/*
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);
}
/*
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);
}
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 arcsin using the identity :
x
arcsin (x) == arctan [ --------------- ]
sqrt(1 - x^2)
*/
void M_arcsin_near_0(M_APM rr, int places, M_APM aa)
{
M_APM tmp5, tmp6;
tmp5 = M_get_stack_var();
tmp6 = M_get_stack_var();
M_cos_to_sin(tmp5, (places + 8), aa);
m_apm_divide(tmp6, (places + 8), aa, tmp5);
M_arctan_near_0(rr, places, tmp6);
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);
}
/*
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 */
}
/*
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);
}
/*
* 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_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_divide_mt(M_APM rr, int places, M_APM aa, M_APM bb)
{
m_apm_enter();
m_apm_divide(rr,places,aa,bb);
m_apm_leave();
}
/*
* 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);
}
