I assume that the latter value is an exact representation of the value of the double because after the 53rd decimal digit, all (1153) following decimal digits are 0.

Two thoughts:

1. The following extract from the mpfr docs suggests that when mpfr does calculations, it does them to full (infinite) precision and only truncates back to the variable's precision (or the current default) once the calculation is complete:

   The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with “infinite accuracy”), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system.

   MPFR does not keep track of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision.

   Which I believe means that calculations that end up in 53-bit precision variables may (will often) yield different results from calculations done using IEEE-754 64-bit floats; because of greater accuracy at intermediate points.

2. MPFR seems to produce far more digits of "precision" than it ought.

   In your example above 1 / 526, you posted a result of: 1.9011406844106463 7711435114653113487293012440204620361328125e-3
   But the actual result (to 100 digits of precision is : 1.9011406844106463 87832699619771863117870722433460076045627376425855513307984790874524714828897338403e-3
   As you can see, only the first 17 digits of the MPFR output are accurate.
   
   As calculated with perl's doubles                    : 1.901140684410646 40000e-003
   
   Ditto your 1 / 535 example: 1.869158878504672 97461032334382480257772840559482574462890625e-3
   Actual (to 100 digits)    : 1.869158878504672 897196261682242990654205607476635514018691588785046728971962616822429906542056074766e-3
   This time only 16 digits of the MPFR output match.
   With perl's doubles       : 1.86915887850467 300000e-003
   

I'm not sure what that all means, but I did validate the 1/526 example manually, as well as calculating them with Math::BigFloat.