/*
* From Knuth, The Art of Computer Programming:
*
* To compute GCD(u,v)
*
* A1:
* if (v == 0) return (u)
* A2:
* t = u mod v
* u = v
* v = t
* goto A1
*/
void m_apm_gcd_traditional(M_APM r, M_APM u, M_APM v)
{
M_APM tmpD, tmpN, tmpU, tmpV;
tmpD = M_get_stack_var();
tmpN = M_get_stack_var();
tmpU = M_get_stack_var();
tmpV = M_get_stack_var();
m_apm_absolute_value(tmpU, u);
m_apm_absolute_value(tmpV, v);
while (TRUE)
{
if (tmpV->m_apm_sign == 0)
break;
m_apm_integer_div_rem(tmpD, tmpN, tmpU, tmpV);
m_apm_copy(tmpU, tmpV);
m_apm_copy(tmpV, tmpN);
}
m_apm_copy(r, tmpU);
M_restore_stack(4);
}
/*
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 */
}
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);
}
}
}
}
MAPM XQNumericLiteral::getValue() const
{
// Use the C API to copy our fake MAPM
MAPM copy;
m_apm_copy(const_cast<M_APM>(copy.c_struct()), const_cast<M_APM>(&value_));
return copy;
}
/*
* 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 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 */
}
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);
}
MAPM::MAPM(M_APM m, bool copy)
{
if(copy) {
create();
m_apm_copy(myVal, m);
}
else {
myVal=(M_APM)m;
ref(myVal);
}
}
M_APM MAPM::val(void)
{
if (myVal->m_apm_refcount==1)
/* Return my private myVal */
return myVal;
/* Otherwise, our copy of myVal is shared--
we need to make a new private copy.
*/
M_APM oldVal=myVal;
myVal=makeNew();
m_apm_copy(myVal,oldVal);
unref(oldVal);
return myVal;
}
/*
* return the nearest integer >= input
*/
void m_apm_ceil(M_APM bb, M_APM aa)
{
M_APM mtmp;
m_apm_copy(bb, aa);
if (m_apm_is_integer(bb)) /* if integer, we're done */
return;
if (bb->m_apm_exponent <= 0) /* if |bb| < 1, result is 0 or 1 */
{
if (bb->m_apm_sign < 0)
M_set_to_zero(bb);
else
m_apm_copy(bb, MM_One);
return;
}
if (bb->m_apm_sign < 0)
{
bb->m_apm_datalength = bb->m_apm_exponent;
M_apm_normalize(bb);
}
else
{
mtmp = M_get_stack_var();
m_apm_copy(mtmp, bb);
mtmp->m_apm_datalength = mtmp->m_apm_exponent;
M_apm_normalize(mtmp);
m_apm_add(bb, mtmp, MM_One);
M_restore_stack(1);
}
}
/*
* 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);
}
}
/*
* functions returns TRUE if the M_APM input number is prime
* FALSE if it is not
*/
int is_number_prime(M_APM input)
{
int ii, ret, index;
char sbuf[32];
/*
* for reference:
*
* table size of 2 to filter multiples of 2 and 3
* table size of 8 to filter multiples of 2, 3 and 5
* table size of 480 to filter multiples of 2,3,5,7, and 11
*
* this increment table will filter out all numbers
* that are multiples of 2,3,5 and 7.
*/
static char incr_table[48] = {
2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2,
6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6,
2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10 };
/*
* since the real algorithm starts at 11 (to syncronize
* with the increment table), we will cheat for numbers < 10.
*/
if (m_apm_compare(input, MM_Ten) <= 0)
{
m_apm_to_integer_string(sbuf, input);
ii = atoi(sbuf);
if (ii == 2 || ii == 3 || ii == 5 || ii == 7)
return(TRUE);
else
return(FALSE);
}
ret = FALSE;
index = 0;
/*
* see if the input number is a
* multiple of 3, 5, or 7.
*/
m_apm_integer_div_rem(M_quot, M_rem, input, MM_Three);
if (m_apm_sign(M_rem) == 0) /* remainder == 0 */
return(ret);
m_apm_integer_div_rem(M_quot, M_rem, input, MM_Five);
if (m_apm_sign(M_rem) == 0)
return(ret);
m_apm_set_long(M_digit, 7L);
m_apm_integer_div_rem(M_quot, M_rem, input, M_digit);
if (m_apm_sign(M_rem) == 0)
return(ret);
ii = m_apm_exponent(input) + 16;
m_apm_sqrt(M_tmp1, ii, input);
m_apm_add(M_limit, MM_Two, M_tmp1);
m_apm_set_long(M_digit, 11L); /* now start at '11' to check */
while (TRUE)
{
if (m_apm_compare(M_digit, M_limit) >= 0)
{
ret = TRUE;
break;
}
m_apm_integer_div_rem(M_quot, M_rem, input, M_digit);
if (m_apm_sign(M_rem) == 0) /* remainder == 0 */
break;
m_apm_set_long(M_tmp1, (long)incr_table[index]);
m_apm_add(M_tmp0, M_digit, M_tmp1);
m_apm_copy(M_digit, M_tmp0);
if (++index == 48)
index = 0;
}
return(ret);
}
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_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_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_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);
}
void m_apm_root4(M_APM x, int dec_places, M_APM y)
{
m_apm_sqrt(x, dec_places, y);
m_apm_copy(y, x);
m_apm_sqrt(x, dec_places, y);
}
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_integer_pow_nr(M_APM rr, M_APM aa, int mexp)
{
M_APM tmp0, tmpy, tmpz;
int nexp, ii;
if (mexp == 0)
{
m_apm_copy(rr, MM_One);
return;
}
else
{
if (mexp < 0)
{
M_apm_log_error_msg(M_APM_RETURN,
"Warning! ... \'m_apm_integer_pow_nr\', Negative exponent");
M_set_to_zero(rr);
return;
}
}
if (mexp == 1)
{
m_apm_copy(rr, aa);
return;
}
if (mexp == 2)
{
m_apm_multiply(rr, aa, aa);
return;
}
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();
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);
if (nexp == 0)
break;
m_apm_copy(tmpy, tmp0);
}
m_apm_multiply(tmp0, tmpz, tmpz);
m_apm_copy(tmpz, tmp0);
}
m_apm_copy(rr, tmp0);
M_restore_stack(3);
}
void m_apm_pow4(M_APM x, M_APM y)
{
m_apm_square(x, y);
m_apm_copy(y, x);
m_apm_square(x, y);
}
void m_apm_to_fixpt_string(char *ss, int dplaces, M_APM mtmp)
{
M_APM ctmp;
void *vp;
int places, i2, ii, jj, kk, xp, dl, numb;
UCHAR *ucp, numdiv, numrem;
char *cpw, *cpd, sbuf[128];
ctmp = M_get_stack_var();
vp = NULL;
cpd = ss;
places = dplaces;
/* just want integer portion if places == 0 */
if (places == 0)
{
if (mtmp->m_apm_sign >= 0)
m_apm_add(ctmp, mtmp, MM_0_5);
else
m_apm_subtract(ctmp, mtmp, MM_0_5);
m_apm_to_integer_string(cpd, ctmp);
M_restore_stack(1);
return;
}
if (places > 0)
M_apm_round_fixpt(ctmp, places, mtmp);
else
m_apm_copy(ctmp, mtmp); /* show ALL digits */
if (ctmp->m_apm_sign == 0) /* result is 0 */
{
if (places < 0)
{
cpd[0] = '0'; /* "0.0" */
cpd[1] = '.';
cpd[2] = '0';
cpd[3] = '\0';
}
else
{
memset(cpd, '0', (places + 2)); /* pre-load string with all '0' */
cpd[1] = '.';
cpd[places + 2] = '\0';
}
M_restore_stack(1);
return;
}
xp = ctmp->m_apm_exponent;
dl = ctmp->m_apm_datalength;
numb = (dl + 1) >> 1;
if (places < 0)
{
if (dl > xp)
jj = dl + 16;
else
jj = xp + 16;
}
else
{
jj = places + 16;
if (xp > 0)
jj += xp;
}
if (jj > 112)
{
if ((vp = (void *)MAPM_MALLOC((jj + 16) * sizeof(char))) == NULL)
{
/* fatal, this does not return */
M_apm_log_error_msg(M_APM_FATAL,
"\'m_apm_to_fixpt_string\', Out of memory");
}
cpw = (char *)vp;
}
else
{
cpw = sbuf;
}
/*
* at this point, the number is non-zero and the the output
* string will contain at least 1 significant digit.
*/
if (ctmp->m_apm_sign == -1) /* negative number */
{
*cpd++ = '-';
}
ucp = ctmp->m_apm_data;
ii = 0;
/* convert MAPM num to ASCII digits and store in working char array */
while (TRUE)
{
M_get_div_rem_10((int)(*ucp++), &numdiv, &numrem);
cpw[ii++] = numdiv + '0';
cpw[ii++] = numrem + '0';
if (--numb == 0)
break;
}
i2 = ii; /* save for later */
if (places < 0) /* show ALL digits */
{
places = dl - xp;
if (places < 1)
places = 1;
}
/* pad with trailing zeros if needed */
kk = xp + places + 2 - ii;
if (kk > 0)
memset(&cpw[ii], '0', kk);
if (xp > 0) /* |num| >= 1, NO lead-in "0.nnn" */
{
ii = xp + places + 1;
jj = 0;
for (kk=0; kk < ii; kk++)
{
if (kk == xp)
cpd[jj++] = '.';
cpd[jj++] = cpw[kk];
}
cpd[ii] = '\0';
}
else /* |num| < 1, have lead-in "0.nnn" */
{
jj = 2 - xp;
ii = 2 + places;
memset(cpd, '0', (ii + 1)); /* pre-load string with all '0' */
cpd[1] = '.'; /* assign decimal point */
for (kk=0; kk < i2; kk++)
{
cpd[jj++] = cpw[kk];
}
cpd[ii] = '\0';
}
if (vp != NULL)
MAPM_FREE(vp);
M_restore_stack(1);
}
/*
* 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_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);
}
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_add(M_APM r, M_APM a, M_APM b)
{
int j, carry, sign, aexp, bexp, adigits, bdigits;
if (M_add_firsttime)
{
M_add_firsttime = FALSE;
M_work1 = m_apm_init();
M_work2 = m_apm_init();
}
if (a->m_apm_sign == 0)
{
m_apm_copy(r,b);
return;
}
if (b->m_apm_sign == 0)
{
m_apm_copy(r,a);
return;
}
if (a->m_apm_sign == 1 && b->m_apm_sign == -1)
{
b->m_apm_sign = 1;
m_apm_subtract(r,a,b);
b->m_apm_sign = -1;
return;
}
if (a->m_apm_sign == -1 && b->m_apm_sign == 1)
{
a->m_apm_sign = 1;
m_apm_subtract(r,b,a);
a->m_apm_sign = -1;
return;
}
sign = a->m_apm_sign; /* signs are the same, result will be same */
aexp = a->m_apm_exponent;
bexp = b->m_apm_exponent;
m_apm_copy(M_work1, a);
m_apm_copy(M_work2, b);
/*
* scale by at least 1 factor of 10 in case the MSB carrys
*/
if (aexp == bexp)
{
M_apm_scale(M_work1, 2); /* shift 2 digits == 1 byte for efficiency */
M_apm_scale(M_work2, 2);
}
else
{
if (aexp > bexp)
{
M_apm_scale(M_work1, 2);
M_apm_scale(M_work2, (aexp + 2 - bexp));
}
else /* aexp < bexp */
{
M_apm_scale(M_work2, 2);
M_apm_scale(M_work1, (bexp + 2 - aexp));
}
}
adigits = M_work1->m_apm_datalength;
bdigits = M_work2->m_apm_datalength;
if (adigits >= bdigits)
{
m_apm_copy(r, M_work1);
j = (bdigits + 1) >> 1;
carry = 0;
while (TRUE)
{
j--;
r->m_apm_data[j] += carry + M_work2->m_apm_data[j];
if (r->m_apm_data[j] >= 100)
{
r->m_apm_data[j] -= 100;
carry = 1;
}
else
carry = 0;
if (j == 0)
break;
}
}
void M_fast_multiply(M_APM rr, M_APM aa, M_APM bb)
{
void *vp;
int ii, k, nexp, sign;
if (M_firsttimef)
{
M_firsttimef = FALSE;
for (k=0; k < M_STACK_SIZE; k++)
mul_stack_data_size[k] = 0;
size_flag = M_get_sizeof_int();
bit_limit = 8 * size_flag + 1;
M_ain = m_apm_init();
M_bin = m_apm_init();
}
exp_stack_ptr = -1;
M_mul_stack_ptr = -1;
m_apm_copy(M_ain, aa);
m_apm_copy(M_bin, bb);
sign = M_ain->m_apm_sign * M_bin->m_apm_sign;
nexp = M_ain->m_apm_exponent + M_bin->m_apm_exponent;
if (M_ain->m_apm_datalength >= M_bin->m_apm_datalength)
ii = M_ain->m_apm_datalength;
else
ii = M_bin->m_apm_datalength;
ii = (ii + 1) >> 1;
ii = M_next_power_of_2(ii);
/* Note: 'ii' must be >= 4 here. this is guaranteed
by the caller: m_apm_multiply
*/
k = 2 * ii; /* required size of result, in bytes */
M_apm_pad(M_ain, k); /* fill out the data so the number of */
M_apm_pad(M_bin, k); /* bytes is an exact power of 2 */
if (k > rr->m_apm_malloclength)
{
if ((vp = MAPM_REALLOC(rr->m_apm_data, (k + 32))) == NULL)
{
/* fatal, this does not return */
M_apm_log_error_msg(M_APM_FATAL, "\'M_fast_multiply\', Out of memory");
}
rr->m_apm_malloclength = k + 28;
rr->m_apm_data = (UCHAR *)vp;
}
#ifdef NO_FFT_MULTIPLY
M_fmul_div_conq(rr->m_apm_data, M_ain->m_apm_data,
M_bin->m_apm_data, ii);
#else
/*
* if the numbers are *really* big, use the divide-and-conquer
* routine first until the numbers are small enough to be handled
* by the FFT algorithm. If the numbers are already small enough,
* call the FFT multiplication now.
*
* Note that 'ii' here is (and must be) an exact power of 2.
*/
if (size_flag == 2) /* if still using 16 bit compilers .... */
{
M_fast_mul_fft(rr->m_apm_data, M_ain->m_apm_data,
M_bin->m_apm_data, ii);
}
else /* >= 32 bit compilers */
{
if (ii > (MAX_FFT_BYTES + 2))
{
M_fmul_div_conq(rr->m_apm_data, M_ain->m_apm_data,
M_bin->m_apm_data, ii);
}
else
{
M_fast_mul_fft(rr->m_apm_data, M_ain->m_apm_data,
M_bin->m_apm_data, ii);
}
}
#endif
rr->m_apm_sign = sign;
rr->m_apm_exponent = nexp;
rr->m_apm_datalength = 4 * ii;
M_apm_normalize(rr);
}
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_factorial(M_APM moutput, M_APM minput)
{
int ii, nmul, ndigits, nd, jj, kk, mm, ct;
M_APM array[NDIM];
M_APM iprod1, iprod2, tmp1, tmp2;
/* return 1 for any input <= 1 */
if (m_apm_compare(minput, MM_One) <= 0)
{
m_apm_copy(moutput, MM_One);
return;
}
ct = 0;
mm = NDIM - 2;
ndigits = 256;
nd = ndigits - 20;
tmp1 = m_apm_init();
tmp2 = m_apm_init();
iprod1 = m_apm_init();
iprod2 = m_apm_init();
array[0] = m_apm_init();
m_apm_copy(tmp2, minput);
/* loop until multiply count-down has reached '2' */
while (TRUE)
{
m_apm_copy(iprod1, MM_One);
/*
* loop until the number of significant digits in this
* partial result is slightly less than 256
*/
while (TRUE)
{
m_apm_multiply(iprod2, iprod1, tmp2);
m_apm_subtract(tmp1, tmp2, MM_One);
m_apm_multiply(iprod1, iprod2, tmp1);
/*
* I know, I know. There just isn't a *clean* way
* to break out of 2 nested loops.
*/
if (m_apm_compare(tmp1, MM_Two) <= 0)
goto PHASE2;
m_apm_subtract(tmp2, tmp1, MM_One);
if (iprod1->m_apm_datalength > nd)
break;
}
if (ct == (NDIM - 1))
{
/*
* if the array has filled up, start multiplying
* some of the partial products now.
*/
m_apm_copy(tmp1, array[mm]);
m_apm_multiply(array[mm], iprod1, tmp1);
if (mm == 0)
{
mm = NDIM - 2;
ndigits = ndigits << 1;
nd = ndigits - 20;
}
else
mm--;
}
else
{
/*
* store this partial product in the array
* and allocate the next array element
*/
m_apm_copy(array[ct], iprod1);
array[++ct] = m_apm_init();
}
}
PHASE2:
m_apm_copy(array[ct], iprod1);
kk = ct;
while (kk != 0)
{
ii = 0;
jj = 0;
nmul = (kk + 1) >> 1;
while (TRUE)
{
/* must use tmp var when ii,jj point to same element */
if (ii == 0)
{
m_apm_copy(tmp1, array[ii]);
m_apm_multiply(array[jj], tmp1, array[ii+1]);
}
else
m_apm_multiply(array[jj], array[ii], array[ii+1]);
if (++jj == nmul)
break;
ii += 2;
}
if ((kk & 1) == 0)
{
jj = kk >> 1;
m_apm_copy(array[jj], array[kk]);
}
kk = kk >> 1;
}
void m_apm_copy_mt(M_APM dest, M_APM src)
{
m_apm_enter();
m_apm_copy(dest,src);
m_apm_leave();
}
