void M_4x_cos(M_APM r, int places, M_APM x)
{
M_APM tmp8, tmp9;
tmp8 = M_get_stack_var();
tmp9 = M_get_stack_var();
/*
* if |x| >= 1.0 use multiple angle identity 4 times
* if |x| < 1.0 use multiple angle identity 3 times
*/
if (x->m_apm_exponent > 0)
{
m_apm_multiply(tmp9, x, MM_5x_256R); /* 1 / (4*4*4*4) */
M_raw_cos(tmp8, (places + 8), tmp9);
M_4x_do_it(tmp9, (places + 8), tmp8);
M_4x_do_it(tmp8, (places + 6), tmp9);
M_4x_do_it(tmp9, (places + 4), tmp8);
M_4x_do_it(r, places, tmp9);
}
else
{
m_apm_multiply(tmp9, x, MM_5x_64R); /* 1 / (4*4*4) */
M_raw_cos(tmp8, (places + 6), tmp9);
M_4x_do_it(tmp9, (places + 4), tmp8);
M_4x_do_it(tmp8, (places + 4), tmp9);
M_4x_do_it(r, places, tmp8);
}
M_restore_stack(2);
}
/*
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 */
}
/*
* 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);
}
}
/*
* 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);
}
/*
* 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;
}
/*
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);
}
/*
* 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);
}
/*
* 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);
}
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);
}
/*
* check if our local copy of PI is precise enough
* for our purpose. if not, calculate PI so it's
* as precise as desired, accurate to 'places' decimal
* places.
*/
void M_check_PI_places(int places)
{
int dplaces;
dplaces = places + 2;
if (dplaces > MM_lc_PI_digits)
{
MM_lc_PI_digits = dplaces + 2;
/* compute PI using the AGM (see right below) */
M_calculate_PI_AGM(MM_lc_PI, (dplaces + 5));
m_apm_multiply(MM_lc_HALF_PI, MM_0_5, MM_lc_PI);
m_apm_multiply(MM_lc_2_PI, MM_Two, MM_lc_PI);
}
}
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 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);
}
/*
* compute r = sqrt(1 - a ^ 2).
*/
void M_cos_to_sin(M_APM r, int places, M_APM a)
{
M_APM tmp1, tmp2;
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
m_apm_multiply(tmp1, a, a);
m_apm_subtract(tmp2, MM_One, tmp1);
m_apm_sqrt(r, places, tmp2);
M_restore_stack(2);
}
/*
* 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_integer_div_rem(M_APM qq, M_APM rr, M_APM aa, M_APM bb)
{
if (aa->m_apm_error || bb->m_apm_error)
{
M_set_to_error(rr);
M_set_to_error(qq);
return;
}
m_apm_integer_divide(qq, aa, bb);
m_apm_multiply(M_div_tmp7, qq, bb);
m_apm_subtract(rr, aa, M_div_tmp7);
}
void m_apm_lcm(M_APM r, M_APM u, M_APM v)
{
M_APM tmpN, tmpG;
tmpN = M_get_stack_var();
tmpG = M_get_stack_var();
m_apm_multiply(tmpN, u, v);
m_apm_gcd(tmpG, u, v);
m_apm_integer_divide(r, tmpN, tmpG);
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_5x_sin(M_APM r, int places, M_APM x)
{
M_APM tmp8, tmp9;
tmp8 = M_get_stack_var();
tmp9 = M_get_stack_var();
m_apm_multiply(tmp9, x, MM_5x_125R); /* 1 / (5*5*5) */
M_raw_sin(tmp8, (places + 6), tmp9);
M_5x_do_it(tmp9, (places + 4), tmp8);
M_5x_do_it(tmp8, (places + 4), tmp9);
M_5x_do_it(r, places, tmp8);
M_restore_stack(2);
}
/*
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 */
}
void m_apm_lcm(M_APM r, M_APM u, M_APM v)
{
M_APM tmpN, tmpG;
if (u->m_apm_error || v->m_apm_error)
{
M_set_to_error(r);
return;
}
tmpN = M_get_stack_var();
tmpG = M_get_stack_var();
m_apm_multiply(tmpN, u, v);
m_apm_gcd(tmpG, u, v);
m_apm_integer_divide(r, tmpN, tmpG);
M_restore_stack(2);
}
/*
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 */
}
/*
* 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);
}
/*
* check if our local copy of log(2) & log(10) is precise
* enough for our purpose. if not, calculate them so it's
* as precise as desired, accurate to at least 'places'.
*/
void M_check_log_places(int places)
{
M_APM tmp6, tmp7, tmp8, tmp9;
int dplaces;
dplaces = places + 4;
if (dplaces > MM_lc_log_digits)
{
MM_lc_log_digits = dplaces + 4;
tmp6 = M_get_stack_var();
tmp7 = M_get_stack_var();
tmp8 = M_get_stack_var();
tmp9 = M_get_stack_var();
dplaces += 6 + (int)log10((double)places);
m_apm_copy(tmp7, MM_One);
tmp7->m_apm_exponent = -places;
M_log_AGM_R_func(tmp8, dplaces, MM_One, tmp7);
m_apm_multiply(tmp6, tmp7, MM_0_5);
M_log_AGM_R_func(tmp9, dplaces, MM_One, tmp6);
m_apm_subtract(MM_lc_log2, tmp9, tmp8); /* log(2) */
tmp7->m_apm_exponent -= 1; /* divide by 10 */
M_log_AGM_R_func(tmp9, dplaces, MM_One, tmp7);
m_apm_subtract(MM_lc_log10, tmp9, tmp8); /* log(10) */
m_apm_reciprocal(MM_lc_log10R, dplaces, MM_lc_log10); /* 1 / log(10) */
M_restore_stack(4);
}
}
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);
}
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);
}
void m_apm_square(M_APM x, M_APM y) /* x = y^2 */
{
m_apm_multiply(x, y, y);
}
