| /* 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 1999-2017 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 |
| http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., |
| 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ |
| |
| /* #define DEBUG */ |
| #define MPFR_NEED_LONGLONG_H /* for count_leading_zeros */ |
| #include "third_party/mpfr/v3_1_6/src/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 (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; |
| } |
| if (MPFR_UNLIKELY(rop != z)) |
| 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 (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 r, s; |
| mpz_t ss; |
| 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); |
| |
| /* 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 (expx <= -2) |
| /* |x| <= 0.25, thus n = round(x/log(2)) = 0 */ |
| n = 0; |
| else |
| { |
| mpfr_init2 (r, sizeof (long) * CHAR_BIT); |
| mpfr_const_log2 (r, MPFR_RNDZ); |
| mpfr_div (r, x, r, MPFR_RNDN); |
| n = mpfr_get_si (r, MPFR_RNDN); |
| mpfr_clear (r); |
| } |
| /* 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 |
| { |
| count_leading_zeros (error_r, (mp_limb_t) SAFE_ABS (unsigned long, n) + 1); |
| error_r = GMP_NUMB_BITS - error_r; |
| /* 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) |
| : __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 + 8; |
| /* if |x| >> 1, take into account the cancelled bits */ |
| if (expx > 0) |
| q += expx; |
| |
| /* Note: due to the mpfr_prec_round below, it is not possible to use |
| the MPFR_GROUP_* macros here. */ |
| |
| mpfr_init2 (r, q + error_r); |
| mpfr_init2 (s, q + error_r); |
| |
| /* 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); |
| |
| if (MPFR_IS_PURE_FP (r)) |
| { |
| while (MPFR_IS_NEG (r)) |
| { /* initial approximation n was too large */ |
| n--; |
| mpfr_add (r, r, s, MPFR_RNDU); |
| } |
| |
| /* 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 (error_r > 0) |
| 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 */ |
| |
| mpz_init (ss); |
| exps = mpfr_get_z_2exp (ss, s); |
| /* 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 (s, ss, MPFR_RNDN); |
| |
| MPFR_SET_EXP(s, MPFR_GET_EXP (s) + exps); |
| mpz_clear (ss); |
| |
| /* 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_set_prec (r, q + error_r); |
| mpfr_set_prec (s, q + error_r); |
| } |
| MPFR_ZIV_FREE (loop); |
| |
| mpfr_clear (r); |
| mpfr_clear (s); |
| |
| 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. |
| NOTE[VL]: the following sentence seems to be obsolete since MY_INIT_MPZ |
| is no longer used (r6919); qn was the number of limbs of q. |
| s must have at least qn+1 limbs (qn should be enough, but currently fails |
| since mpz_mul_2exp(s, s, q-1) reallocates qn+1 limbs) |
| */ |
| 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_ASSERTN (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) */ |
| 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); |
| } |
