/* agm(x,y) is between x and y, so we don't need to save exponent range */ int mpfr_agm (mpfr_ptr r, mpfr_srcptr op2, mpfr_srcptr op1, mp_rnd_t rnd_mode) { int compare, inexact; mp_size_t s; mp_prec_t p, q; mp_limb_t *up, *vp, *tmpp; mpfr_t u, v, tmp; unsigned long n; /* number of iterations */ unsigned long err = 0; MPFR_ZIV_DECL (loop); MPFR_TMP_DECL(marker); MPFR_LOG_FUNC (("op2[%#R]=%R op1[%#R]=%R rnd=%d", op2,op2,op1,op1,rnd_mode), ("r[%#R]=%R inexact=%d", r, r, inexact)); /* Deal with special values */ if (MPFR_ARE_SINGULAR (op1, op2)) { /* If a or b is NaN, the result is NaN */ if (MPFR_IS_NAN(op1) || MPFR_IS_NAN(op2)) { MPFR_SET_NAN(r); MPFR_RET_NAN; } /* now one of a or b is Inf or 0 */ /* If a and b is +Inf, the result is +Inf. Otherwise if a or b is -Inf or 0, the result is NaN */ else if (MPFR_IS_INF(op1) || MPFR_IS_INF(op2)) { if (MPFR_IS_STRICTPOS(op1) && MPFR_IS_STRICTPOS(op2)) { MPFR_SET_INF(r); MPFR_SET_SAME_SIGN(r, op1); MPFR_RET(0); /* exact */ } else { MPFR_SET_NAN(r); MPFR_RET_NAN; } } else /* a and b are neither NaN nor Inf, and one is zero */ { /* If a or b is 0, the result is +0 since a sqrt is positive */ MPFR_ASSERTD (MPFR_IS_ZERO (op1) || MPFR_IS_ZERO (op2)); MPFR_SET_POS (r); MPFR_SET_ZERO (r); MPFR_RET (0); /* exact */ } } MPFR_CLEAR_FLAGS (r); /* If a or b is negative (excluding -Infinity), the result is NaN */ if (MPFR_UNLIKELY(MPFR_IS_NEG(op1) || MPFR_IS_NEG(op2))) { MPFR_SET_NAN(r); MPFR_RET_NAN; } /* Precision of the following calculus */ q = MPFR_PREC(r); p = q + MPFR_INT_CEIL_LOG2(q) + 15; MPFR_ASSERTD (p >= 7); /* see algorithms.tex */ s = (p - 1) / BITS_PER_MP_LIMB + 1; /* b (op2) and a (op1) are the 2 operands but we want b >= a */ compare = mpfr_cmp (op1, op2); if (MPFR_UNLIKELY( compare == 0 )) { mpfr_set (r, op1, rnd_mode); MPFR_RET (0); /* exact */ } else if (compare > 0) { mpfr_srcptr t = op1; op1 = op2; op2 = t; } /* Now b(=op2) >= a (=op1) */ MPFR_TMP_MARK(marker); /* Main loop */ MPFR_ZIV_INIT (loop, p); for (;;) { mp_prec_t eq; /* Init temporary vars */ MPFR_TMP_INIT (up, u, p, s); MPFR_TMP_INIT (vp, v, p, s); MPFR_TMP_INIT (tmpp, tmp, p, s); /* Calculus of un and vn */ mpfr_mul (u, op1, op2, GMP_RNDN); /* Faster since PREC(op) < PREC(u) */ mpfr_sqrt (u, u, GMP_RNDN); mpfr_add (v, op1, op2, GMP_RNDN); /* add with !=prec is still good*/ mpfr_div_2ui (v, v, 1, GMP_RNDN); n = 1; while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2) { mpfr_add (tmp, u, v, GMP_RNDN); mpfr_div_2ui (tmp, tmp, 1, GMP_RNDN); /* See proof in algorithms.tex */ if (4*eq > p) { mpfr_t w; /* tmp = U(k) */ mpfr_init2 (w, (p + 1) / 2); mpfr_sub (w, v, u, GMP_RNDN); /* e = V(k-1)-U(k-1) */ mpfr_sqr (w, w, GMP_RNDN); /* e = e^2 */ mpfr_div_2ui (w, w, 4, GMP_RNDN); /* e*= (1/2)^2*1/4 */ mpfr_div (w, w, tmp, GMP_RNDN); /* 1/4*e^2/U(k) */ mpfr_sub (v, tmp, w, GMP_RNDN); err = MPFR_GET_EXP (tmp) - MPFR_GET_EXP (v); /* 0 or 1 */ mpfr_clear (w); break; } mpfr_mul (u, u, v, GMP_RNDN); mpfr_sqrt (u, u, GMP_RNDN); mpfr_swap (v, tmp); n ++; } /* the error on v is bounded by (18n+51) ulps, or twice if there was an exponent loss in the final subtraction */ err += MPFR_INT_CEIL_LOG2(18 * n + 51); /* 18n+51 should not overflow since n is about log(p) */ /* we should have n+2 <= 2^(p/4) [see algorithms.tex] */ if (MPFR_LIKELY (MPFR_INT_CEIL_LOG2(n + 2) <= p / 4 && MPFR_CAN_ROUND (v, p - err, q, rnd_mode))) break; /* Stop the loop */ /* Next iteration */ MPFR_ZIV_NEXT (loop, p); s = (p - 1) / BITS_PER_MP_LIMB + 1; } MPFR_ZIV_FREE (loop); /* Setting of the result */ inexact = mpfr_set (r, v, rnd_mode); /* Let's clean */ MPFR_TMP_FREE(marker); return inexact; /* agm(u,v) can be exact for u, v rational only for u=v. Proof (due to Nicolas Brisebarre): it suffices to consider u=1 and v<1. Then 1/AGM(1,v) = 2F1(1/2,1/2,1;1-v^2), and a theorem due to G.V. Chudnovsky states that for x a non-zero algebraic number with |x|<1, then 2F1(1/2,1/2,1;x) and 2F1(-1/2,1/2,1;x) are algebraically independent over Q. */ }
int mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode) { int K0, K, precy, m, k, l; int inexact; mpfr_t r, s; mp_limb_t *rp, *sp; mp_size_t sm; mp_exp_t exps, cancel = 0; TMP_DECL (marker); if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x))) { if (MPFR_IS_NAN(x) || MPFR_IS_INF(x)) { MPFR_SET_NAN(y); MPFR_RET_NAN; } else { MPFR_ASSERTD(MPFR_IS_ZERO(x)); return mpfr_set_ui (y, 1, GMP_RNDN); } } mpfr_save_emin_emax (); precy = MPFR_PREC(y); K0 = __gmpfr_isqrt(precy / 2); /* Need K + log2(precy/K) extra bits */ m = precy + 3 * (K0 + 2 * MAX(MPFR_GET_EXP (x), 0)) + 3; TMP_MARK(marker); sm = (m + BITS_PER_MP_LIMB - 1) / BITS_PER_MP_LIMB; MPFR_TMP_INIT(rp, r, m, sm); MPFR_TMP_INIT(sp, s, m, sm); for (;;) { mpfr_mul (r, x, x, GMP_RNDU); /* err <= 1 ulp */ /* we need that |r| < 1 for mpfr_cos2_aux, i.e. up(x^2)/2^(2K) < 1 */ K = K0 + MAX (MPFR_GET_EXP (r), 0); mpfr_div_2ui (r, r, 2 * K, GMP_RNDN); /* r = (x/2^K)^2, err <= 1 ulp */ /* s <- 1 - r/2! + ... + (-1)^l r^l/(2l)! */ l = mpfr_cos2_aux (s, r); MPFR_SET_ONE (r); for (k = 0; k < K; k++) { mpfr_mul (s, s, s, GMP_RNDU); /* err <= 2*olderr */ mpfr_mul_2ui (s, s, 1, GMP_RNDU); /* err <= 4*olderr */ mpfr_sub (s, s, r, GMP_RNDN); } /* absolute error on s is bounded by (2l+1/3)*2^(2K-m) */ for (k = 2 * K, l = 2 * l + 1; l > 1; l = (l + 1) >> 1) k++; /* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */ exps = MPFR_GET_EXP(s); if (MPFR_LIKELY(mpfr_can_round (s, exps + m - k, GMP_RNDN, GMP_RNDZ, precy + (rnd_mode == GMP_RNDN)))) break; m += BITS_PER_MP_LIMB; if (exps < cancel) { m += cancel - exps; cancel = exps; } sm = (m + BITS_PER_MP_LIMB - 1) / BITS_PER_MP_LIMB; MPFR_TMP_INIT(rp, r, m, sm); MPFR_TMP_INIT(sp, s, m, sm); } mpfr_restore_emin_emax (); inexact = mpfr_set (y, s, rnd_mode); /* FIXME: Dont' need check range? */ TMP_FREE(marker); return inexact; }