Last Sunday I shot a few rolls of Ilford HP5+ (both 35 & 120mm) @ the nominal speed of 400 under bright sun conditions.
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In order to keep a good tonality and normal contrast, should I under-develop a little bit the film? Cause, I think if I use normal development time, then the contrast will be unwillingly high.
In a word? NO!!!!!!!!! There is nothing wrong with box speed. The film can handle the latitude of light.
Steve
Well, depends on what end result you want...
I only seriously started experimenting with push/pull developments about a year ago, mostly pull for night shots. I recently had similar conditions to yours (bright sun), and decided to do an experiment by shooting two consecutive sheets of HP5+ from the same scene with different exposures (one normal, one overexposed), to do a "normal" and "pull" development to see what I would get.
The attachments show the (scanned) results, both the negatives, and a bit arbitrary positive conversions I made in you-know-what, but that I think represent the images quite well.
The left images are the "normal" ones, the right the "pull" version.
I measured at EI250, but that is mainly because my light meter is calibrated to match my Dynax 7, which seems to underexpose (at least with my standard BW development techniques) compared to box speed. I get best results if I overexpose by 1/2 - 2/3 of a stop and do a slight pull (10-15%)
I measured an EV9.8 for zone 2/3 in the lower left corner, and EV 16.3 for the brightest patches in the sky, so overall about 6.5 EV values. Not an exceptional range, and a pull might not be necessary, but I decided for "science sake"

to try it anyway...
I exposed the "normal" shot 1 second, F32
I exposed the "pull" shot 8 second, F32 (reciprocity!)
Both shots with a red filter.
I developed the normal shot 10 min. in D76 1:1
I developed the pull shot 6.5 min. in D76 1:1
Notice the pull shot looks considerably flatter, but has clearly more shadow detail in the lower parts of the image in the shadow of the ferns. If I would print this negative, I would probably print it a slightly higher grade (3), to get the image somewhere inbetween the two scanned ones you are seeing here.