I appreciate it's all difficult to understand. Companies who give direct comparisons off the cuff aren't giving you the whole picture. Your HID output in PAR (Photosynthetically Active Radiation) will be a little (or a lot different) depending on your ballast (digital or magnetic), your bulb type and other factors. Your reflector is a crucial element. So you see there isn't an across the board answer. What I can tell you is this: Watt for Watt, with the correct quality light being used (utilising 3 bands minimum, Red, Blue, Orange, with two Blue and Red peaks being hit) and LED light will deliver FAR more PAR than an HPS. This is just fact. The majority of HPS and MH light delivered is to all intents outside the PAR range (the range of light that plants use to photosynthesise). A quality LED delivers nearly 100% efficiency at near 100% PAR. An HPS bulb converts a huge amount of the power it consumes into heat (your bulb is red hot). Only around 30-35% becomes light, and only a fraction of THIS light is PAR (therefore used by your plants). The maths is plain to see there.
Don't listen to suppliers who tell you to swap your 400W HPS for a 90W LED. This is not equivalent. HPS is very good, no doubt about it. It has superb penetration and broad spectrum light leaning towards the red end of the spectrum. However, a 300W or 400W LED will seriously outperform it.
I hope this helps you