diff options
Diffstat (limited to 'src/rootofunity.c')
| -rw-r--r-- | src/rootofunity.c | 16 |
1 files changed, 11 insertions, 5 deletions
diff --git a/src/rootofunity.c b/src/rootofunity.c index 84697c7..e4f533e 100644 --- a/src/rootofunity.c +++ b/src/rootofunity.c @@ -137,7 +137,8 @@ mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_ mpfr_sin_cos (s, c, t, MPFR_RNDN); /* error (1.5*2^{Exp (t) - Exp (s resp. c)} + 0.5) ulp We have 0<t<2*pi, so Exp (t) <= 3. - Unfortunately s or c can be close to 0. + Unfortunately s or c can be close to 0, but with n<2^64, + we lose at most about 64 bits: Where the minimum of s and c over all primitive n-th roots of unity is reached depends on n mod 4. To simplify the argument, we will consider the 4*n-th roots of @@ -148,12 +149,17 @@ mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_ sin (2 * pi * 2^(-64) / 4) > 2^(-64) of exponent at least -63. So the error is bounded above by (1.5*2^66+0.5) ulp < 2^67 ulp. - (This could be made dependent on n, which would be useful for - small n at small precision.) */ + To obtain a more precise bound for smaller n, which is useful + especially at small precision, we may use the error bound of + (1.5*2^(3 - Exp (s or c)) + 0.5) ulp + < 2^(4 - Exp (s or c)) ulp, since Exp (s or c) is at most 0. */ + } - while ( !mpfr_can_round (c, prec - 67, MPFR_RNDN, MPFR_RNDZ, + while ( !mpfr_can_round (c, prec - (4 - mpfr_get_exp (c)), + MPFR_RNDN, MPFR_RNDZ, MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == MPFR_RNDN)) - || !mpfr_can_round (s, prec - 67, MPFR_RNDN, MPFR_RNDZ, + || !mpfr_can_round (s, prec - (4 - mpfr_get_exp (s)), + MPFR_RNDN, MPFR_RNDZ, MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == MPFR_RNDN))); inex_re = mpfr_set (mpc_realref(rop), c, MPC_RND_RE(rnd)); |