B777 FLT CREWBEGINS REQUIRED TURN ON RNAV DEP EARLY WHILE DEPARTING KSI.

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THIRD LEG OF A 4 LEG; 5 DAY IDENT. SCHEDULED REST OF APPROX 13 HRS; BLOCK-TO-BLOCK. RELATIVELY SHORT FLYING TIME PLANNED FOR 1 HR 45 MIN ENRTE. DURING OUR PREDEP BRIEFING; THE CAPT AND I THOROUGHLY DISCUSSED THE ENG-OUT PROC; THE RADIO-OUT PROCS (AND HOW THEY WERE ALIKE); AND THE SID. WE TALKED ABOUT THE REASONS FOR THE ENG-OUT/RADIO-OUT PART OF WHICH IS THE TERRAIN; BUT ALSO BECAUSE OF THE RESTR AREAS DESIGNED TO KEEP YOU FROM FLYING INTO THE NO-MAN'S-LAND/FREE FIRE ZONE A FEW MI N. THE WX WAS GOOD/CLR; WITH NOT MUCH WIND. ON THE RWY; THE CAPT GAVE ME THE AIRPLANE; AND WE GOT CLRNC FOR TKOF. NORMAL TKOF TILL ABOUT 500 FT; WHERE I ANNOUNCED THAT I WAS STARTING MY L TURN. I DON'T KNOW IF THE L TURN WAS CAUGHT BY THE CAPT. SINCE THE WX WAS CLR; WE WERE BOTH LOOKING OUT; NOT ONLY FOR TFC; BUT FOR THE LANDMARKS CONCERNING THE RESTR AREA (FREE FIRE ZONES); AND THE TERRAIN; WHICH REQUIRED THE ENG-OUT/LOST COMS PROCS. UNFORTUNATELY; THE SID DID NOT REQUIRE A L TURN AT THAT POINT. THE CAPT CAUGHT THE ERROR AFTER I WAS ESTABLISHED IN THE TURN; ABOUT THE SAME TIME AS I DID. WE WERE IN A TURN BACK TO THE R; WHEN DEP CTL REQUESTED OUR HDG; AND GAVE US A TURN TO 220 DEGS. THERE WAS NO OTHER TFC ON TKOF OR ON APCH TO EITHER RWY; WHOM WE MIGHT HAVE INTERFERED WITH. AFTER ESTABLISHED IN THE CLB AND ON HDGS; DEP ASKED WHY WE DEVIATED. THE CAPT SAID THAT THERE WAS AN ERROR IN USING THE FMC; NOT SURE THE EXACT WORDS. I WAS KIND OF IN SHOCK AT WHAT JUST OCCURRED! DEP ASKED IF THERE WERE ANY FURTHER PROBS; AND THE CAPT ASSURED HIM THAT THERE WERE NONE. THERE WAS NO FURTHER DISCUSSION ON THE DEV WITH ATC. I APOLOGIZED FOR THE ERROR AND CONFESSED I HAD NO CLUE AS TO WHY. LATER THE CAPT SAID HE WOULD FILL OUT A RPT. I MADE THE ASSUMPTION (FALSE!) THAT IT WAS FOR BOTH OF US. I DON'T KNOW WHY I MADE THE TURN EARLY; CONSIDERABLY EARLY. IT MIGHT HAVE BEEN SOMETHING TO DO WITH THE RESTR AREAS. THERE IS NOTHING THAT I AM CONSCIOUSLY AWARE OF. BELIEVE ME; I HAVE SEARCHED. I KNOW THAT IN FLYING TAG LEGS FOR OVER A YR; I STILL DON'T GET GOOD REST THE NIGHT BEFORE THIS LEG. THIS NIGHT WAS NO BETTER. GOT A COUPLE OF HRS OF INCOMPLETE SLEEP; THOUGH I WAS IN BED FOR 8 HRS WITH LIGHTS OUT. THE TIME OF TKOF IS USUALLY (HOME/BODY TIME) ABOUT 2 HRS BEFORE MY REGULAR GO-TO-BED/SLEEP ROUTINE. HOWEVER; I DON'T RECALL BEING SLEEP DEPRIVED AT THIS TIME. ALSO; THOUGHT ABOUT THE IMPACT OF OUR COMPANY PROBS. I HAVE TRIED TO NOT LET THEM INTERFERE WITH MY PERFORMANCE. THERE IS ALMOST NO INFLUENCE ANY OF US HAVE ON THE ULTIMATE (AND UNKNOWN) OUTCOME. I CAN'T PUT MY FINGER ON ANYTHING LIKE THAT. BUT; I CAN'T GUARANTEE THERE IS NO IMPACT.