Made with the craft one has come to expect of Clint Eastwood’s movies (with their repertory-style crews) and boasting an impressive performance by Bradley Cooper (so beefed-up that he’s almost unrecognizable), this adaptation of the late Chris Kyle’s autobiographical memoir is notable for its deliberately equivocal approach. “American Sniper” is on the one hand hagiographical, celebrating the sacrifices Kyle made in the Iraq war (as well as those he made after his return stateside); but it also raises some troubling questions about his mission, and even more about the devastating impact, both physical and psychological, combat had on him and his comrades. In that respect it aims to pull off somewhat the same trick that “Patton” did: some viewers read that as the ultimate exercise in bellicose patriotism, while others saw it as essentially an anti-war movie. It’s a tightrope act that “Sniper” doesn’t manage quite so successfully, though it deserves credit for trying.

Kyle, of course, became the country’s most renowned sharpshooter, credited with more than a hundred and fifty combat kills in Iraq during four tours of duty there—a feat that earned him the nickname of “Legend” and made him a chief target of insurgents. In 2013, four years after leaving the service, he was shot to death, allegedly by a troubled Marine veteran he was trying to help deal with his PTSD. After a prologue showing him with a woman and her young son in his sights as they approach a group of American soldiers on patrol, Eastwood’s film quickly cuts to a flashback showing Kyle’s childhood aptitude in marksmanship before jumping to his days as a wild-living rodeo rider. It’s 9/11 that leads him to enlist in the Navy SEALs, and to succeed in their rigorous training program despite being older than most of the recruits. It’s during that training period that he also meets, and weds, feisty Taya (Sienna Miller), right before being shipped out on his first tour.

From this point “Sniper” alternates between scenes of combat in Iraq—by far the larger portion of the film—in which Kyle’s phenomenal skill is repeatedly showcased, and shorter episodes in which he returns to the States changed by the experience and increasingly uncomfortable in a peacetime environment. (The contrast is none too subtly shown when Taya and Chris share phone conversations while he’s in the midst of battle.) Eastwood, his usual cinematographer Tom Stern and editors Joel Cox and Gary D. Roach do a fine job with the individual wartime episodes, which range from relatively brief recreations of specific kills to more extended episodes in which Kyle, for example, joins Marines on the ground to assist them in clearing houses. But the focus gradually turns to Kyle’s effort to take down a terrorist leader called The Butcher (Mido Hamada), lieutenant to Al-Quaeda chief Abu al-Zarqawi, which leads him and the special team he heads into a virtual competition with an insurgent sniper, Mustafa (Sammy Sheik), an Olympic medal winner as adept as Kyle at his deadly trade. Even in a climactic mission, when Kyle and his comrades find themselves stranded on a rooftop in enemy-controlled territory, the action is depicted in an unfussy but compelling way that once again demonstrates the smooth professionalism with which the director and his roster of behind-the-scenes cohorts are able to pull off such complicated sequences.

Not nearly as complex as the battle choreography is the film’s depiction of the Iraqi insurgents. There’s one scene, when Kyle’s squad takes over the apartment of a family for surveillance purposes and discovers that the husband has a secret cache of arms under the floorboards, that comes close to showing a degree of understanding of a civilian population complicit in attacks on Americans, and there’s a grudging recognition of Mustafa’s ability, even if it ultimately proves unequal to Kyle’s. But the Iraqis are generally treated as a faceless mass—dismissed by one soldier as “savages”—except in an episode involving a sheik (Navid Negahban) who’s been pressured to cooperate with the US forces and who, along with his young son, suffers terrible retribution at the hands of The Butcher as Kyle watches, unable to intervene. Of course the entire narrative is told from Kyle’s perspective, and so the characterization of those he must take aim at has little room for nuance, except for his own agony at deciding whether to fire at a boy with a grenade. For something more—something akin to the treatment of the Japanese Eastwood himself offered in “Letters from Iwo Jima”—we’ll probably have to wait a decade or two.

The stateside interludes, by contrast, aren’t quite so impressive. Cooper certainly captures the sense of alienation Kyle feels from “normal” American life and the almost irresistible inclination to return to the “abnormal” environment he’s now accustomed to by repeatedly going back to Iraq and the fellow soldiers for whom he feels more responsibility than perhaps he does to his own children. It’s the same issue that Jeremy Renner’s Sgt. James faced in “The Hurt Locker,” though traced here in a more subdued fashion; and like that film, Eastwood’s necessarily raises the issue of the psychological toll that the reality of war takes on its protagonist. But while Kathryn Bigelow, dealing with the matter in purely fictional terms, was able to explore it unflinchingly, here it’s understandably treated more cautiously, and some of the dialogue, especially that given to Taya, is so clichéd that it takes the picture close to soap opera.

Otherwise, however, Miller is quite good as a woman supportive of a husband from whom she can’t help but feel increasingly distant. The actors playing Kyle’s fellow soldiers likewise do admirable if unremarkable jobs. But the linchpin holding the picture together is Cooper, who subtly registers Kyle’s shift from the swaggering, ingratiatingly straightforward young man of the film’s early scenes to a more pensive, if not tortured, individual as the experience of the war—and in particular the realization of what it means to have taken so many lives from a distance while yet regretting not having saved more of his comrades by taking still more—sinks in. It’s easily the most subtle work the actor has done on screen, and together with his current Broadway turn in “The Elephant Man,” evinces his growth from the emptier work of the “Hangover” franchise and even “Silver Linings Playbook.” Eastwood’s own score matches it in its economy and restraint.

“American Sniper” is a respectful, and more than respectable, attempt to celebrate Kyle’s service while also taking account of the effect it had on him and his family. If in the end its achievement doesn’t quite meet its ambition, it remains an admirable effort.