regardless of how many articles I've read and YouTubes I watched. I saw the Youtube mentioned above, rearranged my speakers, and definitely got some better results out of my speakers/room. I read about the Sumiko Speaker Set and will give that method a try tonight. Thank you so much for your contributions.
Thanks for all of your input. I first watched and went by the advice provided in the Youtube video. It was a different approach in that it asked you to start with the speakers literally next to you and move them to the wall. The wisdom is that there would be a point where the speakers disappear. They did. Then the instruction was to move them left/right by about 8 inches. This provided a good result. Then, I tried the Sumiko Speaker Set method. Interesting. It starts with finding the place in your room where the bass comes through the best. So, it starts with one speaker only, and the setup is predicated on excellent bass. You'd think the placement would be right next to the front wall. I am much more interested in imaging and the soundstage than a loud thump. So, the place for the left speaker turned out to be close to the placement for the youtube video method. The Sumiko Speaker Set method then went into seeing if the sound stage was "slanted." if one speaker is too loud, move it back, etc. Then, it went into toeing the speakers. I know we've all done this but I thought I was  no-toe guy until I finally understood that no-toeing can produce a thinner sound. I like a wide sound stage but I understood that I was trading off a richer sound. Then the methodology went into slanting the speaker a little up, which I'm not able to do. Takeaway: I knew speaker placement was important but I was surprised how big a difference very small distance changes make. Obviously, sound is made of waves, so distance makes a difference. I was just surprised by how much. I liked the concept of "slant," which I'd not heard before. In the end, the Sumiko method created a much more 3 dimensional sound stage. both methodologies arrived at the same distance from the front wall. With sumiko, the speakers were about 10 inches further apart and toed in. Sumiko seems to account for the room's acoustics more than the other method. Anyway, the differences were noticeable, the improvement was free, and it was a fun exercise. Thanks for the input.
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When following the Youtube setup, were they perfectly equidistant with L and R toed in symmetrically the exact same amount? If not then how different were the distances and angles?

How exactly was the final Sumiko setup different? 

You're right about tiny distances making a big difference. So what exactly were the differences?
@millercarbon The youtube setup had the speakers equidistant. Frankly, it never occurred to me before the Sumiko Method that speakers would be anything other than equidistant from the listener--the normal equilateral triangle rule. And the youtube didn't mention making the toeing different. Again, I just assumed the toeing would be the same. That's why the "slant" idea was kind of a shock. I haven't experimented with unequal toeing. I assumed you'd want them toed the same to make the sound in the center, like voices, focussed. but now that you mention it, it takes 2 seconds to experiment. I don't know if anyone has tilted their speakers. I can't do that but wondered if that small change made a big difference.

@squeak_king. Thanks!
The reason I ask is the Sumiko setup has you moving one "anchor" speaker for bass and then the other, and then making lots of little adjustments all by ear. Nothing about symmetry, nothing about equidistance. Since I know equidistance is essential for imaging, and since a tape measure is way faster and more accurate than trying to do this by ear, I am curious how close to equidistant you got using that method. 

The next question would be if you did measure, then tweak just the one thing to be equidistant, does that improve the Sumiko setup. Because I know for sure it will, just curious if you tried.
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@millercarbon That's right. You start with the left speaker and find where the bass sounds the best. I did it with the right speaker unplugged. My surprise (shouldn't have been) was that there was a place because of the dimensions of the room where the bass did sound more pronounced. Frankly, I'm not sure that I want everything based on bass, but that's the methodology. And then, yes, you're supposed to move the right speaker around until you get the sound you want. I guess I cheated by starting the speaker in the normal "triangular" position. I did try the "slanted" configuration where one speaker was "behind" the other. I thought it was interesting for one song but moved the speakers equidistant after that. I did find that moving the speakers closer and farther away from one another made significant changes in the soundstage. Moving them 6 inches one way or another was enough to make a difference. The end result was basically an equilateral triangle. I don't know if I was prejudiced by the expectation that it "should" end up that way. Sometimes one song sounded better with the speakers in one position, then the next song sounded better with the speakers in a slightly different position. The hardest measurement is toeing... How do you measure that? I think I had a compass back in junior high and that was a while ago. So, I have to fiddle with more and less toeing. Oh, and my issue is that my speakers are 145 pounds and they're on wheels because they stick out 4+ feet from the wall when I'm listening to them... right in the way of everyone who wants to walk through the den to the kitchen. I wheel them back to the wall when I'm done listening for the night. So, my placements will always be close but never "final." Bottom line, the triangle is a good guide. Placing the first speaker where the bass is pronounced was a revelation. Measuring is a good guide. All of those things get you close. But it pays to make even 6-inch adjustments because you might get surprised. BTW, the method also talks about tilting the speakers up. I can't do that because I have to move my speakers. But it'd be fun to see if tilting makes a big difference.
@tvad I think my speakers only create one sweet spot. How do you get multiple sweet spots or a "wide" sweet spot?
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Hyper precision means throwing away the ancient tape measure and getting a Laser

asymetric toe in is common when setup is a odd shaped rooms - you can see this in my Casa Pacifica system pictures, the right channel is cranked a bit more towards the listener - set equal requires 1db cut.

finally tilt if properly engineered into a time and phase accurate speaker, is hyper critical- consult any Vandersteen manual for a math and science based starting points And tilt settings.

move stuff around, accurate recording of your baselines, allow you to return to them.
@tvad Thanks--I'm learning things all the time, in this case, the effect of toeing with the "cross point" of the speakers in front or behind me. Thanks. And now it looks like I need to play with tilting the speakers. Thanks. 

@tomic601 Interesting about the asymmetric toeing in. I have an asymmetric room. So, I will have to play with that as well. And thanks for the book recommendation.