Fldpi only loads the register, rounding according to the current rounding mode, so you still only have 63 bits of precision when you load the value. For interm

Fldpi only loads the register, rounding according to the current rounding mode, so you still only have 63 bits of precision when you load the value. For intermediate calculations it incorporates the internal extended pi via the GRS bits. And yeah the train wreck of argument reduction was my reference