* M_APM - mapmasin.c
*
* Copyright (C) 1999 - 2007 Michael C. Ring
*
* Permission to use, copy, and distribute this software and its
* documentation for any purpose with or without fee is hereby granted,
* provided that the above copyright notice appear in all copies and
* that both that copyright notice and this permission notice appear
* in supporting documentation.
*
* Permission to modify the software is granted. Permission to distribute
* the modified code is granted. Modifications are to be distributed by
* using the file 'license.txt' as a template to modify the file header.
* 'license.txt' is available in the official MAPM distribution.
*
* This software is provided "as is" without express or implied warranty.
*/
* $Id: mapmasin.c,v 1.28 2007/12/03 01:49:10 mike Exp $
*
* This file contains the 'ARC' family of functions; ARC-SIN, ARC-COS,
* ARC-TAN, and ARC-TAN2.
*
* $Log: mapmasin.c,v $
* Revision 1.28 2007/12/03 01:49:10 mike
* Update license
*
* Revision 1.27 2003/07/24 16:34:02 mike
* update arctan_large_input
*
* Revision 1.26 2003/07/21 20:27:48 mike
* Modify error messages to be in a consistent format.
*
* Revision 1.25 2003/07/21 19:19:26 mike
* add new arctan with large input value
*
* Revision 1.24 2003/05/01 21:58:49 mike
* remove math.h
*
* Revision 1.23 2003/04/09 21:43:00 mike
* optimize iterative asin & acos with lessons learned
* from the new log function
*
* Revision 1.22 2003/03/31 21:58:11 mike
* call generic error handling function
*
* Revision 1.21 2002/11/03 21:41:54 mike
* Updated function parameters to use the modern style
*
* Revision 1.20 2001/02/07 19:07:07 mike
* eliminate MM_skip_limit_PI_check
*
* Revision 1.19 2001/02/06 21:50:56 mike
* don't display accuracy when iteration count maxes out
*
* Revision 1.18 2000/12/02 20:10:09 mike
* add calls to more efficient functions if
* the input args are close to zero
*
* Revision 1.17 2000/09/05 22:18:02 mike
* re-arrange code to eliminate goto from atan2
*
* Revision 1.16 2000/05/28 23:58:41 mike
* minor optimization to arc-tan2
*
* Revision 1.15 2000/05/19 17:13:29 mike
* use local copies of PI variables & recompute
* on the fly as needed
*
* Revision 1.14 2000/03/27 21:43:23 mike
* dtermine how many iterations should be required at
* run time for arc-sin and arc-cos
*
* Revision 1.13 1999/09/21 21:00:33 mike
* make sure the sign of 'sin' from M_cos_to_sin is non-zero
* before assigning it from the original angle.
*
* Revision 1.12 1999/07/21 03:05:06 mike
* added some comments
*
* Revision 1.11 1999/07/19 02:33:39 mike
* reset local precision again
*
* Revision 1.10 1999/07/19 02:18:05 mike
* more fine tuning of local precision
*
* Revision 1.9 1999/07/19 00:08:34 mike
* adjust local precision during iterative loops
*
* Revision 1.8 1999/07/18 22:35:56 mike
* make arc-sin and arc-cos use dynamically changing
* precision to speed up iterative routines for large N
*
* Revision 1.7 1999/07/09 22:52:00 mike
* skip limit PI check when not needed
*
* Revision 1.6 1999/07/09 00:10:39 mike
* use better method for arc sin and arc cos
*
* Revision 1.5 1999/07/08 22:56:20 mike
* replace local MAPM constant with a global
*
* Revision 1.4 1999/06/20 16:55:01 mike
* changed local static variables to MAPM stack variables
*
* Revision 1.3 1999/05/15 02:10:27 mike
* add check for number of decimal places
*
* Revision 1.2 1999/05/10 21:10:21 mike
* added some comments
*
* Revision 1.1 1999/05/10 20:56:31 mike
* Initial revision
*/
#include "m_apm_lc.h"
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)
{
if (iy == 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)
{
m_apm_round(rr, places, tmp5);
}
else
{
M_check_PI_places(places);
if (iy == 1)
{
m_apm_add(tmp7, tmp5, MM_lc_PI);
m_apm_round(rr, places, tmp7);
}
else
{
m_apm_subtract(tmp7, tmp5, MM_lc_PI);
m_apm_round(rr, places, tmp7);
}
}
M_restore_stack(3);
}
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)
{
M_set_to_zero(rr);
return;
}
if (xx->m_apm_exponent <= -4)
{
M_arctan_near_0(rr, places, xx);
return;
}
if (xx->m_apm_exponent >= 4)
{
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);
}
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);
tmp1->m_apm_sign = 1;
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;
M_restore_stack(2);
}
void m_apm_arcsin(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)
{
M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arcsin\', |Argument| > 1");
M_set_to_zero(r);
M_restore_stack(5);
return;
}
if (ii == 0)
{
M_check_PI_places(places);
m_apm_round(r, places, MM_lc_HALF_PI);
r->m_apm_sign = x->m_apm_sign;
M_restore_stack(5);
return;
}
if (m_apm_compare(tmp0, MM_0_85) == 1)
{
M_cos_to_sin(tmp2, (places + 4), x);
m_apm_arccos(r, places, tmp2);
r->m_apm_sign = x->m_apm_sign;
M_restore_stack(5);
return;
}
if (x->m_apm_sign == 0)
{
M_set_to_zero(r);
M_restore_stack(5);
return;
}
if (x->m_apm_exponent <= -4)
{
M_arcsin_near_0(r, places, x);
M_restore_stack(5);
return;
}
tolerance = -(places + 4);
maxp = places + 8 - x->m_apm_exponent;
local_precision = 20 - x->m_apm_exponent;
* 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_asin_guess(current_x, x);
sin(X) - N
X = X - ------------
n+1 cos(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, tmp2, x);
m_apm_divide(tmp0, local_precision, tmp3, tmp1);
m_apm_subtract(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_arcsin\', 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_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)
{
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)
{
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)
{
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)
{
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)
{
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);
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);
}
