When Apple co-founder Steve Jobs told his daughter Lisa Brennan-Jobs that the Apple Lisa computer was not named after her, it was not a cruel lie to a little girl, she insists – he was teaching her "not to ride on his coat-tails".

When Jobs refused to install heat in her bedroom, he was not being callous, she says – he was instilling in her a "value system". When a dying Jobs told Brennan-Jobs that she smelled "like a toilet", it was not a hateful snipe, she maintains – he was merely showing her "honesty".

It's a strange thing to write a devastating memoir with damning details but demand that these things are not, in fact, damning at all. Yet that's exactly what Brennan-Jobs has done in a new memoir, Small Fry. Thanks to a dozen other biographies and films, Apple obsessives already know the broad outlines of Brennan-Jobs' early life: Jobs fathered her at 23, then denied paternity despite a DNA match, and gave little in financial or emotional support even as he became a god of the early computing era. Small Fry is Brennan-Jobs' effort to reclaim her story for herself.

The backdrop to her raw depictions of life with and without Jobs is 1980s Silicon Valley, where artists and hippies mixed with technologists, ideas of how to build the future flourished, and a cascade of trillions of dollars was just beginning to crash onto the landscape. Brennan-Jobs navigated a childhood on welfare with her mother, artist Chrisann Brennan, and an adolescence ensconced in her father's wealth. In passage after passage of Small Fry, Jobs is vicious to his daughter and those around her. Now, in the days before the book is released, Brennan-Jobs is fearful that it will be received as a tell-all exposé, and not the more nuanced portrait of a family that she intended. She worries that the reaction will be about a famous man's legacy rather than a young woman's story – that she will be erased again, this time in her own memoir.

On the eve of publication, what Brennan-Jobs wants readers to know is this: Steve Jobs rejected his daughter for years, but that daughter has absolved him. Triumphantly, she loves him, and she wants the book's scenes of their roller skating and laughing together to be as viral as the scenes of him telling her she will inherit nothing.

Brennan-Jobs' forgiveness is one thing. What's tricky is that she wants the reader to forgive Jobs, too. And she knows that could be a problem. "Have I failed?" she asks. "Have I failed in fully representing the dearness and the pleasure? The dearness of my father, and the outrageous pleasure of being with him when he was in good form?"

After university, Brennan-Jobs left the United States to work in finance in London and Italy; she later shifted into design, and then freelance writing for magazines and literary journals. Now 40, she has long avoided publicity. She has never been profiled, and she has carefully eluded most of her father's chroniclers. (One exception: screenwriter/director Aaron Sorkin, who called her "the heroine" of his 2015 Steve Jobs biopic.) Brennan-Jobs says she did not trust Walter Isaacson, who wrote the definitive, mega-selling biography of her father in 2011.

"I never spoke with Walter, and I never read the book, but I know I came off as cold to my father and not caring whether he felt bad," Brennan-Jobs says, sitting in Cantine, a vegan-friendly cafe in Brooklyn's Carroll Gardens neighbourhood. "I was devastated by it.

"I felt ashamed to be the bad part of a great story," she continues. "And I felt unresolved." And so in Small Fry, she seeks to resolve some of that shame by describing how her childhood unfolded, who key characters were, why it all happened. Brennan-Jobs went back to Silicon Valley and interviewed her family, her friends, her mother's ex-boyfriends, and her father's ex-girlfriend. In her childhood, the region had been green with eucalyptus and full of garage hackers. Now it is the greatest wealth-creation machine in the history of the world, and Jobs remains its towering hero.

Brennan-Jobs began work on what would become Small Fry not long after her father's 2011 death. Years into writing, she felt rushed by her publisher, Penguin Press, and feared being "tarted up" and made to take advantage of her father's legacy. She wanted to be with a smaller publisher who would work with her and give her more time, and switched to Grove, taking what she says was a 90 per cent cut in her advance. (A spokesperson for Penguin declined to comment.)

One result of the delay is that Small Fry is entering the public conversation at a time when, across industries, formerly disempowered or ignored women are having their say about powerful men. A memoir by Steve Jobs' first-born was always going to be a publishing sensation, but Brennan-Jobs has inadvertently timed hers to land when the public is even more attuned to marginalised voices – and when many are having darker thoughts about the world Jobs created with his attention-devouring devices.

None of that, of course, was imaginable when Brennan-Jobs was born on May 17, 1978, on a commune farm in Oregon. Her parents, who had met in high school in Cupertino, California, were both 23. Jobs arrived days after the birth and helped name her, but refused to acknowledge that he was the father. To support her family, Chrisann Brennan cleaned houses and used government assistance. Only after the government sued Jobs did he agree to pay child support.

Small Fry describes how Jobs slowly took a greater interest in his daughter, taking her skating and coming over to her house for visits. Brennan-Jobs moved in with him for a time during high school, when her mother was struggling with money and her temper, but Jobs was cold and had extreme demands for what being a member of the family entailed.

The neighbours next door worried about the teenage Lisa, and one night, when Jobs was out, they moved her from his house and into theirs. Against Jobs' wishes, the neighbours paid for her to finish college. (He later paid them back.)

Brennan-Jobs speaks today of "not wanting to alienate people" she loves, but acknowledges that her memoir might do just that. Aside from Jobs, all the central characters are very much alive. "I hope Thanksgiving's okay," she says.

Her mother is portrayed as a free spirit who nurtured her daughter's creativity, but could be mercurial, hot-tempered and sometimes neglectful. "It was horrendous for me to read," Brennan says. "It was very, very hard. But she got it right."

Jobs' infamous venom is on frequent display in Small Fry. Out one night at dinner, Jobs turns to his daughter's cousin, Sarah, who has just unknowingly offended him by ordering meat. "'Have you ever thought about how awful your voice is?" Jobs asks Sarah. "Please stop talking in that awful voice." He adds, "You should really consider what's wrong with yourself and try to fix it."

Brennan-Jobs describes her father's frequent use of money to confuse or frighten her. "Sometimes he decided not to pay for things at the very last minute," she writes, "walking out of restaurants without paying the bill." When her mother found a beautiful house and asked Jobs to buy it for her and Lisa, he agreed it was nice – but bought it for himself and moved in with his wife, Laurene Powell Jobs.

Brennan says her daughter has, if anything, underplayed the chaos of her childhood. "She didn't go into how bad it really was, if you can believe that," she says. But Small Fry also contains moments of joy that capture Jobs' spontaneity and unparalleled mind.

When Brennan-Jobs goes on a school trip to Japan, he arrives unannounced and pulls her out of the program for a day. Father and daughter sit, talking about God and how he sees consciousness. "I was afraid of him and, at the same time, I felt a quaking, electric love," she writes.

"When I started writing," Brennan-Jobs tells me, "I didn't think he'd be so interesting on the page, and I was almost frustrated that he pulled so much gravity." After Brennan-Jobs moved in with Jobs as a teenager, he forbade her from seeing her mother for six months, as a way to cement her connection to his new family. At the same time, Jobs shifted from neglectful to controlling. When Brennan-Jobs was getting increasingly involved at her high school, starting an opera club and running for freshman-class president, he got upset.

"This isn't working out. You're not succeeding as a member of this family," Jobs says in the memoir. "You're never around. If you want to be a part of this family, you need to put in the time."

To appease her father, Brennan-Jobs transferred to another school that was closer to her father's house. She persisted in becoming editor-in-chief of the school newspaper. Her mentor there, a journalism teacher named Esther Wojcicki, says Small Fry is a faithful account. "The dialogue that she had in there between her and Steve was just exactly right," Wojcicki says. "The book is a gift to all of us."

Early copies of the memoir have circulated among family and friends. Powell Jobs, her children and Jobs' sister, Mona Simpson, give this statement: "Lisa is part of our family, so it was with sadness that we read her book, which differs dramatically from our memories of those times. The portrayal of Steve is not the husband and father we knew. Steve loved Lisa, and he regretted that he was not the father he should have been during her early childhood. It was a great comfort to Steve to have Lisa home with all of us during the last days of his life, and we are all grateful for the years we spent together as a family."

On a hot August day in Brooklyn, Brennan-Jobs and I walk to her studio, a small apartment with brick walls she painted white and a bamboo floor she painted black. While writing Small Fry, she tells me, she covered the mirrors around her work space with paper. "I don't like catching myself in the mirror," she says, "because it's like – 'Oh, self.'"

Brennan-Jobs says she's nervous about how she'll be described physically in a profile, so I ask her to use her own words. "My face is uneven," she says. "I have small eyes. I wish I had dimples, but I don't. I think right now I look jowly." I interject to say she has delicate features, and freckles, and is about five foot two, with slightly reddish brown hair. "My nose," Brennan-Jobs replies, "is not particularly delicate." She is deeply self-deprecating, saying she's horrified to be doing "a celebrity memoir". She says she's sure The New Yorker will not review the book, and that years ago, her first meeting at her publisher Grove only occurred because Elisabeth Schmitz, the editorial director, was doing a favour for a mutual friend.

"My first thought on being pitched the book was, 'I don't do this kind of thing. I don't know how to publish a celebrity memoir,' " says Schmitz, who has acquired literary memoirs like naturalist Helen Macdonald's H Is for Hawk. But something about Brennan-Jobs' writing made her reconsider. "From the first page," she says, "her language is fresh, surprising, unpredictable."

I've read it, and her writing really is compelling. Brennan-Jobs takes the same linguistic knife to herself as she does to others. She writes with disgust about using anecdotes from her childhood to elicit sympathy from others, and she is ashamed to have dropped her father's name during an interview to get into Harvard.

On August 1, Vanity Fair published an excerpt from Small Fry under the digital headline "I Have a Secret. My Father Is Steve Jobs". A few nights later, Brennan-Jobs called me, worried.

She hated the title, and on social media, readers were feasting on the more savage details of her account, especially the "toilet" comment. "He was telling me the truth," Brennan-Jobs tells me, adding that the rosewater perfume she wore had turned. "I wasn't aware of it. Sometimes it's nice of someone to tell you what you smell like."

It was another uncomfortable reminder that even though Small Fry is Brennan-Jobs' story, one written in a precise, literary style, her father's myth looms so large that she cannot control how her words are received. When choosing a narrator for the audio version, she nixed the ones who spoke his lines too harshly or without humour.

So much of Brennan-Jobs' effort with the memoir seems to be to show how brutal Steve Jobs could be – and, in doing so, to reclaim that brutality for herself. And how she wants to reclaim it is to love it.

"You get your inheritance, delivered in a lump of coal or whatever in a sort of awful package," she tells me at one point. "And you have to take a lot of time to turn the awful package on its head, and it reveals something kind of glorious, and then you're set free."

If Brennan-Jobs is alarmed by the reaction to the toilet-water excerpt, she may be unprepared for what happens when readers encounter more disturbing material. Several times in Small Fry, Jobs engages in what seems like inappropriate affection in front of his daughter. Brennan-Jobs describes him embracing Powell Jobs one day, "pulling her in to a kiss, moving his hand closer to her breasts," and up her thigh, "moaning theatrically".

Brennan-Jobs tried to leave, her father stopped her: "'Hey Lis,' he said. 'Stay here. We're having a family moment. It's important that you try to be part of this family.' I sat still, looking away as he moaned and undulated." Brennan-Jobs emphasises in this interview that she never felt threatened by her father, and that to her, these scenes show he was "just awkward".

This kind of display was not an isolated incident, says Brennan-Jobs' mother, who described an upsetting, sexualised conversation between Jobs and their daughter in her 2013 memoir, A Bite in the Apple. One evening, Brennan writes, she let Jobs babysit nine-year-old Lisa. When Brennan came home early, she found Jobs with the girl, "teasing her nonstop about her sexual aspirations", "ridiculing her with sexual innuendos" and "joking about bedroom antics between Lisa and this or that guy".

Brennan, in her memoir, describes feeling scared for her daughter that night, and wanting to place her body between them and get out of there. "I will be clear," Brennan writes. "Steve was not a sexual predator of children. There was something else going on." Still, after that night, Brennan tells me she tried to make sure there was "a chaperone" when Jobs was with his young daughter for long hours. "He was so inappropriate because he didn't know how to do better," Brennan says. In her book, she characterises Jobs as "on a slide whistle between human and inhuman".

One afternoon in August, as Brennan-Jobs and I talk in her kitchen, she makes a juice of dandelion greens, pineapple, turmeric and ginger roots. She eats an extremely healthy diet and knows it mirrors her father's, which veered into esoteric California wellness trends, even as pancreatic cancer took over more of his body.

Brennan-Jobs has a husband, Bill, a longtime Microsoft employee now launching a software start-up. He has two daughters, aged 10 and 12, and he and Brennan-Jobs have a four-month-old son. As she drinks her juice, Bill is nearby with the children, and there's an easygoing energy in the house. "I see my husband and the way he is with his daughters, responsive and alive and sensitive in ways my father would have liked to be," Brennan-Jobs says. "My father would have loved to be a man like that, and he surrounded himself with men like that, but he couldn't be."

Decades after his child-support lawsuit, Jobs erased his paternity again. Small Fry notes that on his corporate bio on the Apple website, the detail-obsessed chief executive was listed as having three children. But, of course, he had four.

The most public torchbearer for Jobs' character and legacy is Powell Jobs. With an inherited fortune of some $US21 billion ($28.6 billion), she has engaged in philanthropy and launched the Emerson Collective, an organisation that pursues liberal political activism and for-profit investments, and owns a majority stake in The Atlantic magazine.

Powell Jobs plays a somewhat "tonic note" in Small Fry, Brennan-Jobs says. Her stepmother brings her into family photos, for example, but many of the descriptions of Powell Jobs are biting. Brennan-Jobs tells me she gave Powell Jobs "the best line" in the book. It appears in a scene where Powell Jobs and Jobs go to a therapy session with a teenage Lisa. Brennan-Jobs cries and says she feels lonely and has wanted them to say good night to her. Powell Jobs responds to the therapist: "We're just cold people."

Toward the end of Jobs' life, he finally apologised to his daughter. Brennan-Jobs calls it her "movie ending". In the book, she writes that Jobs says he is sorry he did not spend more time with her, and for disappearing during her adulthood, forgetting birthdays and not returning notes or calls. In reply, Brennan-Jobs says she knows he was busy. Jobs answers that he acted the way he did because she had offended him.

"It wasn't because I was busy. It was because I was mad you didn't invite me to the Harvard weekend," he says in the book, referring to a matriculation event. He also cries and tells her over and over again, "I owe you one" – a famously articulate communicator unable to summon the basic language of contrition.

Brennan-Jobs may be experiencing a kind of author's remorse as her book makes its way toward store shelves. But details as lethal as these – they sink into Jobs' legend like daggers to the hilt – are more proof than any DNA test that she is her father's daughter. Ultimately, Jobs left his daughter an inheritance in the millions, the same amount as his other children, and she is not involved in the allocation of his financial legacy. If she was in charge of his billions, she says, she would give it away to the Bill and Melinda Gates Foundation – a curious twist given her father's epic rivalry with Apple's arch-nemesis.

"Would it be too perverse?" she asks. "I feel like the Gates Foundation is really doing good stuff, and I think I would just hot-potato it away."

Brennan-Jobs says she wrote Small Fry in part to figure out why he withheld money from her even as his wealth ballooned, and as he spent it more freely on the children he had with Powell Jobs. She says she now sees it was about teaching her that money can corrupt. The ethos "felt true and kind of beautiful and kind of enlightened for somebody like that". The question was "why he would have taken that value system and applied it so severely to me".

"You can have a value system and be unable to totally live it," she adds. "And you can imagine being that rich and famous and how amazing it is if you can hold on to some of your value system. He didn't do it right. He didn't apply it evenly. But I feel grateful for it."

Brennan-Jobs tells me she likes toying with the strange power of being a memoirist writing about trauma because the reader knows she made it out okay. She is here in the privileged position of writing this book, after all. And as a memoirist, even a reluctant one, she gets the final word. One night toward the end of Jobs' life – and the end of the book – he is watching Law and Order in bed. "'Are you going to write about me?'" he asks Lisa. She tells him no. "'Good,' he says, and turns back to the television."

Small Fry by Lisa Brennan-Jobs (Grove Press, $30) is out on September 12.

Edited version of a story first published in The New York Times. © 2018 The New York Times.

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