/*
* 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 */
}
