/* * 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); }
void m_apm_sin_cos(M_APM sinv, M_APM cosv, int places, M_APM aa) { M_APM tmp5, tmp6, tmp7; tmp5 = M_get_stack_var(); tmp6 = M_get_stack_var(); tmp7 = M_get_stack_var(); M_limit_angle_to_pi(tmp5, (places + 6), aa); M_4x_cos(tmp7, (places + 6), tmp5); /* * compute sin(x) = sqrt(1 - cos(x) ^ 2). * * note that the sign of 'sin' will always be positive after the * sqrt call. we need to adjust the sign based on what quadrant * the original angle is in. */ M_cos_to_sin(tmp6, (places + 6), tmp7); if (tmp6->m_apm_sign != 0) tmp6->m_apm_sign = tmp5->m_apm_sign; m_apm_round(sinv, places, tmp6); m_apm_round(cosv, places, tmp7); M_restore_stack(3); }
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); }
/* 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 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 */ }
/* * 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); }
/* * 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); } }
void M_apm_round_fixpt(M_APM btmp, int places, M_APM atmp) { int xp, ii; xp = atmp->m_apm_exponent; ii = xp + places - 1; M_set_to_zero(btmp); /* assume number is too small so the net result is 0 */ if (ii >= 0) { m_apm_round(btmp, ii, atmp); } else { if (ii == -1) /* next digit is significant which may round up */ { if (atmp->m_apm_data[0] >= 50) /* digit >= 5, round up */ { m_apm_copy(btmp, atmp); btmp->m_apm_data[0] = 10; btmp->m_apm_exponent += 1; btmp->m_apm_datalength = 1; M_apm_normalize(btmp); } } } }
/* * 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); }
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 arccos using the identity : arccos (x) == PI / 2 - arcsin (x) */ void M_arccos_near_0(M_APM rr, int places, M_APM aa) { M_APM tmp1, tmp2; tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); M_check_PI_places(places); M_arcsin_near_0(tmp1, (places + 4), aa); m_apm_subtract(tmp2, MM_lc_HALF_PI, tmp1); m_apm_round(rr, places, tmp2); M_restore_stack(2); }
void m_apm_divide(M_APM rr, int places, M_APM aa, M_APM bb) { M_APM tmp0, tmp1; int sn, nexp, dplaces; sn = aa->m_apm_sign * bb->m_apm_sign; if (sn == 0) /* one number is zero, result is zero */ { if (bb->m_apm_sign == 0) { M_apm_log_error_msg(M_APM_RETURN, "Warning! ... \'m_apm_divide\', Divide by 0"); } M_set_to_zero(rr); return; } /* * Use the original 'Knuth' method for smaller divides. On the * author's system, this was the *approx* break even point before * the reciprocal method used below became faster. */ if (places < 250) { M_apm_sdivide(rr, places, aa, bb); return; } /* mimic the decimal place behavior of the original divide */ nexp = aa->m_apm_exponent - bb->m_apm_exponent; if (nexp > 0) dplaces = nexp + places; else dplaces = places; tmp0 = M_get_stack_var(); tmp1 = M_get_stack_var(); m_apm_reciprocal(tmp0, (dplaces + 8), bb); m_apm_multiply(tmp1, tmp0, aa); m_apm_round(rr, dplaces, tmp1); M_restore_stack(2); }
void m_apm_tan(M_APM r, int places, M_APM a) { M_APM tmps, tmpc, tmp0; tmps = M_get_stack_var(); tmpc = M_get_stack_var(); tmp0 = M_get_stack_var(); m_apm_sin_cos(tmps, tmpc, (places + 4), a); /* tan(x) = sin(x) / cos(x) */ m_apm_divide(tmp0, (places + 4), tmps, tmpc); m_apm_round(r, places, tmp0); M_restore_stack(3); }
/* Calls the LOG function. The formula used is : log10(x) = A * log(x) where A = log (e) [0.43429448190325...] 10 */ void m_apm_log10(M_APM rr, int places, M_APM aa) { int dplaces; M_APM tmp8, tmp9; tmp8 = M_get_stack_var(); tmp9 = M_get_stack_var(); dplaces = places + 4; M_check_log_places(dplaces + 45); m_apm_log(tmp9, dplaces, aa); m_apm_multiply(tmp8, tmp9, MM_lc_log10R); m_apm_round(rr, places, tmp8); M_restore_stack(2); /* restore the 2 locals we used here */ }
/* for large input values use : arctan(x) = (PI / 2) - arctan(1 / |x|) and sign of result = sign of original input */ void M_arctan_large_input(M_APM rr, int places, M_APM xx) { M_APM tmp1, tmp2; tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); M_check_PI_places(places); m_apm_divide(tmp1, (places + 6), MM_One, xx); /* 1 / xx */ tmp1->m_apm_sign = 1; /* make positive */ m_apm_arctan(tmp2, (places + 6), tmp1); m_apm_subtract(tmp1, MM_lc_HALF_PI, tmp2); m_apm_round(rr, places, tmp1); rr->m_apm_sign = xx->m_apm_sign; /* fix final sign */ M_restore_stack(2); }
/* * 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); }
void m_apm_reciprocal(M_APM rr, int places, M_APM aa) { M_APM last_x, guess, tmpN, tmp1, tmp2; int ii, bflag, dplaces, nexp, tolerance; if (aa->m_apm_sign == 0) { M_apm_log_error_msg(M_APM_RETURN, "Warning! ... \'m_apm_reciprocal\', Input = 0"); M_set_to_zero(rr); return; } last_x = M_get_stack_var(); guess = M_get_stack_var(); tmpN = M_get_stack_var(); tmp1 = M_get_stack_var(); tmp2 = M_get_stack_var(); m_apm_absolute_value(tmpN, aa); /* normalize the input number (make the exponent 0) so the 'guess' below will not over/under flow on large magnitude exponents. */ nexp = aa->m_apm_exponent; tmpN->m_apm_exponent -= nexp; m_apm_set_double(guess, (1.0 / m_apm_get_double(tmpN))); tolerance = places + 4; dplaces = places + 16; bflag = FALSE; m_apm_negate(last_x, MM_Ten); /* Use the following iteration to calculate the reciprocal : X = X * [ 2 - N * X ] n+1 */ ii = 0; while (TRUE) { m_apm_multiply(tmp1, tmpN, guess); m_apm_subtract(tmp2, MM_Two, tmp1); m_apm_multiply(tmp1, tmp2, guess); if (bflag) break; m_apm_round(guess, dplaces, tmp1); /* force at least 2 iterations so 'last_x' has valid data */ if (ii != 0) { m_apm_subtract(tmp2, guess, last_x); if (tmp2->m_apm_sign == 0) break; /* * if we are within a factor of 4 on the error term, * we will be accurate enough after the *next* iteration * is complete. */ if ((-4 * tmp2->m_apm_exponent) > tolerance) bflag = TRUE; } m_apm_copy(last_x, guess); ii++; } m_apm_round(rr, places, tmp1); rr->m_apm_exponent -= nexp; rr->m_apm_sign = aa->m_apm_sign; M_restore_stack(5); }
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_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); }
void m_apm_sqrt(M_APM rr, int places, M_APM aa) { M_APM last_x, guess, tmpN, tmp7, tmp8, tmp9; int ii, bflag, nexp, tolerance, dplaces; if (aa->m_apm_sign <= 0) { if (aa->m_apm_sign == -1) { M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_sqrt\', Negative argument"); } M_set_to_zero(rr); return; } last_x = M_get_stack_var(); guess = M_get_stack_var(); tmpN = M_get_stack_var(); tmp7 = M_get_stack_var(); tmp8 = M_get_stack_var(); tmp9 = M_get_stack_var(); m_apm_copy(tmpN, aa); /* normalize the input number (make the exponent near 0) so the 'guess' function will not over/under flow on large magnitude exponents. */ nexp = aa->m_apm_exponent / 2; tmpN->m_apm_exponent -= 2 * nexp; M_get_sqrt_guess(guess, tmpN); /* actually gets 1/sqrt guess */ tolerance = places + 4; dplaces = places + 16; bflag = FALSE; m_apm_negate(last_x, MM_Ten); /* Use the following iteration to calculate 1 / sqrt(N) : X = 0.5 * X * [ 3 - N * X^2 ] n+1 */ ii = 0; while (TRUE) { m_apm_multiply(tmp9, tmpN, guess); m_apm_multiply(tmp8, tmp9, guess); m_apm_round(tmp7, dplaces, tmp8); m_apm_subtract(tmp9, MM_Three, tmp7); m_apm_multiply(tmp8, tmp9, guess); m_apm_multiply(tmp9, tmp8, MM_0_5); if (bflag) break; m_apm_round(guess, dplaces, tmp9); /* force at least 2 iterations so 'last_x' has valid data */ if (ii != 0) { m_apm_subtract(tmp7, guess, last_x); if (tmp7->m_apm_sign == 0) break; /* * if we are within a factor of 4 on the error term, * we will be accurate enough after the *next* iteration * is complete. (note that the sign of the exponent on * the error term will be a negative number). */ if ((-4 * tmp7->m_apm_exponent) > tolerance) bflag = TRUE; } m_apm_copy(last_x, guess); ii++; } /* * multiply by the starting number to get the final * sqrt and then adjust the exponent since we found * the sqrt of the normalized number. */ m_apm_multiply(tmp8, tmp9, tmpN); m_apm_round(rr, places, tmp8); rr->m_apm_exponent += nexp; M_restore_stack(6); }
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); }
/* * 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_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_round_mt(M_APM btmp, int places, M_APM atmp) { m_apm_enter(); m_apm_round(btmp,places,atmp); m_apm_leave(); }
void m_apm_integer_pow(M_APM rr, int places, M_APM aa, int mexp) { M_APM tmp0, tmpy, tmpz; int nexp, ii, signflag, local_precision; if (mexp == 0) { m_apm_copy(rr, MM_One); return; } else { if (mexp > 0) { signflag = 0; nexp = mexp; } else { signflag = 1; nexp = -mexp; } } if (aa->m_apm_sign == 0) { M_set_to_zero(rr); return; } tmp0 = M_get_stack_var(); tmpy = M_get_stack_var(); tmpz = M_get_stack_var(); local_precision = places + 8; m_apm_copy(tmpy, MM_One); m_apm_copy(tmpz, aa); while (TRUE) { ii = nexp & 1; nexp = nexp >> 1; if (ii != 0) /* exponent -was- odd */ { m_apm_multiply(tmp0, tmpy, tmpz); m_apm_round(tmpy, local_precision, tmp0); if (nexp == 0) break; } m_apm_multiply(tmp0, tmpz, tmpz); m_apm_round(tmpz, local_precision, tmp0); } if (signflag) { m_apm_reciprocal(rr, places, tmpy); } else { m_apm_round(rr, places, tmpy); } M_restore_stack(3); }
/* Calculate the POW function by calling EXP : Y A X = e where A = Y * log(X) */ void m_apm_pow(M_APM rr, int places, M_APM xx, M_APM yy) { int iflag, pflag; char sbuf[64]; M_APM tmp8, tmp9; /* if yy == 0, return 1 */ if (yy->m_apm_sign == 0) { m_apm_copy(rr, MM_One); return; } /* if xx == 0, return 0 */ if (xx->m_apm_sign == 0) { M_set_to_zero(rr); return; } if (M_size_flag == 0) /* init locals on first call */ { M_size_flag = M_get_sizeof_int(); M_last_log_digits = 0; M_last_xx_input = m_apm_init(); M_last_xx_log = m_apm_init(); } /* * if 'yy' is a small enough integer, call the more * efficient _integer_pow function. */ if (m_apm_is_integer(yy)) { iflag = FALSE; if (M_size_flag == 2) /* 16 bit compilers */ { if (yy->m_apm_exponent <= 4) iflag = TRUE; } else /* >= 32 bit compilers */ { if (yy->m_apm_exponent <= 7) iflag = TRUE; } if (iflag) { m_apm_to_integer_string(sbuf, yy); m_apm_integer_pow(rr, places, xx, atoi(sbuf)); return; } } tmp8 = M_get_stack_var(); tmp9 = M_get_stack_var(); /* * If parameter 'X' is the same this call as it * was the previous call, re-use the saved log * calculation from last time. */ pflag = FALSE; if (M_last_log_digits >= places) { if (m_apm_compare(xx, M_last_xx_input) == 0) pflag = TRUE; } if (pflag) { m_apm_round(tmp9, (places + 8), M_last_xx_log); } else { m_apm_log(tmp9, (places + 8), xx); M_last_log_digits = places + 2; /* save the 'X' input value and the log calculation */ m_apm_copy(M_last_xx_input, xx); m_apm_copy(M_last_xx_log, tmp9); } m_apm_multiply(tmp8, tmp9, yy); m_apm_exp(rr, places, tmp8); M_restore_stack(2); /* restore the 2 locals we used here */ }