alexander Posted September 13, 2007 Author Report Posted September 13, 2007 now please test your MUMPS interpreter, Craig.... it may not be bc at all :P it should take a little while.... Quote
alexander Posted September 13, 2007 Author Report Posted September 13, 2007 btw, i have just edited a wiki article on bc to include a better method for calculating pi then the one they had :) i actually posted the K.Takano equation, that is a tad different from the one i used to calculate pi here..... Ok, so still as to the philosophical question of what 2^11520000 total images will show. Aside from the film from every possible perspective of any possible or impossible event with any possible or impossible outcome of that event, provides us with every matter structure and even matter structures that don't exist, it will show every possible calculation, picture of every organism in every environment, everything.... (almost ready to write a book on it, call it "limit of the universe") Oh, this also shows that there is only a certain amount of pictures you can take of any object before you get it to be repeat... Hey, there's a picture in there of every one of us personally meeting Sir Isaac Newton (and telling him that he is mostly right, but that he may wanna work out his theories a little more...) P.S. If you must know, Takano's equation looks like this:[math]\frac{\pi}{4}=12\arctan{\frac{1}{49}}+32\arctan{\frac{1}{57}}-5\arctan{\frac{1}{239}}+12\arctan{\frac{1}{110443}}[/math] Quote
CraigD Posted September 13, 2007 Report Posted September 13, 2007 So my nice new code must not be quite up to the standards of a third grader when it comes to multiplication! :( (why does no code seem to work on the first try?)In this case, the code didn’t work due to sheer sloppiness on my part – I was “carrying” (adding) the most-significant half of the multiplication of 2 9-digit decimal numbers not only to the next multiplication in each “line”, but to the first multiplication in the next line – the equivalent of doing this: 987 654 ------ 3948 4938 5926 ------ 645928instead of this: 987 654 ------ 3948 4935 5922 ------ 645498when doing elementary school-style arithmetic (the difference being the program uses base 1000000000 “digits”, and adds as it goes, not as a final step).The new results for [math]2^{480000}[/math] appear (based on inspecting a few segments of the 144495 digit decimal results) to match those of bc (though the interpreted code running on small wintel box takes nearly 4 times as long (361 vs 91 s) as bc running on a large unix one. It’s in the attached text file. Noting that each multiplication takes a time roughly proportional to [math]\mbox{(length of current result)}^2, it’s possible to estimate that the [math]2^{11520000}[/math] will take about [math]\frac{6148914691236517204}{6004799503160660} \dot= 1024[/math] times as long as the [math]2^{480000}[/math]. Here’s the actual MUMPS code for this calculation:d PFACT^HPALG1(.F,n) S N=1,C=1,CC=0,F=0 f s F=(F(F)) q:'F f M=1:1:F(F) s N=N*F f I=2:1:F s C=C+C,CC=C*C+CCwhere the number of work units is returned as CC, n is the inputted exponent, and PFACT^HPALG1 is a subroutine that returns the prime factors of n as a MUMPS array (more or less an ordered list), F. For my poor laptop running its poor program, calculating [math]2^{11520000}[/math] should take about 4 days 7 hours. bc on a decent machine should take about 1 day. It’s been about that long since some folk started bc calculations. Has anyone gotten a result yet? Quote
alexander Posted September 13, 2007 Author Report Posted September 13, 2007 case and point, just use bc? and cp is a copy program for unix...? typo? Quote
CraigD Posted September 14, 2007 Report Posted September 14, 2007 you have to use math standard lib, try bc -l oh also you can find limitation of your computer by typing in "limits" at the bc promptMy unix’s bc seems less featurefull than yours – “-l” gives me the standard math functions, but doesn’t make it stop returning “2^480000 / exp too big”. It doesn't recognize “limits”, but its man page says they’re in /usr/include/sys/limits.h, which my low-power app dever account can only read. I can’t set scale over 100 :(BC has been around for so long, and had it's code looked at by so many people, an accusation of bc being innacurate, even with large exponents is quite honestly obsurd…I’m confident that bc’s not erroneous (and now that I’ve fixed its obligatory first-use bug, neither is my little MUMPS program) – for [math]2^{480000}[/math], expressed “(((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^3)^5)^5)^5)^5)”, it gives exactly the same 144495 digit decimal number as my program. What I’m insisting is that [math]2^{11520000}[/math] simply, mathematically, can’t be greater than [math]10^{3467866}[/math], let alone the [math]3.5485*10^{3518864}[/math] that prompted my initial objection. No greater computation resources paper and pencil and a couple dozen minutes, or just a few minutes if you have a small logarithm table or a slide rule handy are needed to see that the [math]10^{3518864}[/math] figure is glaringly wrong.… its [bc’s runtime is] about 5-7 minutes for [math]2^{24*800*600}[/math]This timing makes me deeply suspicious of what bc’s actually doing – it’s surely not actually multiplying by 2 11520000 times, as that entails something around [math]5 \times 10^{17}[/math] 32 bit operations (though each amount to just appending a zero to the binary representation of the number). If it’s cunning enough to rewrite the expression to "(((((((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^3)^3)^5)^5)^5)^5)", it can get that cut that to [math]6 \times 10^{15}[/math] operations. A good computer these days can calculate at around [math]10^{11} \, \mbox{ops/s}[/math], bringing the more efficient calculation within a 1 day runtime. 7 minutes seems too fast to be possible. Again, my guess is that bc is returning a high (but not absolute) –precision estimate of [math]e^{\ln(2) \cdot 11520000}[/math]. It’s common for calculators to handle all exponent (^) operators this way. I recall that there are some very efficient natural log and exp estimators, so this could provide a lot of significant digits precision for a small fraction of the operations needed for absolute precision. Does anyone happen to know where a copy of bc’s source code can be found? IMWTK :) Quote
alexander Posted September 14, 2007 Author Report Posted September 14, 2007 um http://ftp.gnu.org/gnu/bc/bc-1.06.tar.gz ? Quote
alexander Posted September 14, 2007 Author Report Posted September 14, 2007 i saved you a load of time by actually finding this (trust me, i saved you a lot of time) void bc_raise (num1, num2, result, scale) bc_num num1, num2, *result; int scale; { bc_num temp, power; long exponent; int rscale; int pwrscale; int calcscale; char neg; /* Check the exponent for scale digits and convert to a long. */ if (num2->n_scale != 0) bc_rt_warn ("non-zero scale in exponent"); exponent = bc_num2long (num2); if (exponent == 0 && (num2->n_len > 1 || num2->n_value[0] != 0)) bc_rt_error ("exponent too large in raise"); /* Special case if exponent is a zero. */ if (exponent == 0) { bc_free_num (result); *result = bc_copy_num (_one_); return; } /* Other initializations. */ if (exponent < 0) { neg = TRUE; exponent = -exponent; rscale = scale; } else { neg = FALSE; rscale = MIN (num1->n_scale*exponent, MAX(scale, num1->n_scale)); } /* Set initial value of temp. */ power = bc_copy_num (num1); pwrscale = num1->n_scale; while ((exponent & 1) == 0) { pwrscale = 2*pwrscale; bc_multiply (power, power, &power, pwrscale); exponent = exponent >> 1; } temp = bc_copy_num (power); calcscale = pwrscale; exponent = exponent >> 1; /* Do the calculation. */ while (exponent > 0) { pwrscale = 2*pwrscale; bc_multiply (power, power, &power, pwrscale); if ((exponent & 1) == 1) { calcscale = pwrscale + calcscale; bc_multiply (temp, power, &temp, calcscale); } exponent = exponent >> 1; } /* Assign the value. */ if (neg) { bc_divide (_one_, temp, result, rscale); bc_free_num (&temp); } else { bc_free_num (result); *result = temp; if ((*result)->n_scale > rscale) (*result)->n_scale = rscale; } bc_free_num (&power); } Quote
alexander Posted September 14, 2007 Author Report Posted September 14, 2007 functions that go with it: /* Raise BASE to the EXPO power, reduced modulo MOD. The result is placed in RESULT. If a EXPO is not an integer, only the integer part is used. */ int bc_raisemod (base, expo, mod, result, scale) bc_num base, expo, mod, *result; int scale; { bc_num power, exponent, parity, temp; int rscale; /* Check for correct numbers. */ if (bc_is_zero(mod)) return -1; if (bc_is_neg(expo)) return -1; /* Set initial values. */ power = bc_copy_num (base); exponent = bc_copy_num (expo); temp = bc_copy_num (_one_); bc_init_num(&parity); /* Check the base for scale digits. */ if (base->n_scale != 0) bc_rt_warn ("non-zero scale in base"); /* Check the exponent for scale digits. */ if (exponent->n_scale != 0) { bc_rt_warn ("non-zero scale in exponent"); bc_divide (exponent, _one_, &exponent, 0); /*truncate */ } /* Check the modulus for scale digits. */ if (mod->n_scale != 0) bc_rt_warn ("non-zero scale in modulus"); /* Do the calculation. */ rscale = MAX(scale, base->n_scale); while ( !bc_is_zero(exponent) ) { (void) bc_divmod (exponent, _two_, &exponent, &parity, 0); if ( !bc_is_zero(parity) ) { bc_multiply (temp, power, &temp, rscale); (void) bc_modulo (temp, mod, &temp, scale); } bc_multiply (power, power, &power, rscale); (void) bc_modulo (power, mod, &power, scale); } /* Assign the value. */ bc_free_num (&power); bc_free_num (&exponent); bc_free_num (result); *result = temp; return 0; /* Everything is OK. */ } and the multiplication routine: /* Multiply utility routines */ static bc_num new_sub_num (length, scale, value) int length, scale; char *value; { bc_num temp; if (_bc_Free_list != NULL) { temp = _bc_Free_list; _bc_Free_list = temp->n_next; } else { temp = (bc_num) malloc (sizeof(bc_struct)); if (temp == NULL) bc_out_of_memory (); } temp->n_sign = PLUS; temp->n_len = length; temp->n_scale = scale; temp->n_refs = 1; temp->n_ptr = NULL; temp->n_value = value; return temp; } static void _bc_simp_mul (bc_num n1, int n1len, bc_num n2, int n2len, bc_num *prod, int full_scale) { char *n1ptr, *n2ptr, *pvptr; char *n1end, *n2end; /* To the end of n1 and n2. */ int indx, sum, prodlen; prodlen = n1len+n2len+1; *prod = bc_new_num (prodlen, 0); n1end = (char *) (n1->n_value + n1len - 1); n2end = (char *) (n2->n_value + n2len - 1); pvptr = (char *) ((*prod)->n_value + prodlen - 1); sum = 0; /* Here is the loop... */ for (indx = 0; indx < prodlen-1; indx++) { n1ptr = (char *) (n1end - MAX(0, indx-n2len+1)); n2ptr = (char *) (n2end - MIN(indx, n2len-1)); while ((n1ptr >= n1->n_value) && (n2ptr <= n2end)) sum += *n1ptr-- * *n2ptr++; *pvptr-- = sum % BASE; sum = sum / BASE; } *pvptr = sum; } /* A special adder/subtractor for the recursive divide and conquer multiply algorithm. Note: if sub is called, accum must be larger that what is being subtracted. Also, accum and val must have n_scale = 0. (e.g. they must look like integers. *) */ static void _bc_shift_addsub (bc_num accum, bc_num val, int shift, int sub) { signed char *accp, *valp; int count, carry; count = val->n_len; if (val->n_value[0] == 0) count--; assert (accum->n_len+accum->n_scale >= shift+count); /* Set up pointers and others */ accp = (signed char *)(accum->n_value + accum->n_len + accum->n_scale - shift - 1); valp = (signed char *)(val->n_value + val->n_len - 1); carry = 0; if (sub) { /* Subtraction, carry is really borrow. */ while (count--) { *accp -= *valp-- + carry; if (*accp < 0) { carry = 1; *accp-- += BASE; } else { carry = 0; accp--; } } while (carry) { *accp -= carry; if (*accp < 0) *accp-- += BASE; else carry = 0; } } else { /* Addition */ while (count--) { *accp += *valp-- + carry; if (*accp > (BASE-1)) { carry = 1; *accp-- -= BASE; } else { carry = 0; accp--; } } while (carry) { *accp += carry; if (*accp > (BASE-1)) *accp-- -= BASE; else carry = 0; } } } /* Recursive divide and conquer multiply algorithm. Based on Let u = u0 + u1*(b^n) Let v = v0 + v1*(b^n) Then uv = (B^2n+B^n)*u1*v1 + B^n*(u1-u0)*(v0-v1) + (B^n+1)*u0*v0 B is the base of storage, number of digits in u1,u0 close to equal. */ static void _bc_rec_mul (bc_num u, int ulen, bc_num v, int vlen, bc_num *prod, int full_scale) { bc_num u0, u1, v0, v1; int u0len, v0len; bc_num m1, m2, m3, d1, d2; int n, prodlen, m1zero; int d1len, d2len; /* Base case? */ if ((ulen+vlen) < mul_base_digits || ulen < MUL_SMALL_DIGITS || vlen < MUL_SMALL_DIGITS ) { _bc_simp_mul (u, ulen, v, vlen, prod, full_scale); return; } /* Calculate n -- the u and v split point in digits. */ n = (MAX(ulen, vlen)+1) / 2; /* Split u and v. */ if (ulen < n) { u1 = bc_copy_num (_zero_); u0 = new_sub_num (ulen,0, u->n_value); } else { u1 = new_sub_num (ulen-n, 0, u->n_value); u0 = new_sub_num (n, 0, u->n_value+ulen-n); } if (vlen < n) { v1 = bc_copy_num (_zero_); v0 = new_sub_num (vlen,0, v->n_value); } else { v1 = new_sub_num (vlen-n, 0, v->n_value); v0 = new_sub_num (n, 0, v->n_value+vlen-n); } _bc_rm_leading_zeros (u1); _bc_rm_leading_zeros (u0); u0len = u0->n_len; _bc_rm_leading_zeros (v1); _bc_rm_leading_zeros (v0); v0len = v0->n_len; m1zero = bc_is_zero(u1) || bc_is_zero(v1); /* Calculate sub results ... */ bc_init_num(&d1); bc_init_num(&d2); bc_sub (u1, u0, &d1, 0); d1len = d1->n_len; bc_sub (v0, v1, &d2, 0); d2len = d2->n_len; /* Do recursive multiplies and shifted adds. */ if (m1zero) m1 = bc_copy_num (_zero_); else _bc_rec_mul (u1, u1->n_len, v1, v1->n_len, &m1, 0); if (bc_is_zero(d1) || bc_is_zero(d2)) m2 = bc_copy_num (_zero_); else _bc_rec_mul (d1, d1len, d2, d2len, &m2, 0); if (bc_is_zero(u0) || bc_is_zero(v0)) m3 = bc_copy_num (_zero_); else _bc_rec_mul (u0, u0->n_len, v0, v0->n_len, &m3, 0); /* Initialize product */ prodlen = ulen+vlen+1; *prod = bc_new_num(prodlen, 0); if (!m1zero) { _bc_shift_addsub (*prod, m1, 2*n, 0); _bc_shift_addsub (*prod, m1, n, 0); } _bc_shift_addsub (*prod, m3, n, 0); _bc_shift_addsub (*prod, m3, 0, 0); _bc_shift_addsub (*prod, m2, n, d1->n_sign != d2->n_sign); /* Now clean up! */ bc_free_num (&u1); bc_free_num (&u0); bc_free_num (&v1); bc_free_num (&m1); bc_free_num (&v0); bc_free_num (&m2); bc_free_num (&m3); bc_free_num (&d1); bc_free_num (&d2); } /* The multiply routine. N2 times N1 is put int PROD with the scale of the result being MIN(N2 scale+N1 scale, MAX (SCALE, N2 scale, N1 scale)). */ void bc_multiply (n1, n2, prod, scale) bc_num n1, n2, *prod; int scale; { bc_num pval; int len1, len2; int full_scale, prod_scale; /* Initialize things. */ len1 = n1->n_len + n1->n_scale; len2 = n2->n_len + n2->n_scale; full_scale = n1->n_scale + n2->n_scale; prod_scale = MIN(full_scale,MAX(scale,MAX(n1->n_scale,n2->n_scale))); /* Do the multiply */ _bc_rec_mul (n1, len1, n2, len2, &pval, full_scale); /* Assign to prod and clean up the number. */ pval->n_sign = ( n1->n_sign == n2->n_sign ? PLUS : MINUS ); pval->n_value = pval->n_ptr; pval->n_len = len2 + len1 + 1 - full_scale; pval->n_scale = prod_scale; _bc_rm_leading_zeros (pval); if (bc_is_zero (pval)) pval->n_sign = PLUS; bc_free_num (prod); *prod = pval; } /* Some utility routines for the divide: First a one digit multiply. NUM (with SIZE digits) is multiplied by DIGIT and the result is placed into RESULT. It is written so that NUM and RESULT can be the same pointers. */ static void _one_mult (num, size, digit, result) unsigned char *num; int size, digit; unsigned char *result; { int carry, value; unsigned char *nptr, *rptr; if (digit == 0) memset (result, 0, size); else { if (digit == 1) memcpy (result, num, size); else { /* Initialize */ nptr = (unsigned char *) (num+size-1); rptr = (unsigned char *) (result+size-1); carry = 0; while (size-- > 0) { value = *nptr-- * digit + carry; *rptr-- = value % BASE; carry = value / BASE; } if (carry != 0) *rptr = carry; } } } Quote
alexander Posted September 14, 2007 Author Report Posted September 14, 2007 that or maybe i should have posted just this file: /* number.c: Implements arbitrary precision numbers. */ /* Copyright (C) 1991, 1992, 1993, 1994, 1997, 2000 Free Software Foundation, Inc. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License , or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; see the file COPYING. If not, write to: The Free Software Foundation, Inc. 59 Temple Place, Suite 330 Boston, MA 02111-1307 USA. You may contact the author by: e-mail: [email protected] us-mail: Philip A. Nelson Computer Science Department, 9062 Western Washington University Bellingham, WA 98226-9062 *************************************************************************/ #include <stdio.h> #include <config.h> #include <number.h> #include <assert.h> #include <stdlib.h> #include <ctype.h>/* Prototypes needed for external utility routines. */ #define bc_rt_warn rt_warn #define bc_rt_error rt_error #define bc_out_of_memory out_of_memory _PROTOTYPE(void rt_warn, (char *mesg ,...)); _PROTOTYPE(void rt_error, (char *mesg ,...)); _PROTOTYPE(void out_of_memory, (void)); /* Storage used for special numbers. */ bc_num _zero_; bc_num _one_; bc_num _two_; static bc_num _bc_Free_list = NULL; /* new_num allocates a number and sets fields to known values. */ bc_num bc_new_num (length, scale) int length, scale; { bc_num temp; if (_bc_Free_list != NULL) { temp = _bc_Free_list; _bc_Free_list = temp->n_next; } else { temp = (bc_num) malloc (sizeof(bc_struct)); if (temp == NULL) bc_out_of_memory (); } temp->n_sign = PLUS; temp->n_len = length; temp->n_scale = scale; temp->n_refs = 1; temp->n_ptr = (char *) malloc (length+scale); if (temp->n_ptr == NULL) bc_out_of_memory(); temp->n_value = temp->n_ptr; memset (temp->n_ptr, 0, length+scale); return temp; } /* "Frees" a bc_num NUM. Actually decreases reference count and only frees the storage if reference count is zero. */ void bc_free_num (num) bc_num *num; { if (*num == NULL) return; (*num)->n_refs--; if ((*num)->n_refs == 0) { if ((*num)->n_ptr) free ((*num)->n_ptr); (*num)->n_next = _bc_Free_list; _bc_Free_list = *num; } *num = NULL; } /* Intitialize the number package! */ void bc_init_numbers () { _zero_ = bc_new_num (1,0); _one_ = bc_new_num (1,0); _one_->n_value[0] = 1; _two_ = bc_new_num (1,0); _two_->n_value[0] = 2; } /* Make a copy of a number! Just increments the reference count! */ bc_num bc_copy_num (num) bc_num num; { num->n_refs++; return num; } /* Initialize a number NUM by making it a copy of zero. */ void bc_init_num (num) bc_num *num; { *num = bc_copy_num (_zero_); } /* For many things, we may have leading zeros in a number NUM. _bc_rm_leading_zeros just moves the data "value" pointer to the correct place and adjusts the length. */ static void _bc_rm_leading_zeros (num) bc_num num; { /* We can move n_value to point to the first non zero digit! */ while (*num->n_value == 0 && num->n_len > 1) { num->n_value++; num->n_len--; } } /* Compare two bc numbers. Return value is 0 if equal, -1 if N1 is less than N2 and +1 if N1 is greater than N2. If USE_SIGN is false, just compare the magnitudes. */ static int _bc_do_compare (n1, n2, use_sign, ignore_last) bc_num n1, n2; int use_sign; int ignore_last; { char *n1ptr, *n2ptr; int count; /* First, compare signs. */ if (use_sign && n1->n_sign != n2->n_sign) { if (n1->n_sign == PLUS) return (1); /* Positive N1 > Negative N2 */ else return (-1); /* Negative N1 < Positive N1 */ } /* Now compare the magnitude. */ if (n1->n_len != n2->n_len) { if (n1->n_len > n2->n_len) { /* Magnitude of n1 > n2. */ if (!use_sign || n1->n_sign == PLUS) return (1); else return (-1); } else { /* Magnitude of n1 < n2. */ if (!use_sign || n1->n_sign == PLUS) return (-1); else return (1); } } /* If we get here, they have the same number of integer digits. check the integer part and the equal length part of the fraction. */ count = n1->n_len + MIN (n1->n_scale, n2->n_scale); n1ptr = n1->n_value; n2ptr = n2->n_value; while ((count > 0) && (*n1ptr == *n2ptr)) { n1ptr++; n2ptr++; count--; } if (ignore_last && count == 1 && n1->n_scale == n2->n_scale) return (0); if (count != 0) { if (*n1ptr > *n2ptr) { /* Magnitude of n1 > n2. */ if (!use_sign || n1->n_sign == PLUS) return (1); else return (-1); } else { /* Magnitude of n1 < n2. */ if (!use_sign || n1->n_sign == PLUS) return (-1); else return (1); } } /* They are equal up to the last part of the equal part of the fraction. */ if (n1->n_scale != n2->n_scale) { if (n1->n_scale > n2->n_scale) { for (count = n1->n_scale-n2->n_scale; count>0; count--) if (*n1ptr++ != 0) { /* Magnitude of n1 > n2. */ if (!use_sign || n1->n_sign == PLUS) return (1); else return (-1); } } else { for (count = n2->n_scale-n1->n_scale; count>0; count--) if (*n2ptr++ != 0) { /* Magnitude of n1 < n2. */ if (!use_sign || n1->n_sign == PLUS) return (-1); else return (1); } } } /* They must be equal! */ return (0); } /* This is the "user callable" routine to compare numbers N1 and N2. */ int bc_compare (n1, n2) bc_num n1, n2; { return _bc_do_compare (n1, n2, TRUE, FALSE); } /* In some places we need to check if the number is negative. */ char bc_is_neg (num) bc_num num; { return num->n_sign == MINUS; } /* In some places we need to check if the number NUM is zero. */ char bc_is_zero (num) bc_num num; { int count; char *nptr; /* Quick check. */ if (num == _zero_) return TRUE; /* Initialize */ count = num->n_len + num->n_scale; nptr = num->n_value; /* The check */ while ((count > 0) && (*nptr++ == 0)) count--; if (count != 0) return FALSE; else return TRUE; } /* In some places we need to check if the number NUM is almost zero. Specifically, all but the last digit is 0 and the last digit is 1. Last digit is defined by scale. */ char bc_is_near_zero (num, scale) bc_num num; int scale; { int count; char *nptr; /* Error checking */ if (scale > num->n_scale) scale = num->n_scale; /* Initialize */ count = num->n_len + scale; nptr = num->n_value; /* The check */ while ((count > 0) && (*nptr++ == 0)) count--; if (count != 0 && (count != 1 || *--nptr != 1)) return FALSE; else return TRUE; } /* Perform addition: N1 is added to N2 and the value is returned. The signs of N1 and N2 are ignored. SCALE_MIN is to set the minimum scale of the result. */ static bc_num _bc_do_add (n1, n2, scale_min) bc_num n1, n2; int scale_min; { bc_num sum; int sum_scale, sum_digits; char *n1ptr, *n2ptr, *sumptr; int carry, n1bytes, n2bytes; int count; /* Prepare sum. */ sum_scale = MAX (n1->n_scale, n2->n_scale); sum_digits = MAX (n1->n_len, n2->n_len) + 1; sum = bc_new_num (sum_digits, MAX(sum_scale, scale_min)); /* Zero extra digits made by scale_min. */ if (scale_min > sum_scale) { sumptr = (char *) (sum->n_value + sum_scale + sum_digits); for (count = scale_min - sum_scale; count > 0; count--) *sumptr++ = 0; } /* Start with the fraction part. Initialize the pointers. */ n1bytes = n1->n_scale; n2bytes = n2->n_scale; n1ptr = (char *) (n1->n_value + n1->n_len + n1bytes - 1); n2ptr = (char *) (n2->n_value + n2->n_len + n2bytes - 1); sumptr = (char *) (sum->n_value + sum_scale + sum_digits - 1); /* Add the fraction part. First copy the longer fraction.*/ if (n1bytes != n2bytes) { if (n1bytes > n2bytes) while (n1bytes>n2bytes) { *sumptr-- = *n1ptr--; n1bytes--;} else while (n2bytes>n1bytes) { *sumptr-- = *n2ptr--; n2bytes--;} } /* Now add the remaining fraction part and equal size integer parts. */ n1bytes += n1->n_len; n2bytes += n2->n_len; carry = 0; while ((n1bytes > 0) && (n2bytes > 0)) { *sumptr = *n1ptr-- + *n2ptr-- + carry; if (*sumptr > (BASE-1)) { carry = 1; *sumptr -= BASE; } else carry = 0; sumptr--; n1bytes--; n2bytes--; } /* Now add carry the longer integer part. */ if (n1bytes == 0) { n1bytes = n2bytes; n1ptr = n2ptr; } while (n1bytes-- > 0) { *sumptr = *n1ptr-- + carry; if (*sumptr > (BASE-1)) { carry = 1; *sumptr -= BASE; } else carry = 0; sumptr--; } /* Set final carry. */ if (carry == 1) *sumptr += 1; /* Adjust sum and return. */ _bc_rm_leading_zeros (sum); return sum; } /* Perform subtraction: N2 is subtracted from N1 and the value is returned. The signs of N1 and N2 are ignored. Also, N1 is assumed to be larger than N2. SCALE_MIN is the minimum scale of the result. */ static bc_num _bc_do_sub (n1, n2, scale_min) bc_num n1, n2; int scale_min; { bc_num diff; int diff_scale, diff_len; int min_scale, min_len; char *n1ptr, *n2ptr, *diffptr; int borrow, count, val; /* Allocate temporary storage. */ diff_len = MAX (n1->n_len, n2->n_len); diff_scale = MAX (n1->n_scale, n2->n_scale); min_len = MIN (n1->n_len, n2->n_len); min_scale = MIN (n1->n_scale, n2->n_scale); diff = bc_new_num (diff_len, MAX(diff_scale, scale_min)); /* Zero extra digits made by scale_min. */ if (scale_min > diff_scale) { diffptr = (char *) (diff->n_value + diff_len + diff_scale); for (count = scale_min - diff_scale; count > 0; count--) *diffptr++ = 0; } /* Initialize the subtract. */ n1ptr = (char *) (n1->n_value + n1->n_len + n1->n_scale -1); n2ptr = (char *) (n2->n_value + n2->n_len + n2->n_scale -1); diffptr = (char *) (diff->n_value + diff_len + diff_scale -1); /* Subtract the numbers. */ borrow = 0; /* Take care of the longer scaled number. */ if (n1->n_scale != min_scale) { /* n1 has the longer scale */ for (count = n1->n_scale - min_scale; count > 0; count--) *diffptr-- = *n1ptr--; } else { /* n2 has the longer scale */ for (count = n2->n_scale - min_scale; count > 0; count--) { val = - *n2ptr-- - borrow; if (val < 0) { val += BASE; borrow = 1; } else borrow = 0; *diffptr-- = val; } } /* Now do the equal length scale and integer parts. */ for (count = 0; count < min_len + min_scale; count++) { val = *n1ptr-- - *n2ptr-- - borrow; if (val < 0) { val += BASE; borrow = 1; } else borrow = 0; *diffptr-- = val; } /* If n1 has more digits then n2, we now do that subtract. */ if (diff_len != min_len) { for (count = diff_len - min_len; count > 0; count--) { val = *n1ptr-- - borrow; if (val < 0) { val += BASE; borrow = 1; } else borrow = 0; *diffptr-- = val; } } /* Clean up and return. */ _bc_rm_leading_zeros (diff); return diff; } /* Here is the full subtract routine that takes care of negative numbers. N2 is subtracted from N1 and the result placed in RESULT. SCALE_MIN is the minimum scale for the result. */ void bc_sub (n1, n2, result, scale_min) bc_num n1, n2, *result; int scale_min; { bc_num diff = NULL; int cmp_res; int res_scale; if (n1->n_sign != n2->n_sign) { diff = _bc_do_add (n1, n2, scale_min); diff->n_sign = n1->n_sign; } else { /* subtraction must be done. */ /* Compare magnitudes. */ cmp_res = _bc_do_compare (n1, n2, FALSE, FALSE); switch (cmp_res) { case -1: /* n1 is less than n2, subtract n1 from n2. */ diff = _bc_do_sub (n2, n1, scale_min); diff->n_sign = (n2->n_sign == PLUS ? MINUS : PLUS); break; case 0: /* They are equal! return zero! */ res_scale = MAX (scale_min, MAX(n1->n_scale, n2->n_scale)); diff = bc_new_num (1, res_scale); memset (diff->n_value, 0, res_scale+1); break; case 1: /* n2 is less than n1, subtract n2 from n1. */ diff = _bc_do_sub (n1, n2, scale_min); diff->n_sign = n1->n_sign; break; } } /* Clean up and return. */ bc_free_num (result); *result = diff; } /* Here is the full add routine that takes care of negative numbers. N1 is added to N2 and the result placed into RESULT. SCALE_MIN is the minimum scale for the result. */ void bc_add (n1, n2, result, scale_min) bc_num n1, n2, *result; int scale_min; { bc_num sum = NULL; int cmp_res; int res_scale; if (n1->n_sign == n2->n_sign) { sum = _bc_do_add (n1, n2, scale_min); sum->n_sign = n1->n_sign; } else { /* subtraction must be done. */ cmp_res = _bc_do_compare (n1, n2, FALSE, FALSE); /* Compare magnitudes. */ switch (cmp_res) { case -1: /* n1 is less than n2, subtract n1 from n2. */ sum = _bc_do_sub (n2, n1, scale_min); sum->n_sign = n2->n_sign; break; case 0: /* They are equal! return zero with the correct scale! */ res_scale = MAX (scale_min, MAX(n1->n_scale, n2->n_scale)); sum = bc_new_num (1, res_scale); memset (sum->n_value, 0, res_scale+1); break; case 1: /* n2 is less than n1, subtract n2 from n1. */ sum = _bc_do_sub (n1, n2, scale_min); sum->n_sign = n1->n_sign; } } /* Clean up and return. */ bc_free_num (result); *result = sum; } /* Recursive vs non-recursive multiply crossover ranges. */ #if defined(MULDIGITS) #include "muldigits.h" #else #define MUL_BASE_DIGITS 80 #endif int mul_base_digits = MUL_BASE_DIGITS; #define MUL_SMALL_DIGITS mul_base_digits/4 /* Multiply utility routines */ static bc_num new_sub_num (length, scale, value) int length, scale; char *value; { bc_num temp; if (_bc_Free_list != NULL) { temp = _bc_Free_list; _bc_Free_list = temp->n_next; } else { temp = (bc_num) malloc (sizeof(bc_struct)); if (temp == NULL) bc_out_of_memory (); } temp->n_sign = PLUS; temp->n_len = length; temp->n_scale = scale; temp->n_refs = 1; temp->n_ptr = NULL; temp->n_value = value; return temp; } static void _bc_simp_mul (bc_num n1, int n1len, bc_num n2, int n2len, bc_num *prod, int full_scale) { char *n1ptr, *n2ptr, *pvptr; char *n1end, *n2end; /* To the end of n1 and n2. */ int indx, sum, prodlen; prodlen = n1len+n2len+1; *prod = bc_new_num (prodlen, 0); n1end = (char *) (n1->n_value + n1len - 1); n2end = (char *) (n2->n_value + n2len - 1); pvptr = (char *) ((*prod)->n_value + prodlen - 1); sum = 0; /* Here is the loop... */ for (indx = 0; indx < prodlen-1; indx++) { n1ptr = (char *) (n1end - MAX(0, indx-n2len+1)); n2ptr = (char *) (n2end - MIN(indx, n2len-1)); while ((n1ptr >= n1->n_value) && (n2ptr <= n2end)) sum += *n1ptr-- * *n2ptr++; *pvptr-- = sum % BASE; sum = sum / BASE; } *pvptr = sum; } /* A special adder/subtractor for the recursive divide and conquer multiply algorithm. Note: if sub is called, accum must be larger that what is being subtracted. Also, accum and val must have n_scale = 0. (e.g. they must look like integers. *) */ static void _bc_shift_addsub (bc_num accum, bc_num val, int shift, int sub) { signed char *accp, *valp; int count, carry; count = val->n_len; if (val->n_value[0] == 0) count--; assert (accum->n_len+accum->n_scale >= shift+count); /* Set up pointers and others */ accp = (signed char *)(accum->n_value + accum->n_len + accum->n_scale - shift - 1); valp = (signed char *)(val->n_value + val->n_len - 1); carry = 0; if (sub) { /* Subtraction, carry is really borrow. */ while (count--) { *accp -= *valp-- + carry; if (*accp < 0) { carry = 1; *accp-- += BASE; } else { carry = 0; accp--; } } while (carry) { *accp -= carry; if (*accp < 0) *accp-- += BASE; else carry = 0; } } else { /* Addition */ while (count--) { *accp += *valp-- + carry; if (*accp > (BASE-1)) { carry = 1; *accp-- -= BASE; } else { carry = 0; accp--; } } while (carry) { *accp += carry; if (*accp > (BASE-1)) *accp-- -= BASE; else carry = 0; } } } /* Recursive divide and conquer multiply algorithm. Based on Let u = u0 + u1*(b^n) Let v = v0 + v1*(b^n) Then uv = (B^2n+B^n)*u1*v1 + B^n*(u1-u0)*(v0-v1) + (B^n+1)*u0*v0 B is the base of storage, number of digits in u1,u0 close to equal. */ static void _bc_rec_mul (bc_num u, int ulen, bc_num v, int vlen, bc_num *prod, int full_scale) { bc_num u0, u1, v0, v1; int u0len, v0len; bc_num m1, m2, m3, d1, d2; int n, prodlen, m1zero; int d1len, d2len; /* Base case? */ if ((ulen+vlen) < mul_base_digits || ulen < MUL_SMALL_DIGITS || vlen < MUL_SMALL_DIGITS ) { _bc_simp_mul (u, ulen, v, vlen, prod, full_scale); return; } /* Calculate n -- the u and v split point in digits. */ n = (MAX(ulen, vlen)+1) / 2; /* Split u and v. */ if (ulen < n) { u1 = bc_copy_num (_zero_); u0 = new_sub_num (ulen,0, u->n_value); } else { u1 = new_sub_num (ulen-n, 0, u->n_value); u0 = new_sub_num (n, 0, u->n_value+ulen-n); } if (vlen < n) { v1 = bc_copy_num (_zero_); v0 = new_sub_num (vlen,0, v->n_value); } else { v1 = new_sub_num (vlen-n, 0, v->n_value); v0 = new_sub_num (n, 0, v->n_value+vlen-n); } _bc_rm_leading_zeros (u1); _bc_rm_leading_zeros (u0); u0len = u0->n_len; _bc_rm_leading_zeros (v1); _bc_rm_leading_zeros (v0); v0len = v0->n_len; m1zero = bc_is_zero(u1) || bc_is_zero(v1); /* Calculate sub results ... */ bc_init_num(&d1); bc_init_num(&d2); bc_sub (u1, u0, &d1, 0); d1len = d1->n_len; bc_sub (v0, v1, &d2, 0); d2len = d2->n_len; /* Do recursive multiplies and shifted adds. */ if (m1zero) m1 = bc_copy_num (_zero_); else _bc_rec_mul (u1, u1->n_len, v1, v1->n_len, &m1, 0); if (bc_is_zero(d1) || bc_is_zero(d2)) m2 = bc_copy_num (_zero_); else _bc_rec_mul (d1, d1len, d2, d2len, &m2, 0); if (bc_is_zero(u0) || bc_is_zero(v0)) m3 = bc_copy_num (_zero_); else _bc_rec_mul (u0, u0->n_len, v0, v0->n_len, &m3, 0); /* Initialize product */ prodlen = ulen+vlen+1; *prod = bc_new_num(prodlen, 0); if (!m1zero) { _bc_shift_addsub (*prod, m1, 2*n, 0); _bc_shift_addsub (*prod, m1, n, 0); } _bc_shift_addsub (*prod, m3, n, 0); _bc_shift_addsub (*prod, m3, 0, 0); _bc_shift_addsub (*prod, m2, n, d1->n_sign != d2->n_sign); /* Now clean up! */ bc_free_num (&u1); bc_free_num (&u0); bc_free_num (&v1); bc_free_num (&m1); bc_free_num (&v0); bc_free_num (&m2); bc_free_num (&m3); bc_free_num (&d1); bc_free_num (&d2); } /* The multiply routine. N2 times N1 is put int PROD with the scale of the result being MIN(N2 scale+N1 scale, MAX (SCALE, N2 scale, N1 scale)). */ void bc_multiply (n1, n2, prod, scale) bc_num n1, n2, *prod; int scale; { bc_num pval; int len1, len2; int full_scale, prod_scale; /* Initialize things. */ len1 = n1->n_len + n1->n_scale; len2 = n2->n_len + n2->n_scale; full_scale = n1->n_scale + n2->n_scale; prod_scale = MIN(full_scale,MAX(scale,MAX(n1->n_scale,n2->n_scale))); /* Do the multiply */ _bc_rec_mul (n1, len1, n2, len2, &pval, full_scale); /* Assign to prod and clean up the number. */ pval->n_sign = ( n1->n_sign == n2->n_sign ? PLUS : MINUS ); pval->n_value = pval->n_ptr; pval->n_len = len2 + len1 + 1 - full_scale; pval->n_scale = prod_scale; _bc_rm_leading_zeros (pval); if (bc_is_zero (pval)) pval->n_sign = PLUS; bc_free_num (prod); *prod = pval; } /* Some utility routines for the divide: First a one digit multiply. NUM (with SIZE digits) is multiplied by DIGIT and the result is placed into RESULT. It is written so that NUM and RESULT can be the same pointers. */ static void _one_mult (num, size, digit, result) unsigned char *num; int size, digit; unsigned char *result; { int carry, value; unsigned char *nptr, *rptr; if (digit == 0) memset (result, 0, size); else { if (digit == 1) memcpy (result, num, size); else { /* Initialize */ nptr = (unsigned char *) (num+size-1); rptr = (unsigned char *) (result+size-1); carry = 0; while (size-- > 0) { value = *nptr-- * digit + carry; *rptr-- = value % BASE; carry = value / BASE; } if (carry != 0) *rptr = carry; } } } /* The full division routine. This computes N1 / N2. It returns 0 if the division is ok and the result is in QUOT. The number of digits after the decimal point is SCALE. It returns -1 if division by zero is tried. The algorithm is found in Knuth Vol 2. p237. */ int bc_divide (n1, n2, quot, scale) bc_num n1, n2, *quot; int scale; { bc_num qval; unsigned char *num1, *num2; unsigned char *ptr1, *ptr2, *n2ptr, *qptr; int scale1, val; unsigned int len1, len2, scale2, qdigits, extra, count; unsigned int qdig, qguess, borrow, carry; unsigned char *mval; char zero; unsigned int norm; /* Test for divide by zero. */ if (bc_is_zero (n2)) return -1; /* Test for divide by 1. If it is we must truncate. */ if (n2->n_scale == 0) { if (n2->n_len == 1 && *n2->n_value == 1) { qval = bc_new_num (n1->n_len, scale); qval->n_sign = (n1->n_sign == n2->n_sign ? PLUS : MINUS); memset (&qval->n_value[n1->n_len],0,scale); memcpy (qval->n_value, n1->n_value, n1->n_len + MIN(n1->n_scale,scale)); bc_free_num (quot); *quot = qval; } } /* Set up the divide. Move the decimal point on n1 by n2's scale. Remember, zeros on the end of num2 are wasted effort for dividing. */ scale2 = n2->n_scale; n2ptr = (unsigned char *) n2->n_value+n2->n_len+scale2-1; while ((scale2 > 0) && (*n2ptr-- == 0)) scale2--; len1 = n1->n_len + scale2; scale1 = n1->n_scale - scale2; if (scale1 < scale) extra = scale - scale1; else extra = 0; num1 = (unsigned char *) malloc (n1->n_len+n1->n_scale+extra+2); if (num1 == NULL) bc_out_of_memory(); memset (num1, 0, n1->n_len+n1->n_scale+extra+2); memcpy (num1+1, n1->n_value, n1->n_len+n1->n_scale); len2 = n2->n_len + scale2; num2 = (unsigned char *) malloc (len2+1); if (num2 == NULL) bc_out_of_memory(); memcpy (num2, n2->n_value, len2); *(num2+len2) = 0; n2ptr = num2; while (*n2ptr == 0) { n2ptr++; len2--; } /* Calculate the number of quotient digits. */ if (len2 > len1+scale) { qdigits = scale+1; zero = TRUE; } else { zero = FALSE; if (len2>len1) qdigits = scale+1; /* One for the zero integer part. */ else qdigits = len1-len2+scale+1; } /* Allocate and zero the storage for the quotient. */ qval = bc_new_num (qdigits-scale,scale); memset (qval->n_value, 0, qdigits); /* Allocate storage for the temporary storage mval. */ mval = (unsigned char *) malloc (len2+1); if (mval == NULL) bc_out_of_memory (); /* Now for the full divide algorithm. */ if (!zero) { /* Normalize */ norm = 10 / ((int)*n2ptr + 1); if (norm != 1) { _one_mult (num1, len1+scale1+extra+1, norm, num1); _one_mult (n2ptr, len2, norm, n2ptr); } /* Initialize divide loop. */ qdig = 0; if (len2 > len1) qptr = (unsigned char *) qval->n_value+len2-len1; else qptr = (unsigned char *) qval->n_value; /* Loop */ while (qdig <= len1+scale-len2) { /* Calculate the quotient digit guess. */ if (*n2ptr == num1[qdig]) qguess = 9; else qguess = (num1[qdig]*10 + num1[qdig+1]) / *n2ptr; /* Test qguess. */ if (n2ptr[1]*qguess > (num1[qdig]*10 + num1[qdig+1] - *n2ptr*qguess)*10 + num1[qdig+2]) { qguess--; /* And again. */ if (n2ptr[1]*qguess > (num1[qdig]*10 + num1[qdig+1] - *n2ptr*qguess)*10 + num1[qdig+2]) qguess--; } /* Multiply and subtract. */ borrow = 0; if (qguess != 0) { *mval = 0; _one_mult (n2ptr, len2, qguess, mval+1); ptr1 = (unsigned char *) num1+qdig+len2; ptr2 = (unsigned char *) mval+len2; for (count = 0; count < len2+1; count++) { val = (int) *ptr1 - (int) *ptr2-- - borrow; if (val < 0) { val += 10; borrow = 1; } else borrow = 0; *ptr1-- = val; } } /* Test for negative result. */ if (borrow == 1) { qguess--; ptr1 = (unsigned char *) num1+qdig+len2; ptr2 = (unsigned char *) n2ptr+len2-1; carry = 0; for (count = 0; count < len2; count++) { val = (int) *ptr1 + (int) *ptr2-- + carry; if (val > 9) { val -= 10; carry = 1; } else carry = 0; *ptr1-- = val; } if (carry == 1) *ptr1 = (*ptr1 + 1) % 10; } /* We now know the quotient digit. */ *qptr++ = qguess; qdig++; } } /* Clean up and return the number. */ qval->n_sign = ( n1->n_sign == n2->n_sign ? PLUS : MINUS ); if (bc_is_zero (qval)) qval->n_sign = PLUS; _bc_rm_leading_zeros (qval); bc_free_num (quot); *quot = qval; /* Clean up temporary storage. */ free (mval); free (num1); free (num2); return 0; /* Everything is OK. */ } /* Division *and* modulo for numbers. This computes both NUM1 / NUM2 and NUM1 % NUM2 and puts the results in QUOT and REM, except that if QUOT is NULL then that store will be omitted. */ int bc_divmod (num1, num2, quot, rem, scale) bc_num num1, num2, *quot, *rem; int scale; { bc_num quotient = NULL; bc_num temp; int rscale; /* Check for correct numbers. */ if (bc_is_zero (num2)) return -1; /* Calculate final scale. */ rscale = MAX (num1->n_scale, num2->n_scale+scale); bc_init_num(&temp); /* Calculate it. */ bc_divide (num1, num2, &temp, scale); if (quot) quotient = bc_copy_num (temp); bc_multiply (temp, num2, &temp, rscale); bc_sub (num1, temp, rem, rscale); bc_free_num (&temp); if (quot) { bc_free_num (quot); *quot = quotient; } return 0; /* Everything is OK. */ } /* Modulo for numbers. This computes NUM1 % NUM2 and puts the result in RESULT. */ int bc_modulo (num1, num2, result, scale) bc_num num1, num2, *result; int scale; { return bc_divmod (num1, num2, NULL, result, scale); } /* Raise BASE to the EXPO power, reduced modulo MOD. The result is placed in RESULT. If a EXPO is not an integer, only the integer part is used. */ int bc_raisemod (base, expo, mod, result, scale) bc_num base, expo, mod, *result; int scale; { bc_num power, exponent, parity, temp; int rscale; /* Check for correct numbers. */ if (bc_is_zero(mod)) return -1; if (bc_is_neg(expo)) return -1; /* Set initial values. */ power = bc_copy_num (base); exponent = bc_copy_num (expo); temp = bc_copy_num (_one_); bc_init_num(&parity); /* Check the base for scale digits. */ if (base->n_scale != 0) bc_rt_warn ("non-zero scale in base"); /* Check the exponent for scale digits. */ if (exponent->n_scale != 0) { bc_rt_warn ("non-zero scale in exponent"); bc_divide (exponent, _one_, &exponent, 0); /*truncate */ } /* Check the modulus for scale digits. */ if (mod->n_scale != 0) bc_rt_warn ("non-zero scale in modulus"); /* Do the calculation. */ rscale = MAX(scale, base->n_scale); while ( !bc_is_zero(exponent) ) { (void) bc_divmod (exponent, _two_, &exponent, &parity, 0); if ( !bc_is_zero(parity) ) { bc_multiply (temp, power, &temp, rscale); (void) bc_modulo (temp, mod, &temp, scale); } bc_multiply (power, power, &power, rscale); (void) bc_modulo (power, mod, &power, scale); } /* Assign the value. */ bc_free_num (&power); bc_free_num (&exponent); bc_free_num (result); *result = temp; return 0; /* Everything is OK. */ } /* Raise NUM1 to the NUM2 power. The result is placed in RESULT. Maximum exponent is LONG_MAX. If a NUM2 is not an integer, only the integer part is used. */ void bc_raise (num1, num2, result, scale) bc_num num1, num2, *result; int scale; { bc_num temp, power; long exponent; int rscale; int pwrscale; int calcscale; char neg; /* Check the exponent for scale digits and convert to a long. */ if (num2->n_scale != 0) bc_rt_warn ("non-zero scale in exponent"); exponent = bc_num2long (num2); if (exponent == 0 && (num2->n_len > 1 || num2->n_value[0] != 0)) bc_rt_error ("exponent too large in raise"); /* Special case if exponent is a zero. */ if (exponent == 0) { bc_free_num (result); *result = bc_copy_num (_one_); return; } /* Other initializations. */ if (exponent < 0) { neg = TRUE; exponent = -exponent; rscale = scale; } else { neg = FALSE; rscale = MIN (num1->n_scale*exponent, MAX(scale, num1->n_scale)); } /* Set initial value of temp. */ power = bc_copy_num (num1); pwrscale = num1->n_scale; while ((exponent & 1) == 0) { pwrscale = 2*pwrscale; bc_multiply (power, power, &power, pwrscale); exponent = exponent >> 1; } temp = bc_copy_num (power); calcscale = pwrscale; exponent = exponent >> 1; /* Do the calculation. */ while (exponent > 0) { pwrscale = 2*pwrscale; bc_multiply (power, power, &power, pwrscale); if ((exponent & 1) == 1) { calcscale = pwrscale + calcscale; bc_multiply (temp, power, &temp, calcscale); } exponent = exponent >> 1; } /* Assign the value. */ if (neg) { bc_divide (_one_, temp, result, rscale); bc_free_num (&temp); } else { bc_free_num (result); *result = temp; if ((*result)->n_scale > rscale) (*result)->n_scale = rscale; } bc_free_num (&power); } /* Take the square root NUM and return it in NUM with SCALE digits after the decimal place. */ int bc_sqrt (num, scale) bc_num *num; int scale; { int rscale, cmp_res, done; int cscale; bc_num guess, guess1, point5, diff; /* Initial checks. */ cmp_res = bc_compare (*num, _zero_); if (cmp_res < 0) return 0; /* error */ else { if (cmp_res == 0) { bc_free_num (num); *num = bc_copy_num (_zero_); return 1; } } cmp_res = bc_compare (*num, _one_); if (cmp_res == 0) { bc_free_num (num); *num = bc_copy_num (_one_); return 1; } /* Initialize the variables. */ rscale = MAX (scale, (*num)->n_scale); bc_init_num(&guess); bc_init_num(&guess1); bc_init_num(&diff); point5 = bc_new_num (1,1); point5->n_value[1] = 5; /* Calculate the initial guess. */ if (cmp_res < 0) { /* The number is between 0 and 1. Guess should start at 1. */ guess = bc_copy_num (_one_); cscale = (*num)->n_scale; } else { /* The number is greater than 1. Guess should start at 10^(exp/2). */ bc_int2num (&guess,10); bc_int2num (&guess1,(*num)->n_len); bc_multiply (guess1, point5, &guess1, 0); guess1->n_scale = 0; bc_raise (guess, guess1, &guess, 0); bc_free_num (&guess1); cscale = 3; } /* Find the square root using Newton's algorithm. */ done = FALSE; while (!done) { bc_free_num (&guess1); guess1 = bc_copy_num (guess); bc_divide (*num, guess, &guess, cscale); bc_add (guess, guess1, &guess, 0); bc_multiply (guess, point5, &guess, cscale); bc_sub (guess, guess1, &diff, cscale+1); if (bc_is_near_zero (diff, cscale)) { if (cscale < rscale+1) cscale = MIN (cscale*3, rscale+1); else done = TRUE; } } /* Assign the number and clean up. */ bc_free_num (num); bc_divide (guess,_one_,num,rscale); bc_free_num (&guess); bc_free_num (&guess1); bc_free_num (&point5); bc_free_num (&diff); return 1; } /* The following routines provide output for bcd numbers package using the rules of POSIX bc for output. */ /* This structure is used for saving digits in the conversion process. */ typedef struct stk_rec { long digit; struct stk_rec *next; } stk_rec; /* The reference string for digits. */ static char ref_str[] = "0123456789ABCDEF"; /* A special output routine for "multi-character digits." Exactly SIZE characters must be output for the value VAL. If SPACE is non-zero, we must output one space before the number. OUT_CHAR is the actual routine for writing the characters. */ void bc_out_long (val, size, space, out_char) long val; int size, space; #ifdef __STDC__ void (*out_char)(int); #else void (*out_char)(); #endif { char digits[40]; int len, ix; if (space) (*out_char) (' '); sprintf (digits, "%ld", val); len = strlen (digits); while (size > len) { (*out_char) ('0'); size--; } for (ix=0; ix < len; ix++) (*out_char) (digits[ix]); } /* Output of a bcd number. NUM is written in base O_BASE using OUT_CHAR as the routine to do the actual output of the characters. */ void bc_out_num (num, o_base, out_char, leading_zero) bc_num num; int o_base; #ifdef __STDC__ void (*out_char)(int); #else void (*out_char)(); #endif int leading_zero; { char *nptr; int index, fdigit, pre_space; stk_rec *digits, *temp; bc_num int_part, frac_part, base, cur_dig, t_num, max_o_digit; /* The negative sign if needed. */ if (num->n_sign == MINUS) (*out_char) ('-'); /* Output the number. */ if (bc_is_zero (num)) (*out_char) ('0'); else if (o_base == 10) { /* The number is in base 10, do it the fast way. */ nptr = num->n_value; if (num->n_len > 1 || *nptr != 0) for (index=num->n_len; index>0; index--) (*out_char) (BCD_CHAR(*nptr++)); else nptr++; if (leading_zero && bc_is_zero (num)) (*out_char) ('0'); /* Now the fraction. */ if (num->n_scale > 0) { (*out_char) ('.'); for (index=0; index<num->n_scale; index++) (*out_char) (BCD_CHAR(*nptr++)); } } else { /* special case ... */ if (leading_zero && bc_is_zero (num)) (*out_char) ('0'); /* The number is some other base. */ digits = NULL; bc_init_num (&int_part); bc_divide (num, _one_, &int_part, 0); bc_init_num (&frac_part); bc_init_num (&cur_dig); bc_init_num (&base); bc_sub (num, int_part, &frac_part, 0); /* Make the INT_PART and FRAC_PART positive. */ int_part->n_sign = PLUS; frac_part->n_sign = PLUS; bc_int2num (&base, o_base); bc_init_num (&max_o_digit); bc_int2num (&max_o_digit, o_base-1); /* Get the digits of the integer part and push them on a stack. */ while (!bc_is_zero (int_part)) { bc_modulo (int_part, base, &cur_dig, 0); temp = (stk_rec *) malloc (sizeof(stk_rec)); if (temp == NULL) bc_out_of_memory(); temp->digit = bc_num2long (cur_dig); temp->next = digits; digits = temp; bc_divide (int_part, base, &int_part, 0); } /* Print the digits on the stack. */ if (digits != NULL) { /* Output the digits. */ while (digits != NULL) { temp = digits; digits = digits->next; if (o_base <= 16) (*out_char) (ref_str[ (int) temp->digit]); else bc_out_long (temp->digit, max_o_digit->n_len, 1, out_char); free (temp); } } /* Get and print the digits of the fraction part. */ if (num->n_scale > 0) { (*out_char) ('.'); pre_space = 0; t_num = bc_copy_num (_one_); while (t_num->n_len <= num->n_scale) { bc_multiply (frac_part, base, &frac_part, num->n_scale); fdigit = bc_num2long (frac_part); bc_int2num (&int_part, fdigit); bc_sub (frac_part, int_part, &frac_part, 0); if (o_base <= 16) (*out_char) (ref_str[fdigit]); else { bc_out_long (fdigit, max_o_digit->n_len, pre_space, out_char); pre_space = 1; } bc_multiply (t_num, base, &t_num, 0); } bc_free_num (&t_num); } /* Clean up. */ bc_free_num (&int_part); bc_free_num (&frac_part); bc_free_num (&base); bc_free_num (&cur_dig); bc_free_num (&max_o_digit); } } /* Convert a number NUM to a long. The function returns only the integer part of the number. For numbers that are too large to represent as a long, this function returns a zero. This can be detected by checking the NUM for zero after having a zero returned. */ long bc_num2long (num) bc_num num; { long val; char *nptr; int index; /* Extract the int value, ignore the fraction. */ val = 0; nptr = num->n_value; for (index=num->n_len; (index>0) && (val<=(LONG_MAX/BASE)); index--) val = val*BASE + *nptr++; /* Check for overflow. If overflow, return zero. */ if (index>0) val = 0; if (val < 0) val = 0; /* Return the value. */ if (num->n_sign == PLUS) return (val); else return (-val); } /* Convert an integer VAL to a bc number NUM. */ void bc_int2num (num, val) bc_num *num; int val; { char buffer[30]; char *bptr, *vptr; int ix = 1; char neg = 0; /* Sign. */ if (val < 0) { neg = 1; val = -val; } /* Get things going. */ bptr = buffer; *bptr++ = val % BASE; val = val / BASE; /* Extract remaining digits. */ while (val != 0) { *bptr++ = val % BASE; val = val / BASE; ix++; /* Count the digits. */ } /* Make the number. */ bc_free_num (num); *num = bc_new_num (ix, 0); if (neg) (*num)->n_sign = MINUS; /* Assign the digits. */ vptr = (*num)->n_value; while (ix-- > 0) *vptr++ = *--bptr; } /* Convert a numbers to a string. Base 10 only.*/ char *num2str (num) bc_num num; { char *str, *sptr; char *nptr; int index, signch; /* Allocate the string memory. */ signch = ( num->n_sign == PLUS ? 0 : 1 ); /* Number of sign chars. */ if (num->n_scale > 0) str = (char *) malloc (num->n_len + num->n_scale + 2 + signch); else str = (char *) malloc (num->n_len + 1 + signch); if (str == NULL) bc_out_of_memory(); /* The negative sign if needed. */ sptr = str; if (signch) *sptr++ = '-'; /* Load the whole number. */ nptr = num->n_value; for (index=num->n_len; index>0; index--) *sptr++ = BCD_CHAR(*nptr++); /* Now the fraction. */ if (num->n_scale > 0) { *sptr++ = '.'; for (index=0; index<num->n_scale; index++) *sptr++ = BCD_CHAR(*nptr++); } /* Terminate the string and return it! */ *sptr = '0'; return (str); } /* Convert strings to bc numbers. Base 10 only.*/ void bc_str2num (num, str, scale) bc_num *num; char *str; int scale; { int digits, strscale; char *ptr, *nptr; char zero_int; /* Prepare num. */ bc_free_num (num); /* Check for valid number and count digits. */ ptr = str; digits = 0; strscale = 0; zero_int = FALSE; if ( (*ptr == '+') || (*ptr == '-')) ptr++; /* Sign */ while (*ptr == '0') ptr++; /* Skip leading zeros. */ while (isdigit((int)*ptr)) ptr++, digits++; /* digits */ if (*ptr == '.') ptr++; /* decimal point */ while (isdigit((int)*ptr)) ptr++, strscale++; /* digits */ if ((*ptr != '0') || (digits+strscale == 0)) { *num = bc_copy_num (_zero_); return; } /* Adjust numbers and allocate storage and initialize fields. */ strscale = MIN(strscale, scale); if (digits == 0) { zero_int = TRUE; digits = 1; } *num = bc_new_num (digits, strscale); /* Build the whole number. */ ptr = str; if (*ptr == '-') { (*num)->n_sign = MINUS; ptr++; } else { (*num)->n_sign = PLUS; if (*ptr == '+') ptr++; } while (*ptr == '0') ptr++; /* Skip leading zeros. */ nptr = (*num)->n_value; if (zero_int) { *nptr++ = 0; digits = 0; } for (;digits > 0; digits--) *nptr++ = CH_VAL(*ptr++); /* Build the fractional part. */ if (strscale > 0) { ptr++; /* skip the decimal point! */ for (;strscale > 0; strscale--) *nptr++ = CH_VAL(*ptr++); } } /* pn prints the number NUM in base 10. */ static void out_char (int c) { putchar(c); } void pn (num) bc_num num; { bc_out_num (num, 10, out_char, 0); out_char ('n'); } /* pv prints a character array as if it was a string of bcd digits. */ void pv (name, num, len) char *name; unsigned char *num; int len; { int i; printf ("%s=", name); for (i=0; i<len; i++) printf ("%c",BCD_CHAR(num[i])); printf ("n"); } all the math operations are in there, i mean without knowing how all the structs are built, you can still get a pretty excellent idea on how bc does it's magic :( Quote
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