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"When it goes wrong, it really does go wrong – one wrong move, one throwaway comment and months of planning and international diplomacy are in the bin."

No stress then! As one former Downing Street senior staff member recalls, visits to the White House are about as high pressure as it gets.

British prime ministers always lavish praise on the importance of the relationship with the US. So when it's time for an invite to 1600 Pennsylvania Avenue the pressure is really on.

In the run up to the big event, warm words are the order of the day.

Today is no exception. Boris Johnson told the travelling press pack that the relationship between the two countries is as "good as it has been for decades", claiming his relationship with President Biden is "genuinely terrific", and joking the two have bonded over their love of rail – "he's a train nut".

There's bonhomie, and – having witnessed the two at the G7 in Cornwall – it is visible.

The UK camp is encouraged at the American removal of travel restrictions on UK citizens in November. And there's a sense that the president might be ready to make extra commitments on climate finance which would thrill Downing Street. But nothing is that straightforward.

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They'll have to confront tensions over the handling of the withdrawal of forces in Afghanistan. And the UK doesn't expect much progress on trade either, saying American negotiators are "ruthless", playing down expectations of things moving forward.

But first and foremost both sides will want Tuesday's big meeting to go smoothly. No gaffes, no misunderstandings, no embarrassments.

Whatever the personal friendships or frictions between the two main protagonists, it matters because, as one No 10 insider told me, "the relationship matters for the mutual benefit of our people".

The current Downing Street team don't seem to be nervous about the meeting on Tuesday, at least outwardly, but they are conscious of how much it matters, and how previously things have gone awry.

Who could forget the last time a British prime minister held talks in the Oval Office? The press conference back in 2017 was quite the experience.

media captionPresident Trump joked about the special relationship with the UK

What you might not know was that Theresa May's team was worried about the menu – because the ribs on offer at the getting-to-know-you lunch might plaster barbecue sauce all over the then prime minster's face during the huge TV moment. They discovered the meat was "neatly chopped" after all.

Or that former President Trump anti-bacced his hands after shaking hands with Theresa May's team – one of them now jokes "he was ahead of his time".

Or indeed, that "May bossed her foreign secretary around on the plane on the way, talking down to him in front of others, and yet here is the man now in charge". Ouch!

For David Cameron's team there were similar stresses, and frankly high farce. After "endless discussions about photo ops", one member of his team remembers, the two men took Air Force One to a college basketball game, putting David Cameron under unfamiliar pressure when he had to give a half-time interview about the match on a "subject he knew nothing about".

media captionPresident Barack Obama has given British Prime Minister David Cameron a front row seat at the US college basketball championships

Money couldn't buy these kind of images for a British prime minister – putting them on the spot over alley-oops or a 5-out motion, less desirable altogether.

And that's before you factor in the British reaction back then to the mere name of the White House protocol supremo, Randy Bumgardner, to the bemusement of the Americans. Two nations divided perhaps not just by a common language, but by different senses of humour too.

So, there is a lot at stake – the risk of misunderstanding, disagreement on policy, a clash of political personality, or just blunders that cause blushes. As one of those who felt it go wrong to their cost told me, "it's important to be on best behaviour".

Mr Johnson has been to the White House before, as foreign secretary in 2019. But so much has changed then, on this side of the Atlantic and at home. A very different prime minister. A very different president. A very different set of challenges for the world.

And British prime ministers' visits to the White House are, as occasions in themselves, about as momentous as it gets.