3 ms·

Here are some values that are (understandably) not listed on the blog. They happen only due to the limited precision of floating point formats. 128

Here are some values that are (understandably) not listed on the blog. They happen only due to the limited precision of floating point formats. 128 = √(2 / √√(√2 - (2 / √2))) 8192 = √√(2 / ((√2 * √2) - 2)) 16384 = (2 / √√(√2 - (2 / √2))) 67108864 = √(2 / ((√2 * √2) - 2)) 134217728 = (2 / √(√2 - (2 / √2))) 4503599627370496 = (2 / ((√2 * √2) - 2)) 9007199254740992 = (2 / (√2 - (2 / √2))) 6369051672525773 = (√2 / (√2 - (2 / √2))) I found these by accident a long time ago but kept them because they do "work". Try to input one expression in the lil box in https://www.wolframalpha.com/?source=nav https://www.wolframalpha.com/?source=nav and they will quickly evaluate to these values; the charade goes away after you press Enter and get the (mathematically) correct answer. My old solvers from what feels like a previous life: https://madflame991.blogspot.com/2013/02/four-fours.html https://madflame991.blogspot.com/2013/02/four-fours.html https://madflame991.blogspot.com/2013/02/return-of-four-fours.html https://madflame991.blogspot.com/2013/02/return-of-four-four... That was fun

lifthrasiir 2y ago
When everything is an IEEE 754 floating point number, a mathematically "linear" function can indeed be coerced into anything: http://tom7.org/grad/ http://tom7.org/grad/
svat 2y ago
Nice! Looking into them a bit deeper, they all rely on two facts involving quantities that are off by 1 ulp: √2 - 2/√2 as a float64 is exactly equal to 2^{-52} √2 * √2 as a float64 is exactly equal to 2 + 2^{-51} and all other calculations after that are mathematically exact. For example: √(2 / √√(√2 - (2 / √2))) := √(2 / √√(2^{-52})) = √(2 / 2^{-13}) = 2^7 = 128 and √√(2 / ((√2 * √2) - 2)) := √√(2 / 2^{-51}) = √√(2^{52}) = 2^{13} = 8192 In your last one, 6369051672525773 = 2^{52} * 1.4142135623730951454746218587388284504413604736328125 (https://float.exposed/0x4336a09e667f3bcd https://float.exposed/0x4336a09e667f3bcd) where the second value is the exact value of the floating-point representation of √2 (i.e. the closest representable-in-float64 value to √2).

lifthrasiir 2y ago

When everything is an IEEE 754 floating point number, a mathematically "linear" function can indeed be coerced into anything: http://tom7.org/grad/ http://tom7.org/grad/

svat 2y ago

Nice! Looking into them a bit deeper, they all rely on two facts involving quantities that are off by 1 ulp: √2 - 2/√2 as a float64 is exactly equal to 2^{-52} √2 * √2 as a float64 is exactly equal to 2 + 2^{-51} and all other calculations after that are mathematically exact. For example: √(2 / √√(√2 - (2 / √2))) := √(2 / √√(2^{-52})) = √(2 / 2^{-13}) = 2^7 = 128 and √√(2 / ((√2 * √2) - 2)) := √√(2 / 2^{-51}) = √√(2^{52}) = 2^{13} = 8192 In your last one, 6369051672525773 = 2^{52} * 1.4142135623730951454746218587388284504413604736328125 (https://float.exposed/0x4336a09e667f3bcd https://float.exposed/0x4336a09e667f3bcd) where the second value is the exact value of the floating-point representation of √2 (i.e. the closest representable-in-float64 value to √2).