/*
* 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);
}
/*
* 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);
}
static int Blt(lua_State *L)
{
M_APM a=Bget(L,1);
M_APM b=Bget(L,2);
lua_pushboolean(L,m_apm_compare(a,b)<0);
return 1;
}
static int Bcompare(lua_State *L) /** compare(x,y) */
{
M_APM a=Bget(L,1);
M_APM b=Bget(L,2);
lua_pushinteger(L,m_apm_compare(a,b));
return 1;
}
/*
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);
}
int m_apm_compare_mt(M_APM ltmp, M_APM rtmp)
{
int ret;
m_apm_enter();
ret=m_apm_compare(ltmp,rtmp);
m_apm_leave();
return(ret);
}
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);
}
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);
}
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);
}
/*
* 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_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;
}
int MAPM::compare(const MAPM &m) const
{
return m_apm_compare(cval(),m.cval());
}
bool MAPM::operator>=(const MAPM &m) const
{
return m_apm_compare(cval(),m.cval())>=0;
}
void M_apm_sdivide(M_APM r, int places, M_APM a, M_APM b)
{
int j, k, m, b0, sign, nexp, indexr, icompare, iterations;
long trial_numer;
void *vp;
if (M_div_firsttime)
{
M_div_firsttime = FALSE;
M_div_worka = m_apm_init();
M_div_workb = m_apm_init();
M_div_tmp7 = m_apm_init();
M_div_tmp8 = m_apm_init();
M_div_tmp9 = m_apm_init();
}
sign = a->m_apm_sign * b->m_apm_sign;
if (sign == 0) /* one number is zero, result is zero */
{
if (b->m_apm_sign == 0)
{
M_apm_log_error_msg(M_APM_RETURN, "\'M_apm_sdivide\', Divide by 0");
}
M_set_to_zero(r);
return;
}
/*
* Knuth step D1. Since base = 100, base / 2 = 50.
* (also make the working copies positive)
*/
if (b->m_apm_data[0] >= 50)
{
m_apm_absolute_value(M_div_worka, a);
m_apm_absolute_value(M_div_workb, b);
}
else /* 'normal' step D1 */
{
k = 100 / (b->m_apm_data[0] + 1);
m_apm_set_long(M_div_tmp9, (long)k);
m_apm_multiply(M_div_worka, M_div_tmp9, a);
m_apm_multiply(M_div_workb, M_div_tmp9, b);
M_div_worka->m_apm_sign = 1;
M_div_workb->m_apm_sign = 1;
}
/* setup trial denominator for step D3 */
b0 = 100 * (int)M_div_workb->m_apm_data[0];
if (M_div_workb->m_apm_datalength >= 3)
b0 += M_div_workb->m_apm_data[1];
nexp = M_div_worka->m_apm_exponent - M_div_workb->m_apm_exponent;
if (nexp > 0)
iterations = nexp + places + 1;
else
iterations = places + 1;
k = (iterations + 1) >> 1; /* required size of result, in bytes */
if (k > r->m_apm_malloclength)
{
if ((vp = MAPM_REALLOC(r->m_apm_data, (k + 32))) == NULL)
{
/* fatal, this does not return */
M_apm_log_error_msg(M_APM_FATAL, "\'M_apm_sdivide\', Out of memory");
}
r->m_apm_malloclength = k + 28;
r->m_apm_data = (UCHAR *)vp;
}
/* clear the exponent in the working copies */
M_div_worka->m_apm_exponent = 0;
M_div_workb->m_apm_exponent = 0;
/* if numbers are equal, ratio == 1.00000... */
if ((icompare = m_apm_compare(M_div_worka, M_div_workb)) == 0)
{
iterations = 1;
r->m_apm_data[0] = 10;
nexp++;
}
else /* ratio not 1, do the real division */
{
if (icompare == 1) /* numerator > denominator */
{
nexp++; /* to adjust the final exponent */
M_div_worka->m_apm_exponent += 1; /* multiply numerator by 10 */
}
else /* numerator < denominator */
{
M_div_worka->m_apm_exponent += 2; /* multiply numerator by 100 */
}
indexr = 0;
m = 0;
while (TRUE)
{
/*
* Knuth step D3. Only use the 3rd -> 6th digits if the number
* actually has that many digits.
*/
trial_numer = 10000L * (long)M_div_worka->m_apm_data[0];
if (M_div_worka->m_apm_datalength >= 5)
{
trial_numer += 100 * M_div_worka->m_apm_data[1]
+ M_div_worka->m_apm_data[2];
}
else
{
if (M_div_worka->m_apm_datalength >= 3)
trial_numer += 100 * M_div_worka->m_apm_data[1];
}
j = (int)(trial_numer / b0);
/*
* Since the library 'normalizes' all the results, we need
* to look at the exponent of the number to decide if we
* have a lead in 0n or 00.
*/
if ((k = 2 - M_div_worka->m_apm_exponent) > 0)
{
while (TRUE)
{
j /= 10;
if (--k == 0)
break;
}
}
if (j == 100) /* qhat == base ?? */
j = 99; /* if so, decrease by 1 */
m_apm_set_long(M_div_tmp8, (long)j);
m_apm_multiply(M_div_tmp7, M_div_tmp8, M_div_workb);
/*
* Compare our q-hat (j) against the desired number.
* j is either correct, 1 too large, or 2 too large
* per Theorem B on pg 272 of Art of Compter Programming,
* Volume 2, 3rd Edition.
*
* The above statement is only true if using the 2 leading
* digits of the numerator and the leading digit of the
* denominator. Since we are using the (3) leading digits
* of the numerator and the (2) leading digits of the
* denominator, we eliminate the case where our q-hat is
* 2 too large, (and q-hat being 1 too large is quite remote).
*/
if (m_apm_compare(M_div_tmp7, M_div_worka) == 1)
{
j--;
m_apm_subtract(M_div_tmp8, M_div_tmp7, M_div_workb);
m_apm_copy(M_div_tmp7, M_div_tmp8);
}
/*
* Since we know q-hat is correct, step D6 is unnecessary.
*
* Store q-hat, step D5. Since D6 is unnecessary, we can
* do D5 before D4 and decide if we are done.
*/
r->m_apm_data[indexr++] = (UCHAR)j; /* j == 'qhat' */
m += 2;
if (m >= iterations)
break;
/* step D4 */
m_apm_subtract(M_div_tmp9, M_div_worka, M_div_tmp7);
/*
* if the subtraction yields zero, the division is exact
* and we are done early.
*/
if (M_div_tmp9->m_apm_sign == 0)
{
iterations = m;
break;
}
/* multiply by 100 and re-save */
M_div_tmp9->m_apm_exponent += 2;
m_apm_copy(M_div_worka, M_div_tmp9);
}
}
r->m_apm_sign = sign;
r->m_apm_exponent = nexp;
r->m_apm_datalength = iterations;
M_apm_normalize(r);
}
/*
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 */
}
