The following program computes a lower bound on 1+1/1!+1/2!+...+1/100!
using 200-bit precision: `mpfr_t s, t, u;`

declares three
floating-point variables `s`, `t`, `u`;
`mpfr_init2 (t, 200);`

initializes the variable `t` with
200-bit precision; then `mpfr_set_d (t, 1.0, MPFR_RNDD);`

sets the
value of `t` to the double-precision number 1.0 rounded towards
minus infinity (here no rounding is done since 1 can be represented exactly
on 200 bits); the statement `mpfr_mul_ui (t, t, i, MPFR_RNDU);`

multiplies `t` in place by the unsigned integer `i`,
where the result is rounded towards plus infinity;
`mpfr_div (u, u, t, MPFR_RNDD);`

divides `u` by
`t`, rounds the result towards minus infinity and stores it
into `u`; then the statement
`mpfr_out_str (stdout, 10, 0, s, MPFR_RNDD);`

prints the value of
`s` in base 10, rounded towards minus infinity, where the third
argument 0 means that the number of printed digits is automatically chosen
from the precision of `s`; finally the `mpfr_clear`

calls free the space used by the MPFR
variables.

Note: with this program, you need MPFR 3.0 or later.

#include <stdio.h> #include <gmp.h> #include <mpfr.h> int main (void) { unsigned int i; mpfr_t s, t, u; mpfr_init2 (t, 200); mpfr_set_d (t, 1.0, MPFR_RNDD); mpfr_init2 (s, 200); mpfr_set_d (s, 1.0, MPFR_RNDD); mpfr_init2 (u, 200); for (i = 1; i <= 100; i++) { mpfr_mul_ui (t, t, i, MPFR_RNDU); mpfr_set_d (u, 1.0, MPFR_RNDD); mpfr_div (u, u, t, MPFR_RNDD); mpfr_add (s, s, u, MPFR_RNDD); } printf ("Sum is "); mpfr_out_str (stdout, 10, 0, s, MPFR_RNDD); putchar ('\n'); mpfr_clear (s); mpfr_clear (t); mpfr_clear (u); return 0; }

The result of this program is:

$ ./sample Sum is 2.7182818284590452353602874713526624977572470936999595749669131