/* * arcsinh(x) == log [ x + sqrt(x^2 + 1) ] * * also, use arcsinh(-x) == -arcsinh(x) */ void m_apm_arcsinh(M_APM rr, int places, M_APM aa) { M_APM tmp0, tmp1, tmp2; /* result is 0 if input is 0 */ if (aa->m_apm_sign == 0) { M_set_to_zero(rr); return; } tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); m_apm_absolute_value(tmp0, aa); m_apm_multiply(tmp1, tmp0, tmp0); m_apm_add(tmp2, tmp1, MM_One); m_apm_sqrt(tmp1, (places + 6), tmp2); m_apm_add(tmp2, tmp0, tmp1); m_apm_log(rr, places, tmp2); rr->m_apm_sign = aa->m_apm_sign; /* fix final sign */ M_restore_stack(3); }
/* calculate log (1 + x) with the following series: x y = ----- ( |y| < 1 ) x + 2 [ 1 + y ] y^3 y^5 y^7 log [-------] = 2 * [ y + --- + --- + --- ... ] [ 1 - y ] 3 5 7 */ void M_log_near_1(M_APM rr, int places, M_APM xx) { M_APM tmp0, tmp1, tmp2, tmpS, term; int tolerance, dplaces, local_precision; long m1; tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmpS = M_get_stack_var(); term = M_get_stack_var(); tolerance = xx->m_apm_exponent - (places + 6); dplaces = (places + 12) - xx->m_apm_exponent; m_apm_add(tmp0, xx, MM_Two); m_apm_divide(tmpS, (dplaces + 6), xx, tmp0); m_apm_copy(term, tmpS); m_apm_multiply(tmp0, tmpS, tmpS); m_apm_round(tmp2, (dplaces + 6), tmp0); m1 = 3L; while (TRUE) { m_apm_multiply(tmp0, term, tmp2); if ((tmp0->m_apm_exponent < tolerance) || (tmp0->m_apm_sign == 0)) break; local_precision = dplaces + tmp0->m_apm_exponent; if (local_precision < 20) local_precision = 20; m_apm_set_long(tmp1, m1); m_apm_round(term, local_precision, tmp0); m_apm_divide(tmp0, local_precision, term, tmp1); m_apm_add(tmp1, tmpS, tmp0); m_apm_copy(tmpS, tmp1); m1 += 2; } m_apm_multiply(tmp0, MM_Two, tmpS); m_apm_round(rr, places, tmp0); M_restore_stack(5); /* restore the 5 locals we used here */ }
/* * find log(N) * * if places < 360 * solve with cubically convergent algorithm above * * else * * let 'X' be 'close' to the solution (we use ~110 decimal places) * * let Y = N * exp(-X) - 1 * * then * * log(N) = X + log(1 + Y) * * since 'Y' will be small, we can use the efficient log_near_1 algorithm. * */ void M_log_basic_iteration(M_APM rr, int places, M_APM nn) { M_APM tmp0, tmp1, tmp2, tmpX; if (places < 360) { M_log_solve_cubic(rr, places, nn); } else { tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmpX = M_get_stack_var(); M_log_solve_cubic(tmpX, 110, nn); m_apm_negate(tmp0, tmpX); m_apm_exp(tmp1, (places + 8), tmp0); m_apm_multiply(tmp2, tmp1, nn); m_apm_subtract(tmp1, tmp2, MM_One); M_log_near_1(tmp0, (places - 104), tmp1); m_apm_add(tmp1, tmpX, tmp0); m_apm_round(rr, places, tmp1); M_restore_stack(4); } }
/* Calculate arctan using the identity : x arctan (x) == arcsin [ --------------- ] sqrt(1 + x^2) */ void m_apm_arctan(M_APM rr, int places, M_APM xx) { M_APM tmp8, tmp9; if (xx->m_apm_sign == 0) /* input == 0 ?? */ { M_set_to_zero(rr); return; } if (xx->m_apm_exponent <= -4) /* input close to 0 ?? */ { M_arctan_near_0(rr, places, xx); return; } if (xx->m_apm_exponent >= 4) /* large input */ { M_arctan_large_input(rr, places, xx); return; } tmp8 = M_get_stack_var(); tmp9 = M_get_stack_var(); m_apm_multiply(tmp9, xx, xx); m_apm_add(tmp8, tmp9, MM_One); m_apm_sqrt(tmp9, (places + 6), tmp8); m_apm_divide(tmp8, (places + 6), xx, tmp9); m_apm_arcsin(rr, places, tmp8); M_restore_stack(2); }
/* * compute X = (a * X + c) MOD m where c = a */ void m_apm_get_random(M_APM mrnd) { if (M_firsttime2) /* use the system time as the initial seed value */ { M_firsttime2 = FALSE; M_rnd_aa = m_apm_init(); M_rnd_XX = m_apm_init(); M_rnd_mm = m_apm_init(); M_rtmp0 = m_apm_init(); M_rtmp1 = m_apm_init(); /* set the multiplier M_rnd_aa and M_rnd_mm */ m_apm_set_string(M_rnd_aa, "716805947629621"); m_apm_set_string(M_rnd_mm, "1.0E15"); M_get_rnd_seed(M_rnd_XX); } m_apm_multiply(M_rtmp0, M_rnd_XX, M_rnd_aa); m_apm_add(M_rtmp1, M_rtmp0, M_rnd_aa); m_apm_integer_div_rem(M_rtmp0, M_rnd_XX, M_rtmp1, M_rnd_mm); m_apm_copy(mrnd, M_rnd_XX); mrnd->m_apm_exponent -= 15; }
/* * arctanh(x) == 0.5 * log [ (1 + x) / (1 - x) ] * * |x| < 1.0 */ void m_apm_arctanh(M_APM rr, int places, M_APM aa) { M_APM tmp1, tmp2, tmp3; int ii, local_precision; tmp1 = M_get_stack_var(); m_apm_absolute_value(tmp1, aa); ii = m_apm_compare(tmp1, MM_One); if (ii >= 0) /* |x| >= 1.0 */ { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arctanh\', |Argument| >= 1"); M_set_to_zero(rr); M_restore_stack(1); return; } tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); local_precision = places + 8; m_apm_add(tmp1, MM_One, aa); m_apm_subtract(tmp2, MM_One, aa); m_apm_divide(tmp3, local_precision, tmp1, tmp2); m_apm_log(tmp2, local_precision, tmp3); m_apm_multiply(tmp1, tmp2, MM_0_5); m_apm_round(rr, places, tmp1); M_restore_stack(3); }
/* * calculate the multiple angle identity for sin (5x) * * sin (5x) == 16 * sin^5 (x) - 20 * sin^3 (x) + 5 * sin(x) */ void M_5x_do_it(M_APM rr, int places, M_APM xx) { M_APM tmp0, tmp1, t2, t3, t5; tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); t2 = M_get_stack_var(); t3 = M_get_stack_var(); t5 = M_get_stack_var(); m_apm_multiply(tmp1, xx, xx); m_apm_round(t2, (places + 4), tmp1); /* x ^ 2 */ m_apm_multiply(tmp1, t2, xx); m_apm_round(t3, (places + 4), tmp1); /* x ^ 3 */ m_apm_multiply(t5, t2, t3); /* x ^ 5 */ m_apm_multiply(tmp0, xx, MM_Five); m_apm_multiply(tmp1, t5, MM_5x_Sixteen); m_apm_add(t2, tmp0, tmp1); m_apm_multiply(tmp1, t3, MM_5x_Twenty); m_apm_subtract(tmp0, t2, tmp1); m_apm_round(rr, places, tmp0); M_restore_stack(5); }
/* * arccosh(x) == log [ x + sqrt(x^2 - 1) ] * * x >= 1.0 */ void m_apm_arccosh(M_APM rr, int places, M_APM aa) { M_APM tmp1, tmp2; int ii; ii = m_apm_compare(aa, MM_One); if (ii == -1) /* x < 1 */ { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arccosh\', Argument < 1"); M_set_to_zero(rr); return; } tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); m_apm_multiply(tmp1, aa, aa); m_apm_subtract(tmp2, tmp1, MM_One); m_apm_sqrt(tmp1, (places + 6), tmp2); m_apm_add(tmp2, aa, tmp1); m_apm_log(rr, places, tmp2); M_restore_stack(2); }
void M_log_solve_cubic(M_APM rr, int places, M_APM nn) { M_APM tmp0, tmp1, tmp2, tmp3, guess; int ii, maxp, tolerance, local_precision; guess = M_get_stack_var(); tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); M_get_log_guess(guess, nn); tolerance = -(places + 4); maxp = places + 16; local_precision = 18; /* Use the following iteration to solve for log : exp(X) - N X = X - 2 * ------------ n+1 exp(X) + N this is a cubically convergent algorithm (each iteration yields 3X more digits) */ ii = 0; while (TRUE) { m_apm_exp(tmp1, local_precision, guess); m_apm_subtract(tmp3, tmp1, nn); m_apm_add(tmp2, tmp1, nn); m_apm_divide(tmp1, local_precision, tmp3, tmp2); m_apm_multiply(tmp0, MM_Two, tmp1); m_apm_subtract(tmp3, guess, tmp0); if (ii != 0) { if (((3 * tmp0->m_apm_exponent) < tolerance) || (tmp0->m_apm_sign == 0)) break; } m_apm_round(guess, local_precision, tmp3); local_precision *= 3; if (local_precision > maxp) local_precision = maxp; ii = 1; } m_apm_round(rr, places, tmp3); M_restore_stack(5); }
/* compute int *n = round_to_nearest_int(a / log(2)) M_APM b = MAPM version of *n returns 0: OK -1, 1: failure */ int M_exp_compute_nn(int *n, M_APM b, M_APM a) { M_APM tmp0, tmp1; void *vp; char *cp, sbuf[48]; int kk; *n = 0; vp = NULL; cp = sbuf; tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); /* find 'n' and convert it to a normal C int */ /* we just need an approx 1/log(2) for this calculation */ m_apm_multiply(tmp1, a, MM_exp_log2R); /* round to the nearest int */ if (tmp1->m_apm_sign >= 0) { m_apm_add(tmp0, tmp1, MM_0_5); m_apm_floor(tmp1, tmp0); } else { m_apm_subtract(tmp0, tmp1, MM_0_5); m_apm_ceil(tmp1, tmp0); } kk = tmp1->m_apm_exponent; if (kk >= 42) { if ((vp = (void *)MAPM_MALLOC((kk + 16) * sizeof(char))) == NULL) { /* fatal, this does not return */ M_apm_log_error_msg(M_APM_FATAL, "\'M_exp_compute_nn\', Out of memory"); } cp = (char *)vp; } m_apm_to_integer_string(cp, tmp1); *n = atoi(cp); m_apm_set_long(b, (long)(*n)); kk = m_apm_compare(b, tmp1); if (vp != NULL) MAPM_FREE(vp); M_restore_stack(2); return(kk); }
/* calculate the exponential function using the following series : x^2 x^3 x^4 x^5 exp(x) == 1 + x + --- + --- + --- + --- ... 2! 3! 4! 5! */ void M_raw_exp(M_APM rr, int places, M_APM xx) { M_APM tmp0, digit, term; int tolerance, local_precision, prev_exp; long m1; tmp0 = M_get_stack_var(); term = M_get_stack_var(); digit = M_get_stack_var(); local_precision = places + 8; tolerance = -(places + 4); prev_exp = 0; m_apm_add(rr, MM_One, xx); m_apm_copy(term, xx); m1 = 2L; while (TRUE) { m_apm_set_long(digit, m1); m_apm_multiply(tmp0, term, xx); m_apm_divide(term, local_precision, tmp0, digit); m_apm_add(tmp0, rr, term); m_apm_copy(rr, tmp0); if ((term->m_apm_exponent < tolerance) || (term->m_apm_sign == 0)) break; if (m1 != 2L) { local_precision = local_precision + term->m_apm_exponent - prev_exp; if (local_precision < 20) local_precision = 20; } prev_exp = term->m_apm_exponent; m1++; } M_restore_stack(3); /* restore the 3 locals we used here */ }
void M_limit_angle_to_pi(M_APM rr, int places, M_APM aa) { M_APM tmp7, tmp8, tmp9; M_check_PI_places(places); tmp9 = M_get_stack_var(); m_apm_copy(tmp9, MM_lc_PI); if (m_apm_compare(aa, tmp9) == 1) /* > PI */ { tmp7 = M_get_stack_var(); tmp8 = M_get_stack_var(); m_apm_add(tmp7, aa, tmp9); m_apm_integer_divide(tmp9, tmp7, MM_lc_2_PI); m_apm_multiply(tmp8, tmp9, MM_lc_2_PI); m_apm_subtract(tmp9, aa, tmp8); m_apm_round(rr, places, tmp9); M_restore_stack(3); return; } tmp9->m_apm_sign = -1; if (m_apm_compare(aa, tmp9) == -1) /* < -PI */ { tmp7 = M_get_stack_var(); tmp8 = M_get_stack_var(); m_apm_add(tmp7, aa, tmp9); m_apm_integer_divide(tmp9, tmp7, MM_lc_2_PI); m_apm_multiply(tmp8, tmp9, MM_lc_2_PI); m_apm_subtract(tmp9, aa, tmp8); m_apm_round(rr, places, tmp9); M_restore_stack(3); return; } m_apm_copy(rr, aa); M_restore_stack(1); }
/* calculate arctan (x) with the following series: x^3 x^5 x^7 x^9 arctan (x) == x - --- + --- - --- + --- ... 3 5 7 9 */ void M_arctan_near_0(M_APM rr, int places, M_APM aa) { M_APM tmp0, tmpR, tmp2, tmpS, digit, term; int tolerance, local_precision; long m1; tmp0 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmpR = M_get_stack_var(); tmpS = M_get_stack_var(); term = M_get_stack_var(); digit = M_get_stack_var(); tolerance = aa->m_apm_exponent - places - 4; local_precision = places + 8 - aa->m_apm_exponent; m_apm_copy(term, aa); m_apm_copy(tmpS, aa); m_apm_multiply(tmp0, aa, aa); m_apm_round(tmp2, (local_precision + 8), tmp0); m1 = 1; while (TRUE) { m1 += 2; m_apm_set_long(digit, m1); m_apm_multiply(tmp0, term, tmp2); m_apm_round(term, local_precision, tmp0); m_apm_divide(tmp0, local_precision, term, digit); m_apm_subtract(tmpR, tmpS, tmp0); if ((tmp0->m_apm_exponent < tolerance) || (tmp0->m_apm_sign == 0)) { m_apm_round(rr, places, tmpR); break; } m1 += 2; m_apm_set_long(digit, m1); m_apm_multiply(tmp0, term, tmp2); m_apm_round(term, local_precision, tmp0); m_apm_divide(tmp0, local_precision, term, digit); m_apm_add(tmpS, tmpR, tmp0); if ((tmp0->m_apm_exponent < tolerance) || (tmp0->m_apm_sign == 0)) { m_apm_round(rr, places, tmpS); break; } } M_restore_stack(6); /* restore the 6 locals we used here */ }
/* * calculate the multiple angle identity for cos (4x) * * cos (4x) == 8 * [ cos^4 (x) - cos^2 (x) ] + 1 */ void M_4x_do_it(M_APM rr, int places, M_APM xx) { M_APM tmp0, tmp1, t2, t4; tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); t2 = M_get_stack_var(); t4 = M_get_stack_var(); m_apm_multiply(tmp1, xx, xx); m_apm_round(t2, (places + 4), tmp1); /* x ^ 2 */ m_apm_multiply(t4, t2, t2); /* x ^ 4 */ m_apm_subtract(tmp0, t4, t2); m_apm_multiply(tmp1, tmp0, MM_5x_Eight); m_apm_add(tmp0, MM_One, tmp1); m_apm_round(rr, places, tmp0); M_restore_stack(4); }
/* * cosh(x) == 0.5 * [ exp(x) + exp(-x) ] */ void m_apm_cosh(M_APM rr, int places, M_APM aa) { M_APM tmp1, tmp2, tmp3; int local_precision; tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); local_precision = places + 4; m_apm_exp(tmp1, local_precision, aa); m_apm_reciprocal(tmp2, local_precision, tmp1); m_apm_add(tmp3, tmp1, tmp2); m_apm_multiply(tmp1, tmp3, MM_0_5); m_apm_round(rr, places, tmp1); M_restore_stack(3); }
/* * tanh(x) == [ exp(x) - exp(-x) ] / [ exp(x) + exp(-x) ] */ void m_apm_tanh(M_APM rr, int places, M_APM aa) { M_APM tmp1, tmp2, tmp3, tmp4; int local_precision; tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); tmp4 = M_get_stack_var(); local_precision = places + 4; m_apm_exp(tmp1, local_precision, aa); m_apm_reciprocal(tmp2, local_precision, tmp1); m_apm_subtract(tmp3, tmp1, tmp2); m_apm_add(tmp4, tmp1, tmp2); m_apm_divide(tmp1, local_precision, tmp3, tmp4); m_apm_round(rr, places, tmp1); M_restore_stack(4); }
/* * return the nearest integer <= input */ void m_apm_floor(M_APM bb, M_APM aa) { M_APM mtmp; m_apm_copy(bb, aa); if (m_apm_is_integer(bb)) /* if integer, we're done */ return; if (bb->m_apm_exponent <= 0) /* if |bb| < 1, result is -1 or 0 */ { if (bb->m_apm_sign < 0) m_apm_negate(bb, MM_One); else M_set_to_zero(bb); return; } if (bb->m_apm_sign < 0) { mtmp = M_get_stack_var(); m_apm_negate(mtmp, bb); mtmp->m_apm_datalength = mtmp->m_apm_exponent; M_apm_normalize(mtmp); m_apm_add(bb, mtmp, MM_One); bb->m_apm_sign = -1; M_restore_stack(1); } else { bb->m_apm_datalength = bb->m_apm_exponent; M_apm_normalize(bb); } }
void m_apm_add_mt(M_APM r, const M_APM a, const M_APM b) { // m_apm_enter(); m_apm_add(r,a,b); // m_apm_leave(); }
void m_apm_arccos(M_APM r, int places, M_APM x) { M_APM tmp0, tmp1, tmp2, tmp3, current_x; int ii, maxiter, maxp, tolerance, local_precision; current_x = M_get_stack_var(); tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); m_apm_absolute_value(tmp0, x); ii = m_apm_compare(tmp0, MM_One); if (ii == 1) /* |x| > 1 */ { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arccos\', |Argument| > 1"); M_set_to_zero(r); M_restore_stack(5); return; } if (ii == 0) /* |x| == 1, arccos = 0, PI */ { if (x->m_apm_sign == 1) { M_set_to_zero(r); } else { M_check_PI_places(places); m_apm_round(r, places, MM_lc_PI); } M_restore_stack(5); return; } if (m_apm_compare(tmp0, MM_0_85) == 1) /* check if > 0.85 */ { M_cos_to_sin(tmp2, (places + 4), x); if (x->m_apm_sign == 1) { m_apm_arcsin(r, places, tmp2); } else { M_check_PI_places(places); m_apm_arcsin(tmp3, (places + 4), tmp2); m_apm_subtract(tmp1, MM_lc_PI, tmp3); m_apm_round(r, places, tmp1); } M_restore_stack(5); return; } if (x->m_apm_sign == 0) /* input == 0 ?? */ { M_check_PI_places(places); m_apm_round(r, places, MM_lc_HALF_PI); M_restore_stack(5); return; } if (x->m_apm_exponent <= -4) /* input close to 0 ?? */ { M_arccos_near_0(r, places, x); M_restore_stack(5); return; } tolerance = -(places + 4); maxp = places + 8; local_precision = 18; /* * compute the maximum number of iterations * that should be needed to calculate to * the desired accuracy. [ constant below ~= 1 / log(2) ] */ maxiter = (int)(log((double)(places + 2)) * 1.442695) + 3; if (maxiter < 5) maxiter = 5; M_get_acos_guess(current_x, x); /* Use the following iteration to solve for arc-cos : cos(X) - N X = X + ------------ n+1 sin(X) */ ii = 0; while (TRUE) { M_4x_cos(tmp1, local_precision, current_x); M_cos_to_sin(tmp2, local_precision, tmp1); if (tmp2->m_apm_sign != 0) tmp2->m_apm_sign = current_x->m_apm_sign; m_apm_subtract(tmp3, tmp1, x); m_apm_divide(tmp0, local_precision, tmp3, tmp2); m_apm_add(tmp2, current_x, tmp0); m_apm_copy(current_x, tmp2); if (ii != 0) { if (((2 * tmp0->m_apm_exponent) < tolerance) || (tmp0->m_apm_sign == 0)) break; } if (++ii == maxiter) { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arccos\', max iteration count reached"); break; } local_precision *= 2; if (local_precision > maxp) local_precision = maxp; } m_apm_round(r, places, current_x); M_restore_stack(5); }
void m_apm_log(M_APM r, int places, M_APM a) { M_APM tmp0, tmp1, tmp2; int mexp, dplaces; if (a->m_apm_sign <= 0) { M_apm_log_error_msg(M_APM_RETURN, "Warning! ... \'m_apm_log\', Negative argument"); M_set_to_zero(r); return; } tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); dplaces = places + 8; /* * if the input is real close to 1, use the series expansion * to compute the log. * * 0.9999 < a < 1.0001 */ m_apm_subtract(tmp0, a, MM_One); if (tmp0->m_apm_sign == 0) /* is input exactly 1 ?? */ { /* if so, result is 0 */ M_set_to_zero(r); M_restore_stack(3); return; } if (tmp0->m_apm_exponent <= -4) { M_log_near_1(r, places, tmp0); M_restore_stack(3); return; } /* make sure our log(10) is accurate enough for this calculation */ /* (and log(2) which is called from M_log_basic_iteration) */ M_check_log_places(dplaces + 25); mexp = a->m_apm_exponent; if (mexp >= -4 && mexp <= 4) { M_log_basic_iteration(r, places, a); } else { /* * use log (x * y) = log(x) + log(y) * * here we use y = exponent of our base 10 number. * * let 'C' = log(10) = 2.3025850929940.... * * then log(x * y) = log(x) + ( C * base_10_exponent ) */ m_apm_copy(tmp2, a); mexp = tmp2->m_apm_exponent - 2; tmp2->m_apm_exponent = 2; /* force number between 10 & 100 */ M_log_basic_iteration(tmp0, dplaces, tmp2); m_apm_set_long(tmp1, (long)mexp); m_apm_multiply(tmp2, tmp1, MM_lc_log10); m_apm_add(tmp1, tmp2, tmp0); m_apm_round(r, places, tmp1); } M_restore_stack(3); /* restore the 3 locals we used here */ }
void m_apm_arctan2(M_APM rr, int places, M_APM yy, M_APM xx) { M_APM tmp5, tmp6, tmp7; int ix, iy; iy = yy->m_apm_sign; ix = xx->m_apm_sign; if (ix == 0) /* x == 0 */ { if (iy == 0) /* y == 0 */ { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arctan2\', Both Inputs = 0"); M_set_to_zero(rr); return; } M_check_PI_places(places); m_apm_round(rr, places, MM_lc_HALF_PI); rr->m_apm_sign = iy; return; } if (iy == 0) { if (ix == 1) { M_set_to_zero(rr); } else { M_check_PI_places(places); m_apm_round(rr, places, MM_lc_PI); } return; } /* * the special cases have been handled, now do the real work */ tmp5 = M_get_stack_var(); tmp6 = M_get_stack_var(); tmp7 = M_get_stack_var(); m_apm_divide(tmp6, (places + 6), yy, xx); m_apm_arctan(tmp5, (places + 6), tmp6); if (ix == 1) /* 'x' is positive */ { m_apm_round(rr, places, tmp5); } else /* 'x' is negative */ { M_check_PI_places(places); if (iy == 1) /* 'y' is positive */ { m_apm_add(tmp7, tmp5, MM_lc_PI); m_apm_round(rr, places, tmp7); } else /* 'y' is negative */ { m_apm_subtract(tmp7, tmp5, MM_lc_PI); m_apm_round(rr, places, tmp7); } } M_restore_stack(3); }
int main(int argc, char *argv[]) { char version_info[80]; int ct; /* declare the M_APM variables ... */ M_APM aa_mapm; M_APM bb_mapm; M_APM cc_mapm; M_APM dd_mapm; if (argc < 2) { m_apm_lib_short_version(version_info); fprintf(stdout, "Usage: primenum number\t\t\t[Version 1.3, MAPM Version %s]\n", version_info); fprintf(stdout, " find the first 10 prime numbers starting with \'number\'\n"); exit(4); } /* now initialize the M_APM variables ... */ aa_mapm = m_apm_init(); bb_mapm = m_apm_init(); cc_mapm = m_apm_init(); dd_mapm = m_apm_init(); init_working_mapm(); m_apm_set_string(dd_mapm, argv[1]); /* * if input < 3, set start point = 3 */ if (m_apm_compare(dd_mapm, MM_Three) == -1) { m_apm_copy(dd_mapm, MM_Three); } /* * make sure we start with an odd integer */ m_apm_integer_divide(aa_mapm, dd_mapm, MM_Two); m_apm_multiply(bb_mapm, MM_Two, aa_mapm); m_apm_add(aa_mapm, MM_One, bb_mapm); ct = 0; while (TRUE) { if (is_number_prime(aa_mapm)) { m_apm_to_integer_string(buffer, aa_mapm); fprintf(stdout,"%s\n",buffer); if (++ct == 10) break; } m_apm_add(cc_mapm, MM_Two, aa_mapm); m_apm_copy(aa_mapm, cc_mapm); } free_working_mapm(); m_apm_free(aa_mapm); m_apm_free(bb_mapm); m_apm_free(cc_mapm); m_apm_free(dd_mapm); m_apm_free_all_mem(); exit(0); }
/* * functions returns TRUE if the M_APM input number is prime * FALSE if it is not */ int is_number_prime(M_APM input) { int ii, ret, index; char sbuf[32]; /* * for reference: * * table size of 2 to filter multiples of 2 and 3 * table size of 8 to filter multiples of 2, 3 and 5 * table size of 480 to filter multiples of 2,3,5,7, and 11 * * this increment table will filter out all numbers * that are multiples of 2,3,5 and 7. */ static char incr_table[48] = { 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10 }; /* * since the real algorithm starts at 11 (to syncronize * with the increment table), we will cheat for numbers < 10. */ if (m_apm_compare(input, MM_Ten) <= 0) { m_apm_to_integer_string(sbuf, input); ii = atoi(sbuf); if (ii == 2 || ii == 3 || ii == 5 || ii == 7) return(TRUE); else return(FALSE); } ret = FALSE; index = 0; /* * see if the input number is a * multiple of 3, 5, or 7. */ m_apm_integer_div_rem(M_quot, M_rem, input, MM_Three); if (m_apm_sign(M_rem) == 0) /* remainder == 0 */ return(ret); m_apm_integer_div_rem(M_quot, M_rem, input, MM_Five); if (m_apm_sign(M_rem) == 0) return(ret); m_apm_set_long(M_digit, 7L); m_apm_integer_div_rem(M_quot, M_rem, input, M_digit); if (m_apm_sign(M_rem) == 0) return(ret); ii = m_apm_exponent(input) + 16; m_apm_sqrt(M_tmp1, ii, input); m_apm_add(M_limit, MM_Two, M_tmp1); m_apm_set_long(M_digit, 11L); /* now start at '11' to check */ while (TRUE) { if (m_apm_compare(M_digit, M_limit) >= 0) { ret = TRUE; break; } m_apm_integer_div_rem(M_quot, M_rem, input, M_digit); if (m_apm_sign(M_rem) == 0) /* remainder == 0 */ break; m_apm_set_long(M_tmp1, (long)incr_table[index]); m_apm_add(M_tmp0, M_digit, M_tmp1); m_apm_copy(M_digit, M_tmp0); if (++index == 48) index = 0; } return(ret); }
void m_apm_to_fixpt_string(char *ss, int dplaces, M_APM mtmp) { M_APM ctmp; void *vp; int places, i2, ii, jj, kk, xp, dl, numb; UCHAR *ucp, numdiv, numrem; char *cpw, *cpd, sbuf[128]; ctmp = M_get_stack_var(); vp = NULL; cpd = ss; places = dplaces; /* just want integer portion if places == 0 */ if (places == 0) { if (mtmp->m_apm_sign >= 0) m_apm_add(ctmp, mtmp, MM_0_5); else m_apm_subtract(ctmp, mtmp, MM_0_5); m_apm_to_integer_string(cpd, ctmp); M_restore_stack(1); return; } if (places > 0) M_apm_round_fixpt(ctmp, places, mtmp); else m_apm_copy(ctmp, mtmp); /* show ALL digits */ if (ctmp->m_apm_sign == 0) /* result is 0 */ { if (places < 0) { cpd[0] = '0'; /* "0.0" */ cpd[1] = '.'; cpd[2] = '0'; cpd[3] = '\0'; } else { memset(cpd, '0', (places + 2)); /* pre-load string with all '0' */ cpd[1] = '.'; cpd[places + 2] = '\0'; } M_restore_stack(1); return; } xp = ctmp->m_apm_exponent; dl = ctmp->m_apm_datalength; numb = (dl + 1) >> 1; if (places < 0) { if (dl > xp) jj = dl + 16; else jj = xp + 16; } else { jj = places + 16; if (xp > 0) jj += xp; } if (jj > 112) { if ((vp = (void *)MAPM_MALLOC((jj + 16) * sizeof(char))) == NULL) { /* fatal, this does not return */ M_apm_log_error_msg(M_APM_FATAL, "\'m_apm_to_fixpt_string\', Out of memory"); } cpw = (char *)vp; } else { cpw = sbuf; } /* * at this point, the number is non-zero and the the output * string will contain at least 1 significant digit. */ if (ctmp->m_apm_sign == -1) /* negative number */ { *cpd++ = '-'; } ucp = ctmp->m_apm_data; ii = 0; /* convert MAPM num to ASCII digits and store in working char array */ while (TRUE) { M_get_div_rem_10((int)(*ucp++), &numdiv, &numrem); cpw[ii++] = numdiv + '0'; cpw[ii++] = numrem + '0'; if (--numb == 0) break; } i2 = ii; /* save for later */ if (places < 0) /* show ALL digits */ { places = dl - xp; if (places < 1) places = 1; } /* pad with trailing zeros if needed */ kk = xp + places + 2 - ii; if (kk > 0) memset(&cpw[ii], '0', kk); if (xp > 0) /* |num| >= 1, NO lead-in "0.nnn" */ { ii = xp + places + 1; jj = 0; for (kk=0; kk < ii; kk++) { if (kk == xp) cpd[jj++] = '.'; cpd[jj++] = cpw[kk]; } cpd[ii] = '\0'; } else /* |num| < 1, have lead-in "0.nnn" */ { jj = 2 - xp; ii = 2 + places; memset(cpd, '0', (ii + 1)); /* pre-load string with all '0' */ cpd[1] = '.'; /* assign decimal point */ for (kk=0; kk < i2; kk++) { cpd[jj++] = cpw[kk]; } cpd[ii] = '\0'; } if (vp != NULL) MAPM_FREE(vp); M_restore_stack(1); }
/* * Calculate PI using the AGM (Arithmetic-Geometric Mean) * * Init : A0 = 1 * B0 = 1 / sqrt(2) * Sum = 1 * * Iterate: n = 1... * * * A = 0.5 * [ A + B ] * n n-1 n-1 * * * B = sqrt [ A * B ] * n n-1 n-1 * * * * C = 0.5 * [ A - B ] * n n-1 n-1 * * * 2 n+1 * Sum = Sum - C * 2 * n * * * At the end when C is 'small enough' : * n * * 2 * PI = 4 * A / Sum * n+1 * * -OR- * * 2 * PI = ( A + B ) / Sum * n n * */ void M_calculate_PI_AGM(M_APM outv, int places) { M_APM tmp1, tmp2, a0, b0, c0, a1, b1, sum, pow_2; int dplaces, nn; tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); a0 = M_get_stack_var(); b0 = M_get_stack_var(); c0 = M_get_stack_var(); a1 = M_get_stack_var(); b1 = M_get_stack_var(); sum = M_get_stack_var(); pow_2 = M_get_stack_var(); dplaces = places + 16; m_apm_copy(a0, MM_One); m_apm_copy(sum, MM_One); m_apm_copy(pow_2, MM_Four); m_apm_sqrt(b0, dplaces, MM_0_5); /* sqrt(0.5) */ while (TRUE) { m_apm_add(tmp1, a0, b0); m_apm_multiply(a1, MM_0_5, tmp1); m_apm_multiply(tmp1, a0, b0); m_apm_sqrt(b1, dplaces, tmp1); m_apm_subtract(tmp1, a0, b0); m_apm_multiply(c0, MM_0_5, tmp1); /* * the net 'PI' calculated from this iteration will * be accurate to ~4 X the value of (c0)'s exponent. * this was determined experimentally. */ nn = -4 * c0->m_apm_exponent; m_apm_multiply(tmp1, c0, c0); m_apm_multiply(tmp2, tmp1, pow_2); m_apm_subtract(tmp1, sum, tmp2); m_apm_round(sum, dplaces, tmp1); if (nn >= dplaces) break; m_apm_copy(a0, a1); m_apm_copy(b0, b1); m_apm_multiply(tmp1, pow_2, MM_Two); m_apm_copy(pow_2, tmp1); } m_apm_add(tmp1, a1, b1); m_apm_multiply(tmp2, tmp1, tmp1); m_apm_divide(tmp1, dplaces, tmp2, sum); m_apm_round(outv, places, tmp1); M_restore_stack(9); }
void m_apm_exp(M_APM r, int places, M_APM x) { M_APM tmp7, tmp8, tmp9; int dplaces, nn, ii; if (MM_firsttime1) { MM_firsttime1 = FALSE; MM_exp_log2R = m_apm_init(); MM_exp_512R = m_apm_init(); m_apm_set_string(MM_exp_log2R, "1.44269504089"); /* ~ 1 / log(2) */ m_apm_set_string(MM_exp_512R, "1.953125E-3"); /* 1 / 512 */ } tmp7 = M_get_stack_var(); tmp8 = M_get_stack_var(); tmp9 = M_get_stack_var(); if (x->m_apm_sign == 0) /* if input == 0, return '1' */ { m_apm_copy(r, MM_One); M_restore_stack(3); return; } if (x->m_apm_exponent <= -3) /* already small enough so call _raw directly */ { M_raw_exp(tmp9, (places + 6), x); m_apm_round(r, places, tmp9); M_restore_stack(3); return; } /* From David H. Bailey's MPFUN Fortran package : exp (t) = (1 + r + r^2 / 2! + r^3 / 3! + r^4 / 4! ...) ^ q * 2 ^ n where q = 256, r = t' / q, t' = t - n Log(2) and where n is chosen so that -0.5 Log(2) < t' <= 0.5 Log(2). Reducing t mod Log(2) and dividing by 256 insures that -0.001 < r <= 0.001, which accelerates convergence in the above series. I use q = 512 and also limit how small 'r' can become. The 'r' used here is limited in magnitude from 1.95E-4 < |r| < 1.35E-3. Forcing 'r' into a narrow range keeps the algorithm 'well behaved'. ( the range is [0.1 / 512] to [log(2) / 512] ) */ if (M_exp_compute_nn(&nn, tmp7, x) != 0) { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_exp\', Input too large, Overflow"); M_set_to_zero(r); M_restore_stack(3); return; } dplaces = places + 8; /* check to make sure our log(2) is accurate enough */ M_check_log_places(dplaces); m_apm_multiply(tmp8, tmp7, MM_lc_log2); m_apm_subtract(tmp7, x, tmp8); /* * guarantee that |tmp7| is between 0.1 and 0.9999999.... * (in practice, the upper limit only reaches log(2), 0.693... ) */ while (TRUE) { if (tmp7->m_apm_sign != 0) { if (tmp7->m_apm_exponent == 0) break; } if (tmp7->m_apm_sign >= 0) { nn++; m_apm_subtract(tmp8, tmp7, MM_lc_log2); m_apm_copy(tmp7, tmp8); } else { nn--; m_apm_add(tmp8, tmp7, MM_lc_log2); m_apm_copy(tmp7, tmp8); } } m_apm_multiply(tmp9, tmp7, MM_exp_512R); /* perform the series expansion ... */ M_raw_exp(tmp8, dplaces, tmp9); /* * raise result to the 512 power * * note : x ^ 512 = (((x ^ 2) ^ 2) ^ 2) ... 9 times */ ii = 9; while (TRUE) { m_apm_multiply(tmp9, tmp8, tmp8); m_apm_round(tmp8, dplaces, tmp9); if (--ii == 0) break; } /* now compute 2 ^ N */ m_apm_integer_pow(tmp7, dplaces, MM_Two, nn); m_apm_multiply(tmp9, tmp7, tmp8); m_apm_round(r, places, tmp9); M_restore_stack(3); /* restore the 3 locals we used here */ }
void M_log_AGM_R_func(M_APM rr, int places, M_APM aa, M_APM bb) { M_APM tmp1, tmp2, tmp3, tmp4, tmpC2, sum, pow_2, tmpA0, tmpB0; int tolerance, dplaces; tmpA0 = M_get_stack_var(); tmpB0 = M_get_stack_var(); tmpC2 = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); tmp3 = M_get_stack_var(); tmp4 = M_get_stack_var(); sum = M_get_stack_var(); pow_2 = M_get_stack_var(); tolerance = places + 8; dplaces = places + 16; m_apm_copy(tmpA0, aa); m_apm_copy(tmpB0, bb); m_apm_copy(pow_2, MM_0_5); m_apm_multiply(tmp1, aa, aa); /* 0.5 * [ a ^ 2 - b ^ 2 ] */ m_apm_multiply(tmp2, bb, bb); m_apm_subtract(tmp3, tmp1, tmp2); m_apm_multiply(sum, MM_0_5, tmp3); while (TRUE) { m_apm_subtract(tmp1, tmpA0, tmpB0); /* C n+1 = 0.5 * [ An - Bn ] */ m_apm_multiply(tmp4, MM_0_5, tmp1); /* C n+1 */ m_apm_multiply(tmpC2, tmp4, tmp4); /* C n+1 ^ 2 */ /* do the AGM */ m_apm_add(tmp1, tmpA0, tmpB0); m_apm_multiply(tmp3, MM_0_5, tmp1); m_apm_multiply(tmp2, tmpA0, tmpB0); m_apm_sqrt(tmpB0, dplaces, tmp2); m_apm_round(tmpA0, dplaces, tmp3); /* end AGM */ m_apm_multiply(tmp2, MM_Two, pow_2); m_apm_copy(pow_2, tmp2); m_apm_multiply(tmp1, tmpC2, pow_2); m_apm_add(tmp3, sum, tmp1); if ((tmp1->m_apm_sign == 0) || ((-2 * tmp1->m_apm_exponent) > tolerance)) break; m_apm_round(sum, dplaces, tmp3); } m_apm_subtract(tmp4, MM_One, tmp3); m_apm_reciprocal(rr, places, tmp4); M_restore_stack(9); }
void m_apm_add_mt(M_APM r, M_APM a, M_APM b) { m_apm_enter(); m_apm_add(r,a,b); m_apm_leave(); }