new exp bit-burst implementation

elefun.h

@@ -68,6 +68,7 @@ void elefun_exp_sum_bs_powtab(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp,
 void elefun_exp_sum_bs_simple(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp,
     const fmpz_t x, mp_bitcnt_t r, long N);
 
+void elefun_exp_fmpr_bb(fmprb_t z, const fmpr_t x, long prec, int m1);
 
 
 void _elefun_cos_minpoly_roots(fmprb_ptr alpha, long d, ulong n, long prec);

elefun/exp_fmpr_bb.c

@@ -0,0 +1,181 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB 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 General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2014 Fredrik Johansson
+
+******************************************************************************/
+
+#include "elefun.h"
+
+/*
+Determine N such that the error is bounded by 2^-prec when summing the
+Taylor series of exp(x) up to term x^N inclusive. We choose an N with
+many trailing zeros to improve efficiency of the binary splitting.
+*/
+static long
+bs_num_terms(long mag, long prec)
+{
+    long N;
+
+    N = elefun_exp_taylor_bound(mag, prec);
+    /* Convert from N exclusive to N inclusive. */
+    N--;
+
+    if (N > 10000)
+        while (N % 128 != 0)
+            N++;
+
+    if (N > 1000)
+        while (N % 16 != 0)
+            N++;
+
+    if (N > 100)
+        while (N % 2 != 0)
+            N++;
+
+    return N;
+}
+
+void
+elefun_exp_fmpr_bb(fmprb_t z, const fmpr_t x, long prec, int m1)
+{
+    long k, iter, bits, r, mag, q, wp, N;
+    long argred_bits, start_bits;
+    mp_bitcnt_t Qexp[1];
+    int inexact;
+    fmpz_t t, u, T, Q;
+
+    if (fmpr_is_zero(x))
+    {
+        fmprb_one(z);
+        return;
+    }
+
+    mag = fmpr_abs_bound_lt_2exp_si(x);
+
+    /* We assume that this function only gets called with something
+       reasonable as input (huge/tiny input will be handled by
+       the main exp wrapper). */
+    if (mag > 200 || mag < -2 * prec - 100)
+    {
+        printf("elefun_exp_fmpr_bb: unexpectedly large/small input\n");
+        abort();
+    }
+
+    if (prec < 100000000)
+    {
+        argred_bits = 16;
+        start_bits = 32;
+    }
+    else
+    {
+        argred_bits = 32;
+        start_bits = 64;
+    }
+
+    /* Argument reduction: exp(x) -> exp(x/2^q). This improves efficiency
+       of the first iteration in the bit-burst algorithm. */
+    q = FLINT_MAX(0, mag + argred_bits);
+
+    /* Determine working precision. */
+    wp = prec + 10 + 2 * q + 2 * FLINT_BIT_COUNT(prec);
+    if (m1 && mag < 0)
+        wp += (-mag);
+
+    fmpz_init(t);
+    fmpz_init(u);
+    fmpz_init(Q);
+    fmpz_init(T);
+
+    /* Convert x/2^q to a fixed-point number. */
+    inexact = fmpr_get_fmpz_fixed_si(t, x, -wp + q);
+
+    /* Aliasing of z and x is safe now that only use t. */
+    /* Start with z = 1. */
+    fmprb_one(z);
+
+    /* Bit-burst loop. */
+    for (iter = 0, bits = start_bits; !fmpz_is_zero(t);
+        iter++, bits *= 2)
+    {
+        /* Extract bits. */
+        r = FLINT_MIN(bits, wp);
+        fmpz_tdiv_q_2exp(u, t, wp - r);
+
+        /* Binary splitting (+1 fixed-point ulp truncation error). */
+        mag = fmpz_bits(u) - r;
+        N = bs_num_terms(mag, wp);
+        elefun_exp_sum_bs_powtab(T, Q, Qexp, u, r, N);
+
+        /* T = T / Q  (+1 fixed-point ulp error). */
+        if (*Qexp >= wp)
+        {
+            fmpz_tdiv_q_2exp(T, T, *Qexp - wp);
+            fmpz_tdiv_q(T, T, Q);
+        }
+        else
+        {
+            fmpz_mul_2exp(T, T, wp - *Qexp);
+            fmpz_tdiv_q(T, T, Q);
+        }
+
+        /* T = 1 + T */
+        fmpz_one(Q);
+        fmpz_mul_2exp(Q, Q, wp);
+        fmpz_add(T, T, Q);
+
+        /* Now T = exp(u) with at most 2 fixed-point ulp error. */
+        /* Set z = z * T. */
+        {
+            fmprb_t w;
+            fmprb_init(w);
+            fmpr_set_fmpz(fmprb_midref(w), T);
+            fmpr_mul_2exp_si(fmprb_midref(w), fmprb_midref(w), -wp);
+            fmpr_set_si_2exp_si(fmprb_radref(w), 2, -wp);
+            fmprb_mul(z, z, w, wp);
+            fmprb_clear(w);
+        }
+
+        /* Remove used bits. */
+        fmpz_mul_2exp(u, u, wp - r);
+        fmpz_sub(t, t, u);
+    }
+
+    /* We have exp(x + eps) - exp(x) < 2*eps (by assumption that the argument
+       reduction is large enough). */
+    if (inexact)
+        fmprb_add_error_2exp_si(z, -wp + 1);
+
+    fmpz_clear(t);
+    fmpz_clear(u);
+    fmpz_clear(Q);
+    fmpz_clear(T);
+
+    /* exp(x) = exp(x/2^q)^(2^q) */
+    for (k = 0; k < q; k++)
+        fmprb_mul(z, z, z, wp);
+
+    if (m1)
+        fmprb_sub_ui(z, z, 1, wp);
+
+    fmprb_set_round(z, z, prec);
+}
+

elefun/test/t-exp_fmpr_bb.c

@@ -0,0 +1,96 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB 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 General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2014 Fredrik Johansson
+
+******************************************************************************/
+
+#include "elefun.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("exp_fmpr_bb....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 10000; iter++)
+    {
+        fmprb_t x, y, z;
+        long prec = 2 + n_randint(state, 5000);
+
+        fmprb_init(x);
+        fmprb_init(y);
+        fmprb_init(z);
+
+        fmprb_randtest(x, state, 1 + n_randint(state, 5000), 3);
+        fmpr_zero(fmprb_radref(x));
+
+        if (n_randint(state, 2))
+            fmprb_mul_2exp_si(x, x, 1 + n_randint(state, 40));
+        else
+            fmprb_mul_2exp_si(x, x, -n_randint(state, 1.5 * prec));
+
+        elefun_exp_via_mpfr(y, x, prec + 100);
+        elefun_exp_fmpr_bb(z, fmprb_midref(x), prec, 0);
+
+        if (!fmprb_contains(z, y))
+        {
+            printf("FAIL: containment\n\n");
+            printf("prec = %ld\n\n", prec);
+            printf("x = "); fmprb_print(x); printf("\n\n");
+            printf("y = "); fmprb_print(y); printf("\n\n");
+            printf("z = "); fmprb_print(z); printf("\n\n");
+            abort();
+        }
+
+        if (fmprb_rel_accuracy_bits(z) < prec - 2)
+        {
+            printf("FAIL: poor accuracy\n\n");
+            printf("prec = %ld,  acc = %ld\n\n", prec, fmprb_rel_accuracy_bits(z));
+            printf("x = "); fmprb_print(x); printf("\n\n");
+            printf("y = "); fmprb_print(y); printf("\n\n");
+            printf("z = "); fmprb_print(z); printf("\n\n");
+            abort();
+        }
+
+        elefun_exp_fmpr_bb(x, fmprb_midref(x), prec, 0);
+
+        if (!fmprb_overlaps(x, z))
+        {
+            printf("FAIL: aliasing\n\n");
+            abort();
+        }
+
+        fmprb_clear(x);
+        fmprb_clear(y);
+        fmprb_clear(z);
+    }
+
+    flint_randclear(state);
+    flint_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+

fmprb/exp.c

@@ -170,11 +170,18 @@ fmprb_exp_fmpr(fmprb_t z, const fmpr_t x, long prec, long maglim, int m1)
     /* standard case */
     if (range_check_mpfr(mag) && *mag > small_cutoff)
     {
-        if (m1)
-            r = fmpr_expm1(fmprb_midref(z), x, prec, FMPR_RND_DOWN);
+        if (prec > 4000)
+        {
+            elefun_exp_fmpr_bb(z, x, prec, m1);
+        }
         else
-            r = fmpr_exp(fmprb_midref(z), x, prec, FMPR_RND_DOWN);
-        fmpr_set_error_result(fmprb_radref(z), fmprb_midref(z), r);
+        {
+            if (m1)
+                r = fmpr_expm1(fmprb_midref(z), x, prec, FMPR_RND_DOWN);
+            else
+                r = fmpr_exp(fmprb_midref(z), x, prec, FMPR_RND_DOWN);
+            fmpr_set_error_result(fmprb_radref(z), fmprb_midref(z), r);
+        }
     }
     /* close to zero */
     else if (fmpz_sgn(mag) < 0)