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 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);
}
/*
* 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);
}
}
/*
* 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);
}
/*
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);
}
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 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 */
}
/*
* 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);
}
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);
}
/*
* 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);
}
}
/*
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);
}
/*
* sinh(x) == 0.5 * [ exp(x) - exp(-x) ]
*/
void m_apm_sinh(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_subtract(tmp3, tmp1, tmp2);
m_apm_multiply(tmp1, tmp3, MM_0_5);
m_apm_round(rr, places, tmp1);
M_restore_stack(3);
}
/*
* 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);
}
/*
for large input values use :
arctan(x) = (PI / 2) - arctan(1 / |x|)
and sign of result = sign of original input
*/
void M_arctan_large_input(M_APM rr, int places, M_APM xx)
{
M_APM tmp1, tmp2;
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
M_check_PI_places(places);
m_apm_divide(tmp1, (places + 6), MM_One, xx); /* 1 / xx */
tmp1->m_apm_sign = 1; /* make positive */
m_apm_arctan(tmp2, (places + 6), tmp1);
m_apm_subtract(tmp1, MM_lc_HALF_PI, tmp2);
m_apm_round(rr, places, tmp1);
rr->m_apm_sign = xx->m_apm_sign; /* fix final sign */
M_restore_stack(2);
}
/*
* tanh(x) == [ exp(x) - exp(-x) ] / [ exp(x) + exp(-x) ]
*/
void m_apm_tanh(M_APM rr, int places, M_APM aa)
{
M_APM tmp1, tmp2, tmp3, tmp4;
int local_precision;
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
tmp3 = M_get_stack_var();
tmp4 = M_get_stack_var();
local_precision = places + 4;
m_apm_exp(tmp1, local_precision, aa);
m_apm_reciprocal(tmp2, local_precision, tmp1);
m_apm_subtract(tmp3, tmp1, tmp2);
m_apm_add(tmp4, tmp1, tmp2);
m_apm_divide(tmp1, local_precision, tmp3, tmp4);
m_apm_round(rr, places, tmp1);
M_restore_stack(4);
}
void m_apm_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_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);
}
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_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);
}
/*
* From Knuth, The Art of Computer Programming:
*
* This is the binary GCD algorithm as described
* in the book (Algorithm B)
*/
void m_apm_gcd(M_APM r, M_APM u, M_APM v)
{
M_APM tmpM, tmpN, tmpT, tmpU, tmpV;
int kk, kr, mm;
long pow_2;
/* 'is_integer' will return 0 || 1 */
if ((m_apm_is_integer(u) + m_apm_is_integer(v)) != 2)
{
M_apm_log_error_msg(M_APM_RETURN,
"Warning! \'m_apm_gcd\', Non-integer input");
M_set_to_zero(r);
return;
}
if (u->m_apm_sign == 0)
{
m_apm_absolute_value(r, v);
return;
}
if (v->m_apm_sign == 0)
{
m_apm_absolute_value(r, u);
return;
}
tmpM = M_get_stack_var();
tmpN = M_get_stack_var();
tmpT = 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);
/* Step B1 */
kk = 0;
while (TRUE)
{
mm = 1;
if (m_apm_is_odd(tmpU))
break;
mm = 0;
if (m_apm_is_odd(tmpV))
break;
m_apm_multiply(tmpN, MM_0_5, tmpU);
m_apm_copy(tmpU, tmpN);
m_apm_multiply(tmpN, MM_0_5, tmpV);
m_apm_copy(tmpV, tmpN);
kk++;
}
/* Step B2 */
if (mm)
{
m_apm_negate(tmpT, tmpV);
goto B4;
}
m_apm_copy(tmpT, tmpU);
/* Step: */
B3:
m_apm_multiply(tmpN, MM_0_5, tmpT);
m_apm_copy(tmpT, tmpN);
/* Step: */
B4:
if (m_apm_is_even(tmpT))
goto B3;
/* Step B5 */
if (tmpT->m_apm_sign == 1)
m_apm_copy(tmpU, tmpT);
else
m_apm_negate(tmpV, tmpT);
/* Step B6 */
m_apm_subtract(tmpT, tmpU, tmpV);
if (tmpT->m_apm_sign != 0)
goto B3;
/*
* result = U * 2 ^ kk
*/
if (kk == 0)
m_apm_copy(r, tmpU);
else
{
if (kk == 1)
m_apm_multiply(r, tmpU, MM_Two);
if (kk == 2)
m_apm_multiply(r, tmpU, MM_Four);
if (kk >= 3)
{
mm = kk / 28;
kr = kk % 28;
pow_2 = 1L << kr;
if (mm == 0)
{
m_apm_set_long(tmpN, pow_2);
m_apm_multiply(r, tmpU, tmpN);
}
else
{
m_apm_copy(tmpN, MM_One);
m_apm_set_long(tmpM, 0x10000000L); /* 2 ^ 28 */
while (TRUE)
{
m_apm_multiply(tmpT, tmpN, tmpM);
m_apm_copy(tmpN, tmpT);
if (--mm == 0)
break;
}
if (kr == 0)
{
m_apm_multiply(r, tmpU, tmpN);
}
else
{
m_apm_set_long(tmpM, pow_2);
m_apm_multiply(tmpT, tmpN, tmpM);
m_apm_multiply(r, tmpU, tmpT);
}
}
}
}
M_restore_stack(5);
}
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_subtract_mt(M_APM r, M_APM a, M_APM b)
{
m_apm_enter();
m_apm_subtract(r,a,b);
m_apm_leave();
}
/*
* Calculate PI using the AGM (Arithmetic-Geometric Mean)
*
* Init : A0 = 1
* B0 = 1 / sqrt(2)
* Sum = 1
*
* Iterate: n = 1...
*
*
* A = 0.5 * [ A + B ]
* n n-1 n-1
*
*
* B = sqrt [ A * B ]
* n n-1 n-1
*
*
*
* C = 0.5 * [ A - B ]
* n n-1 n-1
*
*
* 2 n+1
* Sum = Sum - C * 2
* n
*
*
* At the end when C is 'small enough' :
* n
*
* 2
* PI = 4 * A / Sum
* n+1
*
* -OR-
*
* 2
* PI = ( A + B ) / Sum
* n n
*
*/
void M_calculate_PI_AGM(M_APM outv, int places)
{
M_APM tmp1, tmp2, a0, b0, c0, a1, b1, sum, pow_2;
int dplaces, nn;
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
a0 = M_get_stack_var();
b0 = M_get_stack_var();
c0 = M_get_stack_var();
a1 = M_get_stack_var();
b1 = M_get_stack_var();
sum = M_get_stack_var();
pow_2 = M_get_stack_var();
dplaces = places + 16;
m_apm_copy(a0, MM_One);
m_apm_copy(sum, MM_One);
m_apm_copy(pow_2, MM_Four);
m_apm_sqrt(b0, dplaces, MM_0_5); /* sqrt(0.5) */
while (TRUE)
{
m_apm_add(tmp1, a0, b0);
m_apm_multiply(a1, MM_0_5, tmp1);
m_apm_multiply(tmp1, a0, b0);
m_apm_sqrt(b1, dplaces, tmp1);
m_apm_subtract(tmp1, a0, b0);
m_apm_multiply(c0, MM_0_5, tmp1);
/*
* the net 'PI' calculated from this iteration will
* be accurate to ~4 X the value of (c0)'s exponent.
* this was determined experimentally.
*/
nn = -4 * c0->m_apm_exponent;
m_apm_multiply(tmp1, c0, c0);
m_apm_multiply(tmp2, tmp1, pow_2);
m_apm_subtract(tmp1, sum, tmp2);
m_apm_round(sum, dplaces, tmp1);
if (nn >= dplaces)
break;
m_apm_copy(a0, a1);
m_apm_copy(b0, b1);
m_apm_multiply(tmp1, pow_2, MM_Two);
m_apm_copy(pow_2, tmp1);
}
m_apm_add(tmp1, a1, b1);
m_apm_multiply(tmp2, tmp1, tmp1);
m_apm_divide(tmp1, dplaces, tmp2, sum);
m_apm_round(outv, places, tmp1);
M_restore_stack(9);
}
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_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_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_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;
}
}
