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diff --git a/Build/source/libs/mpfr/mpfr-src/src/exp_2.c b/Build/source/libs/mpfr/mpfr-src/src/exp_2.c
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-/* mpfr_exp_2 -- exponential of a floating-point number
- using algorithms in O(n^(1/2)*M(n)) and O(n^(1/3)*M(n))
-
-Copyright 1999-2020 Free Software Foundation, Inc.
-Contributed by the AriC and Caramba projects, INRIA.
-
-This file is part of the GNU MPFR Library.
-
-The GNU MPFR Library is free software; you can redistribute it and/or modify
-it under the terms of the GNU Lesser General Public License as published by
-the Free Software Foundation; either version 3 of the License, or (at your
-option) any later version.
-
-The GNU MPFR Library is distributed in the hope that it will be useful, but
-WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
-or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
-License for more details.
-
-You should have received a copy of the GNU Lesser General Public License
-along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
-https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
-51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
-
-#define MPFR_NEED_LONGLONG_H /* MPFR_INT_CEIL_LOG2 */
-#include "mpfr-impl.h"
-
-static unsigned long
-mpfr_exp2_aux (mpz_t, mpfr_srcptr, mpfr_prec_t, mpfr_exp_t *);
-static unsigned long
-mpfr_exp2_aux2 (mpz_t, mpfr_srcptr, mpfr_prec_t, mpfr_exp_t *);
-static mpfr_exp_t
-mpz_normalize (mpz_t, mpz_t, mpfr_exp_t);
-static mpfr_exp_t
-mpz_normalize2 (mpz_t, mpz_t, mpfr_exp_t, mpfr_exp_t);
-
-/* if k = the number of bits of z > q, divides z by 2^(k-q) and returns k-q.
- Otherwise do nothing and return 0.
- */
-static mpfr_exp_t
-mpz_normalize (mpz_t rop, mpz_t z, mpfr_exp_t q)
-{
- size_t k;
-
- MPFR_MPZ_SIZEINBASE2 (k, z);
- MPFR_ASSERTD (k == (mpfr_uexp_t) k);
- if (MPFR_LIKELY(q < 0 || (mpfr_uexp_t) k > (mpfr_uexp_t) q))
- {
- mpz_fdiv_q_2exp (rop, z, (unsigned long) ((mpfr_uexp_t) k - q));
- return (mpfr_exp_t) k - q;
- }
- mpz_set (rop, z);
- return 0;
-}
-
-/* if expz > target, shift z by (expz-target) bits to the left.
- if expz < target, shift z by (target-expz) bits to the right.
- Returns target.
-*/
-static mpfr_exp_t
-mpz_normalize2 (mpz_t rop, mpz_t z, mpfr_exp_t expz, mpfr_exp_t target)
-{
- if (MPFR_LIKELY(target > expz))
- mpz_fdiv_q_2exp (rop, z, target - expz);
- else
- mpz_mul_2exp (rop, z, expz - target);
- return target;
-}
-
-/* use Brent's formula exp(x) = (1+r+r^2/2!+r^3/3!+...)^(2^K)*2^n
- where x = n*log(2)+(2^K)*r
- together with the Paterson-Stockmeyer O(t^(1/2)) algorithm for the
- evaluation of power series. The resulting complexity is O(n^(1/3)*M(n)).
- This function returns with the exact flags due to exp.
-*/
-int
-mpfr_exp_2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
-{
- long n;
- unsigned long K, k, l, err; /* FIXME: Which type ? */
- int error_r;
- mpfr_exp_t exps, expx;
- mpfr_prec_t q, precy;
- int inexact;
- mpfr_t s, r;
- mpz_t ss;
- MPFR_GROUP_DECL(group);
- MPFR_ZIV_DECL (loop);
-
- MPFR_LOG_FUNC
- (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
- ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
- inexact));
-
- expx = MPFR_GET_EXP (x);
- precy = MPFR_PREC(y);
-
- /* First perform argument reduction: if x' = x - n*log(2) we have
- exp(x) = exp(x)*2^n. We should take n near from x/log(2) but it does not
- need to be exact.
- Warning: we cannot use the 'double' type here, since on 64-bit machines
- x may be as large as 2^62*log(2) without overflow, and then x/log(2)
- is about 2^62: not every integer of that size can be represented as a
- 'double', thus the argument reduction would fail. */
- if (MPFR_UNLIKELY(expx <= -2))
- /* |x| <= 0.25, thus n = round(x/log(2)) = 0 */
- n = 0;
- else
- {
- mp_limb_t r_limb[(sizeof (long) -1) / sizeof(mp_limb_t) + 1];
- /* Note: we use precision sizeof (long) * CHAR_BIT - 1 here since it is
- more efficient that full limb precision.
- The value of n will depend on whether MPFR_LONG_WITHIN_LIMB is
- defined or not. For instance, for r = 0.111E0, one gets n = 0
- in the former case and n = 1 in the latter case. */
- MPFR_TMP_INIT1(r_limb, r, sizeof (long) * CHAR_BIT - 1);
- mpfr_div (r, x, __gmpfr_const_log2_RNDD, MPFR_RNDN);
-#ifdef MPFR_LONG_WITHIN_LIMB
- /* The following code assume an unsigned long can fit in a mp_limb_t */
- {
- mp_limb_t a;
- mpfr_exp_t exp;
- /* Read the long directly (faster than using mpfr_get_si
- since it fits, it is not singular, it can't be zero
- and there is no conversion to do) */
- MPFR_ASSERTD (MPFR_NOTZERO (r));
- exp = MPFR_GET_EXP (r);
- MPFR_ASSERTD (exp <= GMP_NUMB_BITS);
- if (exp >= 1)
- {
- a = MPFR_MANT(r)[0] >> (GMP_NUMB_BITS - exp);
- n = MPFR_IS_POS (r) ? a : a <= LONG_MAX ? - (long) a : LONG_MIN;
- }
- else
- n = 0;
- }
-#else
- /* Use generic way to get the long */
- n = mpfr_get_si (r, MPFR_RNDN);
-#endif
- }
- /* we have |x| <= (|n|+1)*log(2) */
- MPFR_LOG_MSG (("d(x)=%1.30e n=%ld\n", mpfr_get_d1(x), n));
-
- /* error_r bounds the cancelled bits in x - n*log(2) */
- if (MPFR_UNLIKELY (n == 0))
- error_r = 0;
- else
- {
- error_r = mpfr_nbits_ulong (SAFE_ABS (unsigned long, n) + 1);
- /* we have |x| <= 2^error_r * log(2) */
- }
-
- /* for the O(n^(1/2)*M(n)) method, the Taylor series computation of
- n/K terms costs about n/(2K) multiplications when computed in fixed
- point */
- K = (precy < MPFR_EXP_2_THRESHOLD) ? __gmpfr_isqrt ((precy + 1) / 2) + 3
- : __gmpfr_cuberoot (4*precy);
- l = (precy - 1) / K + 1;
- err = K + MPFR_INT_CEIL_LOG2 (2 * l + 18);
- /* add K extra bits, i.e. failure probability <= 1/2^K = O(1/precy) */
- q = precy + err + K + 10;
- /* if |x| >> 1, take into account the cancelled bits */
- if (expx > 0)
- q += expx;
-
- /* Even with to the mpfr_prec_round below, it is possible to use
- the MPFR_GROUP_* macros here because mpfr_prec_round is only
- called in a special case. */
- MPFR_GROUP_INIT_2(group, q + error_r, r, s);
- mpz_init (ss);
-
- /* the algorithm consists in computing an upper bound of exp(x) using
- a precision of q bits, and see if we can round to MPFR_PREC(y) taking
- into account the maximal error. Otherwise we increase q. */
- MPFR_ZIV_INIT (loop, q);
- for (;;)
- {
- MPFR_LOG_MSG (("n=%ld K=%lu l=%lu q=%lu error_r=%d\n",
- n, K, l, (unsigned long) q, error_r));
-
- /* First reduce the argument to r = x - n * log(2),
- so that r is small in absolute value. We want an upper
- bound on r to get an upper bound on exp(x). */
-
- /* if n<0, we have to get an upper bound of log(2)
- in order to get an upper bound of r = x-n*log(2) */
- mpfr_const_log2 (s, (n >= 0) ? MPFR_RNDZ : MPFR_RNDU);
- /* s is within 1 ulp(s) of log(2) */
-
- mpfr_mul_ui (r, s, (n < 0) ? -n : n, (n >= 0) ? MPFR_RNDZ : MPFR_RNDU);
- /* r is within 3 ulps of |n|*log(2) */
- if (n < 0)
- MPFR_CHANGE_SIGN (r);
- /* r <= n*log(2), within 3 ulps */
-
- MPFR_LOG_VAR (x);
- MPFR_LOG_VAR (r);
-
- mpfr_sub (r, x, r, MPFR_RNDU);
-
- while (MPFR_IS_PURE_FP(r) && MPFR_IS_NEG (r))
- { /* initial approximation n was too large */
- n--;
- mpfr_add (r, r, s, MPFR_RNDU);
- }
-
- /* if r is 0, we cannot round correctly */
- if (MPFR_LIKELY(MPFR_IS_PURE_FP (r)))
- {
- /* since there was a cancellation in x - n*log(2), the low error_r
- bits from r are zero and thus non significant, thus we can reduce
- the working precision */
- if (MPFR_LIKELY(error_r > 0))
- {
- MPFR_ASSERTD( MPFR_PREC(r) > q);
- /* since MPFR_PREC(r) > q, there is no reallocation to do,
- and it is safe to use it with MPFR_GROUP functions */
- mpfr_prec_round (r, q, MPFR_RNDU);
- }
- /* the error on r is at most 3 ulps (3 ulps if error_r = 0,
- and 1 + 3/2 if error_r > 0) */
- MPFR_LOG_VAR (r);
- MPFR_ASSERTD (MPFR_IS_POS (r));
- mpfr_div_2ui (r, r, K, MPFR_RNDU); /* r = (x-n*log(2))/2^K, exact */
-
- /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! */
- MPFR_ASSERTD (MPFR_IS_PURE_FP (r) && MPFR_EXP (r) < 0);
- l = (precy < MPFR_EXP_2_THRESHOLD)
- ? mpfr_exp2_aux (ss, r, q, &exps) /* naive method */
- : mpfr_exp2_aux2 (ss, r, q, &exps); /* Paterson/Stockmeyer meth */
-
- MPFR_LOG_MSG (("l=%lu q=%lu (K+l)*q^2=%1.3e\n",
- l, (unsigned long) q, (K + l) * (double) q * q));
-
- for (k = 0; k < K; k++)
- {
- mpz_mul (ss, ss, ss);
- exps *= 2;
- exps += mpz_normalize (ss, ss, q);
- }
- mpfr_set_z_2exp (s, ss, exps, MPFR_RNDN);
-
- /* error is at most 2^K*l, plus 2 to take into account of
- the error of 3 ulps on r */
- err = K + MPFR_INT_CEIL_LOG2 (l) + 2;
-
- MPFR_LOG_MSG (("before mult. by 2^n:\n", 0));
- MPFR_LOG_VAR (s);
- MPFR_LOG_MSG (("err=%lu bits\n", K));
-
- if (MPFR_LIKELY (MPFR_CAN_ROUND (s, q - err, precy, rnd_mode)))
- {
- MPFR_CLEAR_FLAGS ();
- inexact = mpfr_mul_2si (y, s, n, rnd_mode);
- break;
- }
- }
- MPFR_ZIV_NEXT (loop, q);
- MPFR_GROUP_REPREC_2(group, q+error_r, r, s);
- }
- MPFR_ZIV_FREE (loop);
- mpz_clear (ss);
- MPFR_GROUP_CLEAR (group);
-
- return inexact;
-}
-
-/* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
- using naive method with O(l) multiplications.
- Return the number of iterations l.
- The absolute error on s is less than 3*l*(l+1)*2^(-q).
- Version using fixed-point arithmetic with mpz instead
- of mpfr for internal computations.
-*/
-static unsigned long
-mpfr_exp2_aux (mpz_t s, mpfr_srcptr r, mpfr_prec_t q, mpfr_exp_t *exps)
-{
- unsigned long l;
- mpfr_exp_t dif, expt, expr;
- mpz_t t, rr;
- mp_size_t sbit, tbit;
-
- MPFR_ASSERTD (MPFR_IS_PURE_FP (r));
-
- expt = 0;
- *exps = 1 - (mpfr_exp_t) q; /* s = 2^(q-1) */
- mpz_init (t);
- mpz_init (rr);
- mpz_set_ui (t, 1);
- mpz_set_ui (s, 1);
- mpz_mul_2exp (s, s, q-1);
- expr = mpfr_get_z_2exp (rr, r); /* no error here */
-
- l = 0;
- for (;;)
- {
- l++;
- mpz_mul(t, t, rr);
- expt += expr;
- MPFR_MPZ_SIZEINBASE2 (sbit, s);
- MPFR_MPZ_SIZEINBASE2 (tbit, t);
- dif = *exps + sbit - expt - tbit;
- /* truncates the bits of t which are < ulp(s) = 2^(1-q) */
- expt += mpz_normalize (t, t, (mpfr_exp_t) q - dif);
- /* error at most 2^(1-q) */
- if (l > 1)
- {
- /* GMP doesn't optimize the case of power of 2 */
- if (IS_POW2(l))
- {
- int bits = MPFR_INT_CEIL_LOG2(l);
- mpz_fdiv_q_2exp (t, t, bits); /* error at most 2^(1-q) */
- }
- else
- {
- mpz_fdiv_q_ui (t, t, l); /* error at most 2^(1-q) */
- }
- /* the error wrt t^l/l! is here at most 3*l*ulp(s) */
- MPFR_ASSERTD (expt == *exps);
- }
- if (mpz_sgn (t) == 0)
- break;
- mpz_add (s, s, t); /* no error here: exact */
- /* ensures rr has the same size as t: after several shifts, the error
- on rr is still at most ulp(t)=ulp(s) */
- MPFR_MPZ_SIZEINBASE2 (tbit, t);
- expr += mpz_normalize (rr, rr, tbit);
- }
-
- mpz_clear (t);
- mpz_clear (rr);
-
- return 3 * l * (l + 1);
-}
-
-/* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
- using Paterson-Stockmeyer algorithm with O(sqrt(l)) multiplications.
- Return l.
- Uses m multiplications of full size and 2l/m of decreasing size,
- i.e. a total equivalent to about m+l/m full multiplications,
- i.e. 2*sqrt(l) for m=sqrt(l).
- NOTE[VL]: The following sentence seems to be obsolete since MY_INIT_MPZ
- is no longer used (r6919); sizer was the number of limbs of r.
- Version using mpz. ss must have at least (sizer+1) limbs.
- The error is bounded by (l^2+4*l) ulps where l is the return value.
-*/
-static unsigned long
-mpfr_exp2_aux2 (mpz_t s, mpfr_srcptr r, mpfr_prec_t q, mpfr_exp_t *exps)
-{
- mpfr_exp_t expr, *expR, expt;
- mpfr_prec_t ql;
- unsigned long l, m, i;
- mpz_t t, *R, rr, tmp;
- mp_size_t sbit, rrbit;
- MPFR_TMP_DECL(marker);
-
- /* estimate value of l */
- MPFR_ASSERTD (MPFR_GET_EXP (r) < 0);
- l = q / (- MPFR_GET_EXP (r));
- m = __gmpfr_isqrt (l);
- /* we access R[2], thus we need m >= 2 */
- if (m < 2)
- m = 2;
-
- MPFR_TMP_MARK(marker);
- R = (mpz_t*) MPFR_TMP_ALLOC ((m + 1) * sizeof (mpz_t)); /* R[i] is r^i */
- expR = (mpfr_exp_t*) MPFR_TMP_ALLOC((m + 1) * sizeof (mpfr_exp_t));
- /* expR[i] is the exponent for R[i] */
- mpz_init (tmp);
- mpz_init (rr);
- mpz_init (t);
- mpz_set_ui (s, 0);
- *exps = 1 - q; /* 1 ulp = 2^(1-q) */
- for (i = 0 ; i <= m ; i++)
- mpz_init (R[i]);
- expR[1] = mpfr_get_z_2exp (R[1], r); /* exact operation: no error */
- expR[1] = mpz_normalize2 (R[1], R[1], expR[1], 1 - q); /* error <= 1 ulp */
- mpz_mul (t, R[1], R[1]); /* err(t) <= 2 ulps */
- mpz_fdiv_q_2exp (R[2], t, q - 1); /* err(R[2]) <= 3 ulps */
- expR[2] = 1 - q;
- for (i = 3 ; i <= m ; i++)
- {
- if ((i & 1) == 1)
- mpz_mul (t, R[i-1], R[1]); /* err(t) <= 2*i-2 */
- else
- mpz_mul (t, R[i/2], R[i/2]);
- mpz_fdiv_q_2exp (R[i], t, q - 1); /* err(R[i]) <= 2*i-1 ulps */
- expR[i] = 1 - q;
- }
- mpz_set_ui (R[0], 1);
- mpz_mul_2exp (R[0], R[0], q-1);
- expR[0] = 1-q; /* R[0]=1 */
- mpz_set_ui (rr, 1);
- expr = 0; /* rr contains r^l/l! */
- /* by induction: err(rr) <= 2*l ulps */
-
- l = 0;
- ql = q; /* precision used for current giant step */
- do
- {
- /* all R[i] must have exponent 1-ql */
- if (l != 0)
- for (i = 0 ; i < m ; i++)
- expR[i] = mpz_normalize2 (R[i], R[i], expR[i], 1 - ql);
- /* the absolute error on R[i]*rr is still 2*i-1 ulps */
- expt = mpz_normalize2 (t, R[m-1], expR[m-1], 1 - ql);
- /* err(t) <= 2*m-1 ulps */
- /* computes t = 1 + r/(l+1) + ... + r^(m-1)*l!/(l+m-1)!
- using Horner's scheme */
- for (i = m-1 ; i-- != 0 ; )
- {
- mpz_fdiv_q_ui (t, t, l+i+1); /* err(t) += 1 ulp */
- mpz_add (t, t, R[i]);
- }
- /* now err(t) <= (3m-2) ulps */
-
- /* now multiplies t by r^l/l! and adds to s */
- mpz_mul (t, t, rr);
- expt += expr;
- expt = mpz_normalize2 (t, t, expt, *exps);
- /* err(t) <= (3m-1) + err_rr(l) <= (3m-2) + 2*l */
- MPFR_ASSERTD (expt == *exps);
- mpz_add (s, s, t); /* no error here */
-
- /* updates rr, the multiplication of the factors l+i could be done
- using binary splitting too, but it is not sure it would save much */
- mpz_mul (t, rr, R[m]); /* err(t) <= err(rr) + 2m-1 */
- expr += expR[m];
- mpz_set_ui (tmp, 1);
- for (i = 1 ; i <= m ; i++)
- mpz_mul_ui (tmp, tmp, l + i);
- mpz_fdiv_q (t, t, tmp); /* err(t) <= err(rr) + 2m */
- l += m;
- if (MPFR_UNLIKELY (mpz_sgn (t) == 0))
- break;
- expr += mpz_normalize (rr, t, ql); /* err_rr(l+1) <= err_rr(l) + 2m+1 */
- if (MPFR_UNLIKELY (mpz_sgn (rr) == 0))
- rrbit = 1;
- else
- MPFR_MPZ_SIZEINBASE2 (rrbit, rr);
- MPFR_MPZ_SIZEINBASE2 (sbit, s);
- ql = q - *exps - sbit + expr + rrbit;
- /* TODO: Wrong cast. I don't want what is right, but this is
- certainly wrong */
- }
- while ((size_t) expr + rrbit > (size_t) -q);
-
- for (i = 0 ; i <= m ; i++)
- mpz_clear (R[i]);
- MPFR_TMP_FREE(marker);
- mpz_clear (rr);
- mpz_clear (t);
- mpz_clear (tmp);
-
- return l * (l + 4);
-}