/* * ***************************************************************************** * * SPDX-License-Identifier: BSD-2-Clause * * Copyright (c) 2018-2021 Gavin D. Howard and contributors. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions are met: * * * Redistributions of source code must retain the above copyright notice, this * list of conditions and the following disclaimer. * * * Redistributions in binary form must reproduce the above copyright notice, * this list of conditions and the following disclaimer in the documentation * and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE * POSSIBILITY OF SUCH DAMAGE. * * ***************************************************************************** * * Code for the number type. * */ #include #include #include #include #include #include #include #include #include #include // Before you try to understand this code, see the development manual // (manuals/development.md#numbers). static void bc_num_m(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale); /** * Multiply two numbers and throw a math error if they overflow. * @param a The first operand. * @param b The second operand. * @return The product of the two operands. */ static inline size_t bc_num_mulOverflow(size_t a, size_t b) { size_t res = a * b; if (BC_ERR(BC_VM_MUL_OVERFLOW(a, b, res))) bc_err(BC_ERR_MATH_OVERFLOW); return res; } /** * Conditionally negate @a n based on @a neg. Algorithm taken from * https://graphics.stanford.edu/~seander/bithacks.html#ConditionalNegate . * @param n The value to turn into a signed value and negate. * @param neg The condition to negate or not. */ static inline ssize_t bc_num_neg(size_t n, bool neg) { return (((ssize_t) n) ^ -((ssize_t) neg)) + neg; } /** * Compare a BcNum against zero. * @param n The number to compare. * @return -1 if the number is less than 0, 1 if greater, and 0 if equal. */ ssize_t bc_num_cmpZero(const BcNum* n) { return bc_num_neg((n)->len != 0, BC_NUM_NEG(n)); } /** * Return the number of integer limbs in a BcNum. This is the opposite of rdx. * @param n The number to return the amount of integer limbs for. * @return The amount of integer limbs in @a n. */ static inline size_t bc_num_int(const BcNum* n) { return n->len ? n->len - BC_NUM_RDX_VAL(n) : 0; } /** * Expand a number's allocation capacity to at least req limbs. * @param n The number to expand. * @param req The number limbs to expand the allocation capacity to. */ static void bc_num_expand(BcNum* restrict n, size_t req) { assert(n != NULL); req = req >= BC_NUM_DEF_SIZE ? req : BC_NUM_DEF_SIZE; if (req > n->cap) { BC_SIG_LOCK; n->num = bc_vm_realloc(n->num, BC_NUM_SIZE(req)); n->cap = req; BC_SIG_UNLOCK; } } /** * Set a number to 0 with the specified scale. * @param n The number to set to zero. * @param scale The scale to set the number to. */ static void bc_num_setToZero(BcNum* restrict n, size_t scale) { assert(n != NULL); n->scale = scale; n->len = n->rdx = 0; } void bc_num_zero(BcNum* restrict n) { bc_num_setToZero(n, 0); } void bc_num_one(BcNum* restrict n) { bc_num_zero(n); n->len = 1; n->num[0] = 1; } /** * "Cleans" a number, which means reducing the length if the most significant * limbs are zero. * @param n The number to clean. */ static void bc_num_clean(BcNum* restrict n) { // Reduce the length. while (BC_NUM_NONZERO(n) && !n->num[n->len - 1]) { n->len -= 1; } // Special cases. if (BC_NUM_ZERO(n)) n->rdx = 0; else { // len must be at least as much as rdx. size_t rdx = BC_NUM_RDX_VAL(n); if (n->len < rdx) n->len = rdx; } } /** * Returns the log base 10 of @a i. I could have done this with floating-point * math, and in fact, I originally did. However, that was the only * floating-point code in the entire codebase, and I decided I didn't want any. * This is fast enough. Also, it might handle larger numbers better. * @param i The number to return the log base 10 of. * @return The log base 10 of @a i. */ static size_t bc_num_log10(size_t i) { size_t len; for (len = 1; i; i /= BC_BASE, ++len) { continue; } assert(len - 1 <= BC_BASE_DIGS + 1); return len - 1; } /** * Returns the number of decimal digits in a limb that are zero starting at the * most significant digits. This basically returns how much of the limb is used. * @param n The number. * @return The number of decimal digits that are 0 starting at the most * significant digits. */ static inline size_t bc_num_zeroDigits(const BcDig* n) { assert(*n >= 0); assert(((size_t) *n) < BC_BASE_POW); return BC_BASE_DIGS - bc_num_log10((size_t) *n); } /** * Return the total number of integer digits in a number. This is the opposite * of scale, like bc_num_int() is the opposite of rdx. * @param n The number. * @return The number of integer digits in @a n. */ static size_t bc_num_intDigits(const BcNum* n) { size_t digits = bc_num_int(n) * BC_BASE_DIGS; if (digits > 0) digits -= bc_num_zeroDigits(n->num + n->len - 1); return digits; } /** * Returns the number of limbs of a number that are non-zero starting at the * most significant limbs. This expects that there are *no* integer limbs in the * number because it is specifically to figure out how many zero limbs after the * decimal place to ignore. If there are zero limbs after non-zero limbs, they * are counted as non-zero limbs. * @param n The number. * @return The number of non-zero limbs after the decimal point. */ static size_t bc_num_nonZeroLen(const BcNum* restrict n) { size_t i, len = n->len; assert(len == BC_NUM_RDX_VAL(n)); for (i = len - 1; i < len && !n->num[i]; --i) { continue; } assert(i + 1 > 0); return i + 1; } /** * Performs a one-limb add with a carry. * @param a The first limb. * @param b The second limb. * @param carry An in/out parameter; the carry in from the previous add and the * carry out from this add. * @return The resulting limb sum. */ static BcDig bc_num_addDigits(BcDig a, BcDig b, bool* carry) { assert(((BcBigDig) BC_BASE_POW) * 2 == ((BcDig) BC_BASE_POW) * 2); assert(a < BC_BASE_POW); assert(b < BC_BASE_POW); a += b + *carry; *carry = (a >= BC_BASE_POW); if (*carry) a -= BC_BASE_POW; assert(a >= 0); assert(a < BC_BASE_POW); return a; } /** * Performs a one-limb subtract with a carry. * @param a The first limb. * @param b The second limb. * @param carry An in/out parameter; the carry in from the previous subtract * and the carry out from this subtract. * @return The resulting limb difference. */ static BcDig bc_num_subDigits(BcDig a, BcDig b, bool* carry) { assert(a < BC_BASE_POW); assert(b < BC_BASE_POW); b += *carry; *carry = (a < b); if (*carry) a += BC_BASE_POW; assert(a - b >= 0); assert(a - b < BC_BASE_POW); return a - b; } /** * Add two BcDig arrays and store the result in the first array. * @param a The first operand and out array. * @param b The second operand. * @param len The length of @a b. */ static void bc_num_addArrays(BcDig* restrict a, const BcDig* restrict b, size_t len) { size_t i; bool carry = false; for (i = 0; i < len; ++i) { a[i] = bc_num_addDigits(a[i], b[i], &carry); } // Take care of the extra limbs in the bigger array. for (; carry; ++i) { a[i] = bc_num_addDigits(a[i], 0, &carry); } } /** * Subtract two BcDig arrays and store the result in the first array. * @param a The first operand and out array. * @param b The second operand. * @param len The length of @a b. */ static void bc_num_subArrays(BcDig* restrict a, const BcDig* restrict b, size_t len) { size_t i; bool carry = false; for (i = 0; i < len; ++i) { a[i] = bc_num_subDigits(a[i], b[i], &carry); } // Take care of the extra limbs in the bigger array. for (; carry; ++i) { a[i] = bc_num_subDigits(a[i], 0, &carry); } } /** * Multiply a BcNum array by a one-limb number. This is a faster version of * multiplication for when we can use it. * @param a The BcNum to multiply by the one-limb number. * @param b The one limb of the one-limb number. * @param c The return parameter. */ static void bc_num_mulArray(const BcNum* restrict a, BcBigDig b, BcNum* restrict c) { size_t i; BcBigDig carry = 0; assert(b <= BC_BASE_POW); // Make sure the return parameter is big enough. if (a->len + 1 > c->cap) bc_num_expand(c, a->len + 1); // We want the entire return parameter to be zero for cleaning later. // NOLINTNEXTLINE memset(c->num, 0, BC_NUM_SIZE(c->cap)); // Actual multiplication loop. for (i = 0; i < a->len; ++i) { BcBigDig in = ((BcBigDig) a->num[i]) * b + carry; c->num[i] = in % BC_BASE_POW; carry = in / BC_BASE_POW; } assert(carry < BC_BASE_POW); // Finishing touches. c->num[i] = (BcDig) carry; c->len = a->len; c->len += (carry != 0); bc_num_clean(c); // Postconditions. assert(!BC_NUM_NEG(c) || BC_NUM_NONZERO(c)); assert(BC_NUM_RDX_VAL(c) <= c->len || !c->len); assert(!c->len || c->num[c->len - 1] || BC_NUM_RDX_VAL(c) == c->len); } /** * Divide a BcNum array by a one-limb number. This is a faster version of divide * for when we can use it. * @param a The BcNum to multiply by the one-limb number. * @param b The one limb of the one-limb number. * @param c The return parameter for the quotient. * @param rem The return parameter for the remainder. */ static void bc_num_divArray(const BcNum* restrict a, BcBigDig b, BcNum* restrict c, BcBigDig* rem) { size_t i; BcBigDig carry = 0; assert(c->cap >= a->len); // Actual division loop. for (i = a->len - 1; i < a->len; --i) { BcBigDig in = ((BcBigDig) a->num[i]) + carry * BC_BASE_POW; assert(in / b < BC_BASE_POW); c->num[i] = (BcDig) (in / b); carry = in % b; } // Finishing touches. c->len = a->len; bc_num_clean(c); *rem = carry; // Postconditions. assert(!BC_NUM_NEG(c) || BC_NUM_NONZERO(c)); assert(BC_NUM_RDX_VAL(c) <= c->len || !c->len); assert(!c->len || c->num[c->len - 1] || BC_NUM_RDX_VAL(c) == c->len); } /** * Compare two BcDig arrays and return >0 if @a b is greater, <0 if @a b is * less, and 0 if equal. Both @a a and @a b must have the same length. * @param a The first array. * @param b The second array. * @param len The minimum length of the arrays. */ static ssize_t bc_num_compare(const BcDig* restrict a, const BcDig* restrict b, size_t len) { size_t i; BcDig c = 0; for (i = len - 1; i < len && !(c = a[i] - b[i]); --i) { continue; } return bc_num_neg(i + 1, c < 0); } ssize_t bc_num_cmp(const BcNum* a, const BcNum* b) { size_t i, min, a_int, b_int, diff, ardx, brdx; BcDig* max_num; BcDig* min_num; bool a_max, neg = false; ssize_t cmp; assert(a != NULL && b != NULL); // Same num? Equal. if (a == b) return 0; // Easy cases. if (BC_NUM_ZERO(a)) return bc_num_neg(b->len != 0, !BC_NUM_NEG(b)); if (BC_NUM_ZERO(b)) return bc_num_cmpZero(a); if (BC_NUM_NEG(a)) { if (BC_NUM_NEG(b)) neg = true; else return -1; } else if (BC_NUM_NEG(b)) return 1; // Get the number of int limbs in each number and get the difference. a_int = bc_num_int(a); b_int = bc_num_int(b); a_int -= b_int; // If there's a difference, then just return the comparison. if (a_int) return neg ? -((ssize_t) a_int) : (ssize_t) a_int; // Get the rdx's and figure out the max. ardx = BC_NUM_RDX_VAL(a); brdx = BC_NUM_RDX_VAL(b); a_max = (ardx > brdx); // Set variables based on the above. if (a_max) { min = brdx; diff = ardx - brdx; max_num = a->num + diff; min_num = b->num; } else { min = ardx; diff = brdx - ardx; max_num = b->num + diff; min_num = a->num; } // Do a full limb-by-limb comparison. cmp = bc_num_compare(max_num, min_num, b_int + min); // If we found a difference, return it based on state. if (cmp) return bc_num_neg((size_t) cmp, !a_max == !neg); // If there was no difference, then the final step is to check which number // has greater or lesser limbs beyond the other's. for (max_num -= diff, i = diff - 1; i < diff; --i) { if (max_num[i]) return bc_num_neg(1, !a_max == !neg); } return 0; } void bc_num_truncate(BcNum* restrict n, size_t places) { size_t nrdx, places_rdx; if (!places) return; // Grab these needed values; places_rdx is the rdx equivalent to places like // rdx is to scale. nrdx = BC_NUM_RDX_VAL(n); places_rdx = nrdx ? nrdx - BC_NUM_RDX(n->scale - places) : 0; // We cannot truncate more places than we have. assert(places <= n->scale && (BC_NUM_ZERO(n) || places_rdx <= n->len)); n->scale -= places; BC_NUM_RDX_SET(n, nrdx - places_rdx); // Only when the number is nonzero do we need to do the hard stuff. if (BC_NUM_NONZERO(n)) { size_t pow; // This calculates how many decimal digits are in the least significant // limb. pow = n->scale % BC_BASE_DIGS; pow = pow ? BC_BASE_DIGS - pow : 0; pow = bc_num_pow10[pow]; n->len -= places_rdx; // We have to move limbs to maintain invariants. The limbs must begin at // the beginning of the BcNum array. // NOLINTNEXTLINE memmove(n->num, n->num + places_rdx, BC_NUM_SIZE(n->len)); // Clear the lower part of the last digit. if (BC_NUM_NONZERO(n)) n->num[0] -= n->num[0] % (BcDig) pow; bc_num_clean(n); } } void bc_num_extend(BcNum* restrict n, size_t places) { size_t nrdx, places_rdx; if (!places) return; // Easy case with zero; set the scale. if (BC_NUM_ZERO(n)) { n->scale += places; return; } // Grab these needed values; places_rdx is the rdx equivalent to places like // rdx is to scale. nrdx = BC_NUM_RDX_VAL(n); places_rdx = BC_NUM_RDX(places + n->scale) - nrdx; // This is the hard case. We need to expand the number, move the limbs, and // set the limbs that were just cleared. if (places_rdx) { bc_num_expand(n, bc_vm_growSize(n->len, places_rdx)); // NOLINTNEXTLINE memmove(n->num + places_rdx, n->num, BC_NUM_SIZE(n->len)); // NOLINTNEXTLINE memset(n->num, 0, BC_NUM_SIZE(places_rdx)); } // Finally, set scale and rdx. BC_NUM_RDX_SET(n, nrdx + places_rdx); n->scale += places; n->len += places_rdx; assert(BC_NUM_RDX_VAL(n) == BC_NUM_RDX(n->scale)); } /** * Retires (finishes) a multiplication or division operation. */ static void bc_num_retireMul(BcNum* restrict n, size_t scale, bool neg1, bool neg2) { // Make sure scale is correct. if (n->scale < scale) bc_num_extend(n, scale - n->scale); else bc_num_truncate(n, n->scale - scale); bc_num_clean(n); // We need to ensure rdx is correct. if (BC_NUM_NONZERO(n)) n->rdx = BC_NUM_NEG_VAL(n, !neg1 != !neg2); } /** * Splits a number into two BcNum's. This is used in Karatsuba. * @param n The number to split. * @param idx The index to split at. * @param a An out parameter; the low part of @a n. * @param b An out parameter; the high part of @a n. */ static void bc_num_split(const BcNum* restrict n, size_t idx, BcNum* restrict a, BcNum* restrict b) { // We want a and b to be clear. assert(BC_NUM_ZERO(a)); assert(BC_NUM_ZERO(b)); // The usual case. if (idx < n->len) { // Set the fields first. b->len = n->len - idx; a->len = idx; a->scale = b->scale = 0; BC_NUM_RDX_SET(a, 0); BC_NUM_RDX_SET(b, 0); assert(a->cap >= a->len); assert(b->cap >= b->len); // Copy the arrays. This is not necessary for safety, but it is faster, // for some reason. // NOLINTNEXTLINE memcpy(b->num, n->num + idx, BC_NUM_SIZE(b->len)); // NOLINTNEXTLINE memcpy(a->num, n->num, BC_NUM_SIZE(idx)); bc_num_clean(b); } // If the index is weird, just skip the split. else bc_num_copy(a, n); bc_num_clean(a); } /** * Copies a number into another, but shifts the rdx so that the result number * only sees the integer part of the source number. * @param n The number to copy. * @param r The result number with a shifted rdx, len, and num. */ static void bc_num_shiftRdx(const BcNum* restrict n, BcNum* restrict r) { size_t rdx = BC_NUM_RDX_VAL(n); r->len = n->len - rdx; r->cap = n->cap - rdx; r->num = n->num + rdx; BC_NUM_RDX_SET_NEG(r, 0, BC_NUM_NEG(n)); r->scale = 0; } /** * Shifts a number so that all of the least significant limbs of the number are * skipped. This must be undone by bc_num_unshiftZero(). * @param n The number to shift. */ static size_t bc_num_shiftZero(BcNum* restrict n) { size_t i; // If we don't have an integer, that is a problem, but it's also a bug // because the caller should have set everything up right. assert(!BC_NUM_RDX_VAL(n) || BC_NUM_ZERO(n)); for (i = 0; i < n->len && !n->num[i]; ++i) { continue; } n->len -= i; n->num += i; return i; } /** * Undo the damage done by bc_num_unshiftZero(). This must be called like all * cleanup functions: after a label used by setjmp() and longjmp(). * @param n The number to unshift. * @param places_rdx The amount the number was originally shift. */ static void bc_num_unshiftZero(BcNum* restrict n, size_t places_rdx) { n->len += places_rdx; n->num -= places_rdx; } /** * Shifts the digits right within a number by no more than BC_BASE_DIGS - 1. * This is the final step on shifting numbers with the shift operators. It * depends on the caller to shift the limbs properly because all it does is * ensure that the rdx point is realigned to be between limbs. * @param n The number to shift digits in. * @param dig The number of places to shift right. */ static void bc_num_shift(BcNum* restrict n, BcBigDig dig) { size_t i, len = n->len; BcBigDig carry = 0, pow; BcDig* ptr = n->num; assert(dig < BC_BASE_DIGS); // Figure out the parameters for division. pow = bc_num_pow10[dig]; dig = bc_num_pow10[BC_BASE_DIGS - dig]; // Run a series of divisions and mods with carries across the entire number // array. This effectively shifts everything over. for (i = len - 1; i < len; --i) { BcBigDig in, temp; in = ((BcBigDig) ptr[i]); temp = carry * dig; carry = in % pow; ptr[i] = ((BcDig) (in / pow)) + (BcDig) temp; } assert(!carry); } /** * Shift a number left by a certain number of places. This is the workhorse of * the left shift operator. * @param n The number to shift left. * @param places The amount of places to shift @a n left by. */ static void bc_num_shiftLeft(BcNum* restrict n, size_t places) { BcBigDig dig; size_t places_rdx; bool shift; if (!places) return; // Make sure to grow the number if necessary. if (places > n->scale) { size_t size = bc_vm_growSize(BC_NUM_RDX(places - n->scale), n->len); if (size > SIZE_MAX - 1) bc_err(BC_ERR_MATH_OVERFLOW); } // If zero, we can just set the scale and bail. if (BC_NUM_ZERO(n)) { if (n->scale >= places) n->scale -= places; else n->scale = 0; return; } // When I changed bc to have multiple digits per limb, this was the hardest // code to change. This and shift right. Make sure you understand this // before attempting anything. // This is how many limbs we will shift. dig = (BcBigDig) (places % BC_BASE_DIGS); shift = (dig != 0); // Convert places to a rdx value. places_rdx = BC_NUM_RDX(places); // If the number is not an integer, we need special care. The reason an // integer doesn't is because left shift would only extend the integer, // whereas a non-integer might have its fractional part eliminated or only // partially eliminated. if (n->scale) { size_t nrdx = BC_NUM_RDX_VAL(n); // If the number's rdx is bigger, that's the hard case. if (nrdx >= places_rdx) { size_t mod = n->scale % BC_BASE_DIGS, revdig; // We want mod to be in the range [1, BC_BASE_DIGS], not // [0, BC_BASE_DIGS). mod = mod ? mod : BC_BASE_DIGS; // We need to reverse dig to get the actual number of digits. revdig = dig ? BC_BASE_DIGS - dig : 0; // If the two overflow BC_BASE_DIGS, we need to move an extra place. if (mod + revdig > BC_BASE_DIGS) places_rdx = 1; else places_rdx = 0; } else places_rdx -= nrdx; } // If this is non-zero, we need an extra place, so expand, move, and set. if (places_rdx) { bc_num_expand(n, bc_vm_growSize(n->len, places_rdx)); // NOLINTNEXTLINE memmove(n->num + places_rdx, n->num, BC_NUM_SIZE(n->len)); // NOLINTNEXTLINE memset(n->num, 0, BC_NUM_SIZE(places_rdx)); n->len += places_rdx; } // Set the scale appropriately. if (places > n->scale) { n->scale = 0; BC_NUM_RDX_SET(n, 0); } else { n->scale -= places; BC_NUM_RDX_SET(n, BC_NUM_RDX(n->scale)); } // Finally, shift within limbs. if (shift) bc_num_shift(n, BC_BASE_DIGS - dig); bc_num_clean(n); } void bc_num_shiftRight(BcNum* restrict n, size_t places) { BcBigDig dig; size_t places_rdx, scale, scale_mod, int_len, expand; bool shift; if (!places) return; // If zero, we can just set the scale and bail. if (BC_NUM_ZERO(n)) { n->scale += places; bc_num_expand(n, BC_NUM_RDX(n->scale)); return; } // Amount within a limb we have to shift by. dig = (BcBigDig) (places % BC_BASE_DIGS); shift = (dig != 0); scale = n->scale; // Figure out how the scale is affected. scale_mod = scale % BC_BASE_DIGS; scale_mod = scale_mod ? scale_mod : BC_BASE_DIGS; // We need to know the int length and rdx for places. int_len = bc_num_int(n); places_rdx = BC_NUM_RDX(places); // If we are going to shift past a limb boundary or not, set accordingly. if (scale_mod + dig > BC_BASE_DIGS) { expand = places_rdx - 1; places_rdx = 1; } else { expand = places_rdx; places_rdx = 0; } // Clamp expanding. if (expand > int_len) expand -= int_len; else expand = 0; // Extend, expand, and zero. bc_num_extend(n, places_rdx * BC_BASE_DIGS); bc_num_expand(n, bc_vm_growSize(expand, n->len)); // NOLINTNEXTLINE memset(n->num + n->len, 0, BC_NUM_SIZE(expand)); // Set the fields. n->len += expand; n->scale = 0; BC_NUM_RDX_SET(n, 0); // Finally, shift within limbs. if (shift) bc_num_shift(n, dig); n->scale = scale + places; BC_NUM_RDX_SET(n, BC_NUM_RDX(n->scale)); bc_num_clean(n); assert(BC_NUM_RDX_VAL(n) <= n->len && n->len <= n->cap); assert(BC_NUM_RDX_VAL(n) == BC_NUM_RDX(n->scale)); } /** * Invert @a into @a b at the current scale. * @param a The number to invert. * @param b The return parameter. This must be preallocated. * @param scale The current scale. */ static inline void bc_num_inv(BcNum* a, BcNum* b, size_t scale) { assert(BC_NUM_NONZERO(a)); bc_num_div(&vm.one, a, b, scale); } /** * Tests if a number is a integer with scale or not. Returns true if the number * is not an integer. If it is, its integer shifted form is copied into the * result parameter for use where only integers are allowed. * @param n The integer to test and shift. * @param r The number to store the shifted result into. This number should * *not* be allocated. * @return True if the number is a non-integer, false otherwise. */ static bool bc_num_nonInt(const BcNum* restrict n, BcNum* restrict r) { bool zero; size_t i, rdx = BC_NUM_RDX_VAL(n); if (!rdx) { // NOLINTNEXTLINE memcpy(r, n, sizeof(BcNum)); return false; } zero = true; for (i = 0; zero && i < rdx; ++i) { zero = (n->num[i] == 0); } if (BC_ERR(!zero)) return true; bc_num_shiftRdx(n, r); return false; } #if BC_ENABLE_EXTRA_MATH /** * Execute common code for an operater that needs an integer for the second * operand and return the integer operand as a BcBigDig. * @param a The first operand. * @param b The second operand. * @param c The result operand. * @return The second operand as a hardware integer. */ static BcBigDig bc_num_intop(const BcNum* a, const BcNum* b, BcNum* restrict c) { BcNum temp; if (BC_ERR(bc_num_nonInt(b, &temp))) bc_err(BC_ERR_MATH_NON_INTEGER); bc_num_copy(c, a); return bc_num_bigdig(&temp); } #endif // BC_ENABLE_EXTRA_MATH /** * This is the actual implementation of add *and* subtract. Since this function * doesn't need to use scale (per the bc spec), I am hijacking it to say whether * it's doing an add or a subtract. And then I convert substraction to addition * of negative second operand. This is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param sub Non-zero for a subtract, zero for an add. */ static void bc_num_as(BcNum* a, BcNum* b, BcNum* restrict c, size_t sub) { BcDig* ptr_c; BcDig* ptr_l; BcDig* ptr_r; size_t i, min_rdx, max_rdx, diff, a_int, b_int, min_len, max_len, max_int; size_t len_l, len_r, ardx, brdx; bool b_neg, do_sub, do_rev_sub, carry, c_neg; if (BC_NUM_ZERO(b)) { bc_num_copy(c, a); return; } if (BC_NUM_ZERO(a)) { bc_num_copy(c, b); c->rdx = BC_NUM_NEG_VAL(c, BC_NUM_NEG(b) != sub); return; } // Invert sign of b if it is to be subtracted. This operation must // precede the tests for any of the operands being zero. b_neg = (BC_NUM_NEG(b) != sub); // Figure out if we will actually add the numbers if their signs are equal // or subtract. do_sub = (BC_NUM_NEG(a) != b_neg); a_int = bc_num_int(a); b_int = bc_num_int(b); max_int = BC_MAX(a_int, b_int); // Figure out which number will have its last limbs copied (for addition) or // subtracted (for subtraction). ardx = BC_NUM_RDX_VAL(a); brdx = BC_NUM_RDX_VAL(b); min_rdx = BC_MIN(ardx, brdx); max_rdx = BC_MAX(ardx, brdx); diff = max_rdx - min_rdx; max_len = max_int + max_rdx; if (do_sub) { // Check whether b has to be subtracted from a or a from b. if (a_int != b_int) do_rev_sub = (a_int < b_int); else if (ardx > brdx) { do_rev_sub = (bc_num_compare(a->num + diff, b->num, b->len) < 0); } else do_rev_sub = (bc_num_compare(a->num, b->num + diff, a->len) <= 0); } else { // The result array of the addition might come out one element // longer than the bigger of the operand arrays. max_len += 1; do_rev_sub = (a_int < b_int); } assert(max_len <= c->cap); // Cache values for simple code later. if (do_rev_sub) { ptr_l = b->num; ptr_r = a->num; len_l = b->len; len_r = a->len; } else { ptr_l = a->num; ptr_r = b->num; len_l = a->len; len_r = b->len; } ptr_c = c->num; carry = false; // This is true if the numbers have a different number of limbs after the // decimal point. if (diff) { // If the rdx values of the operands do not match, the result will // have low end elements that are the positive or negative trailing // elements of the operand with higher rdx value. if ((ardx > brdx) != do_rev_sub) { // !do_rev_sub && ardx > brdx || do_rev_sub && brdx > ardx // The left operand has BcDig values that need to be copied, // either from a or from b (in case of a reversed subtraction). // NOLINTNEXTLINE memcpy(ptr_c, ptr_l, BC_NUM_SIZE(diff)); ptr_l += diff; len_l -= diff; } else { // The right operand has BcDig values that need to be copied // or subtracted from zero (in case of a subtraction). if (do_sub) { // do_sub (do_rev_sub && ardx > brdx || // !do_rev_sub && brdx > ardx) for (i = 0; i < diff; i++) { ptr_c[i] = bc_num_subDigits(0, ptr_r[i], &carry); } } else { // !do_sub && brdx > ardx // NOLINTNEXTLINE memcpy(ptr_c, ptr_r, BC_NUM_SIZE(diff)); } // Future code needs to ignore the limbs we just did. ptr_r += diff; len_r -= diff; } // The return value pointer needs to ignore what we just did. ptr_c += diff; } // This is the length that can be directly added/subtracted. min_len = BC_MIN(len_l, len_r); // After dealing with possible low array elements that depend on only one // operand above, the actual add or subtract can be performed as if the rdx // of both operands was the same. // // Inlining takes care of eliminating constant zero arguments to // addDigit/subDigit (checked in disassembly of resulting bc binary // compiled with gcc and clang). if (do_sub) { // Actual subtraction. for (i = 0; i < min_len; ++i) { ptr_c[i] = bc_num_subDigits(ptr_l[i], ptr_r[i], &carry); } // Finishing the limbs beyond the direct subtraction. for (; i < len_l; ++i) { ptr_c[i] = bc_num_subDigits(ptr_l[i], 0, &carry); } } else { // Actual addition. for (i = 0; i < min_len; ++i) { ptr_c[i] = bc_num_addDigits(ptr_l[i], ptr_r[i], &carry); } // Finishing the limbs beyond the direct addition. for (; i < len_l; ++i) { ptr_c[i] = bc_num_addDigits(ptr_l[i], 0, &carry); } // Addition can create an extra limb. We take care of that here. ptr_c[i] = bc_num_addDigits(0, 0, &carry); } assert(carry == false); // The result has the same sign as a, unless the operation was a // reverse subtraction (b - a). c_neg = BC_NUM_NEG(a) != (do_sub && do_rev_sub); BC_NUM_RDX_SET_NEG(c, max_rdx, c_neg); c->len = max_len; c->scale = BC_MAX(a->scale, b->scale); bc_num_clean(c); } /** * The simple multiplication that karatsuba dishes out to when the length of the * numbers gets low enough. This doesn't use scale because it treats the * operands as though they are integers. * @param a The first operand. * @param b The second operand. * @param c The return parameter. */ static void bc_num_m_simp(const BcNum* a, const BcNum* b, BcNum* restrict c) { size_t i, alen = a->len, blen = b->len, clen; BcDig* ptr_a = a->num; BcDig* ptr_b = b->num; BcDig* ptr_c; BcBigDig sum = 0, carry = 0; assert(sizeof(sum) >= sizeof(BcDig) * 2); assert(!BC_NUM_RDX_VAL(a) && !BC_NUM_RDX_VAL(b)); // Make sure c is big enough. clen = bc_vm_growSize(alen, blen); bc_num_expand(c, bc_vm_growSize(clen, 1)); // If we don't memset, then we might have uninitialized data use later. ptr_c = c->num; // NOLINTNEXTLINE memset(ptr_c, 0, BC_NUM_SIZE(c->cap)); // This is the actual multiplication loop. It uses the lattice form of long // multiplication (see the explanation on the web page at // https://knilt.arcc.albany.edu/What_is_Lattice_Multiplication or the // explanation at Wikipedia). for (i = 0; i < clen; ++i) { ssize_t sidx = (ssize_t) (i - blen + 1); size_t j, k; // These are the start indices. j = (size_t) BC_MAX(0, sidx); k = BC_MIN(i, blen - 1); // On every iteration of this loop, a multiplication happens, then the // sum is automatically calculated. for (; j < alen && k < blen; ++j, --k) { sum += ((BcBigDig) ptr_a[j]) * ((BcBigDig) ptr_b[k]); if (sum >= ((BcBigDig) BC_BASE_POW) * BC_BASE_POW) { carry += sum / BC_BASE_POW; sum %= BC_BASE_POW; } } // Calculate the carry. if (sum >= BC_BASE_POW) { carry += sum / BC_BASE_POW; sum %= BC_BASE_POW; } // Store and set up for next iteration. ptr_c[i] = (BcDig) sum; assert(ptr_c[i] < BC_BASE_POW); sum = carry; carry = 0; } // This should always be true because there should be no carry on the last // digit; multiplication never goes above the sum of both lengths. assert(!sum); c->len = clen; } /** * Does a shifted add or subtract for Karatsuba below. This calls either * bc_num_addArrays() or bc_num_subArrays(). * @param n An in/out parameter; the first operand and return parameter. * @param a The second operand. * @param shift The amount to shift @a n by when adding/subtracting. * @param op The function to call, either bc_num_addArrays() or * bc_num_subArrays(). */ static void bc_num_shiftAddSub(BcNum* restrict n, const BcNum* restrict a, size_t shift, BcNumShiftAddOp op) { assert(n->len >= shift + a->len); assert(!BC_NUM_RDX_VAL(n) && !BC_NUM_RDX_VAL(a)); op(n->num + shift, a->num, a->len); } /** * Implements the Karatsuba algorithm. */ static void bc_num_k(const BcNum* a, const BcNum* b, BcNum* restrict c) { size_t max, max2, total; BcNum l1, h1, l2, h2, m2, m1, z0, z1, z2, temp; BcDig* digs; BcDig* dig_ptr; BcNumShiftAddOp op; bool aone = BC_NUM_ONE(a); assert(BC_NUM_ZERO(c)); if (BC_NUM_ZERO(a) || BC_NUM_ZERO(b)) return; if (aone || BC_NUM_ONE(b)) { bc_num_copy(c, aone ? b : a); if ((aone && BC_NUM_NEG(a)) || BC_NUM_NEG(b)) BC_NUM_NEG_TGL(c); return; } // Shell out to the simple algorithm with certain conditions. if (a->len < BC_NUM_KARATSUBA_LEN || b->len < BC_NUM_KARATSUBA_LEN) { bc_num_m_simp(a, b, c); return; } // We need to calculate the max size of the numbers that can result from the // operations. max = BC_MAX(a->len, b->len); max = BC_MAX(max, BC_NUM_DEF_SIZE); max2 = (max + 1) / 2; // Calculate the space needed for all of the temporary allocations. We do // this to just allocate once. total = bc_vm_arraySize(BC_NUM_KARATSUBA_ALLOCS, max); BC_SIG_LOCK; // Allocate space for all of the temporaries. digs = dig_ptr = bc_vm_malloc(BC_NUM_SIZE(total)); // Set up the temporaries. bc_num_setup(&l1, dig_ptr, max); dig_ptr += max; bc_num_setup(&h1, dig_ptr, max); dig_ptr += max; bc_num_setup(&l2, dig_ptr, max); dig_ptr += max; bc_num_setup(&h2, dig_ptr, max); dig_ptr += max; bc_num_setup(&m1, dig_ptr, max); dig_ptr += max; bc_num_setup(&m2, dig_ptr, max); // Some temporaries need the ability to grow, so we allocate them // separately. max = bc_vm_growSize(max, 1); bc_num_init(&z0, max); bc_num_init(&z1, max); bc_num_init(&z2, max); max = bc_vm_growSize(max, max) + 1; bc_num_init(&temp, max); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // First, set up c. bc_num_expand(c, max); c->len = max; // NOLINTNEXTLINE memset(c->num, 0, BC_NUM_SIZE(c->len)); // Split the parameters. bc_num_split(a, max2, &l1, &h1); bc_num_split(b, max2, &l2, &h2); // Do the subtraction. bc_num_sub(&h1, &l1, &m1, 0); bc_num_sub(&l2, &h2, &m2, 0); // The if statements below are there for efficiency reasons. The best way to // understand them is to understand the Karatsuba algorithm because now that // the ollocations and splits are done, the algorithm is pretty // straightforward. if (BC_NUM_NONZERO(&h1) && BC_NUM_NONZERO(&h2)) { assert(BC_NUM_RDX_VALID_NP(h1)); assert(BC_NUM_RDX_VALID_NP(h2)); bc_num_m(&h1, &h2, &z2, 0); bc_num_clean(&z2); bc_num_shiftAddSub(c, &z2, max2 * 2, bc_num_addArrays); bc_num_shiftAddSub(c, &z2, max2, bc_num_addArrays); } if (BC_NUM_NONZERO(&l1) && BC_NUM_NONZERO(&l2)) { assert(BC_NUM_RDX_VALID_NP(l1)); assert(BC_NUM_RDX_VALID_NP(l2)); bc_num_m(&l1, &l2, &z0, 0); bc_num_clean(&z0); bc_num_shiftAddSub(c, &z0, max2, bc_num_addArrays); bc_num_shiftAddSub(c, &z0, 0, bc_num_addArrays); } if (BC_NUM_NONZERO(&m1) && BC_NUM_NONZERO(&m2)) { assert(BC_NUM_RDX_VALID_NP(m1)); assert(BC_NUM_RDX_VALID_NP(m1)); bc_num_m(&m1, &m2, &z1, 0); bc_num_clean(&z1); op = (BC_NUM_NEG_NP(m1) != BC_NUM_NEG_NP(m2)) ? bc_num_subArrays : bc_num_addArrays; bc_num_shiftAddSub(c, &z1, max2, op); } err: BC_SIG_MAYLOCK; free(digs); bc_num_free(&temp); bc_num_free(&z2); bc_num_free(&z1); bc_num_free(&z0); BC_LONGJMP_CONT; } /** * Does checks for Karatsuba. It also changes things to ensure that the * Karatsuba and simple multiplication can treat the numbers as integers. This * is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_m(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcNum cpa, cpb; size_t ascale, bscale, ardx, brdx, azero = 0, bzero = 0, zero, len, rscale; assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_zero(c); ascale = a->scale; bscale = b->scale; // This sets the final scale according to the bc spec. scale = BC_MAX(scale, ascale); scale = BC_MAX(scale, bscale); rscale = ascale + bscale; scale = BC_MIN(rscale, scale); // If this condition is true, we can use bc_num_mulArray(), which would be // much faster. if ((a->len == 1 || b->len == 1) && !a->rdx && !b->rdx) { BcNum* operand; BcBigDig dig; // Set the correct operands. if (a->len == 1) { dig = (BcBigDig) a->num[0]; operand = b; } else { dig = (BcBigDig) b->num[0]; operand = a; } bc_num_mulArray(operand, dig, c); // Need to make sure the sign is correct. if (BC_NUM_NONZERO(c)) { c->rdx = BC_NUM_NEG_VAL(c, BC_NUM_NEG(a) != BC_NUM_NEG(b)); } return; } assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); BC_SIG_LOCK; // We need copies because of all of the mutation needed to make Karatsuba // think the numbers are integers. bc_num_init(&cpa, a->len + BC_NUM_RDX_VAL(a)); bc_num_init(&cpb, b->len + BC_NUM_RDX_VAL(b)); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; bc_num_copy(&cpa, a); bc_num_copy(&cpb, b); assert(BC_NUM_RDX_VALID_NP(cpa)); assert(BC_NUM_RDX_VALID_NP(cpb)); BC_NUM_NEG_CLR_NP(cpa); BC_NUM_NEG_CLR_NP(cpb); assert(BC_NUM_RDX_VALID_NP(cpa)); assert(BC_NUM_RDX_VALID_NP(cpb)); // These are what makes them appear like integers. ardx = BC_NUM_RDX_VAL_NP(cpa) * BC_BASE_DIGS; bc_num_shiftLeft(&cpa, ardx); brdx = BC_NUM_RDX_VAL_NP(cpb) * BC_BASE_DIGS; bc_num_shiftLeft(&cpb, brdx); // We need to reset the jump here because azero and bzero are used in the // cleanup, and local variables are not guaranteed to be the same after a // jump. BC_SIG_LOCK; BC_UNSETJMP; // We want to ignore zero limbs. azero = bc_num_shiftZero(&cpa); bzero = bc_num_shiftZero(&cpb); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; bc_num_clean(&cpa); bc_num_clean(&cpb); bc_num_k(&cpa, &cpb, c); // The return parameter needs to have its scale set. This is the start. It // also needs to be shifted by the same amount as a and b have limbs after // the decimal point. zero = bc_vm_growSize(azero, bzero); len = bc_vm_growSize(c->len, zero); bc_num_expand(c, len); // Shift c based on the limbs after the decimal point in a and b. bc_num_shiftLeft(c, (len - c->len) * BC_BASE_DIGS); bc_num_shiftRight(c, ardx + brdx); bc_num_retireMul(c, scale, BC_NUM_NEG(a), BC_NUM_NEG(b)); err: BC_SIG_MAYLOCK; bc_num_unshiftZero(&cpb, bzero); bc_num_free(&cpb); bc_num_unshiftZero(&cpa, azero); bc_num_free(&cpa); BC_LONGJMP_CONT; } /** * Returns true if the BcDig array has non-zero limbs, false otherwise. * @param a The array to test. * @param len The length of the array. * @return True if @a has any non-zero limbs, false otherwise. */ static bool bc_num_nonZeroDig(BcDig* restrict a, size_t len) { size_t i; bool nonzero = false; for (i = len - 1; !nonzero && i < len; --i) { nonzero = (a[i] != 0); } return nonzero; } /** * Compares a BcDig array against a BcNum. This is especially suited for * division. Returns >0 if @a a is greater than @a b, <0 if it is less, and =0 * if they are equal. * @param a The array. * @param b The number. * @param len The length to assume the arrays are. This is always less than the * actual length because of how this is implemented. */ static ssize_t bc_num_divCmp(const BcDig* a, const BcNum* b, size_t len) { ssize_t cmp; if (b->len > len && a[len]) cmp = bc_num_compare(a, b->num, len + 1); else if (b->len <= len) { if (a[len]) cmp = 1; else cmp = bc_num_compare(a, b->num, len); } else cmp = -1; return cmp; } /** * Extends the two operands of a division by BC_BASE_DIGS minus the number of * digits in the divisor estimate. In other words, it is shifting the numbers in * order to force the divisor estimate to fill the limb. * @param a The first operand. * @param b The second operand. * @param divisor The divisor estimate. */ static void bc_num_divExtend(BcNum* restrict a, BcNum* restrict b, BcBigDig divisor) { size_t pow; assert(divisor < BC_BASE_POW); pow = BC_BASE_DIGS - bc_num_log10((size_t) divisor); bc_num_shiftLeft(a, pow); bc_num_shiftLeft(b, pow); } /** * Actually does division. This is a rewrite of my original code by Stefan Esser * from FreeBSD. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_d_long(BcNum* restrict a, BcNum* restrict b, BcNum* restrict c, size_t scale) { BcBigDig divisor; size_t len, end, i, rdx; BcNum cpb; bool nonzero = false; assert(b->len < a->len); len = b->len; end = a->len - len; assert(len >= 1); // This is a final time to make sure c is big enough and that its array is // properly zeroed. bc_num_expand(c, a->len); // NOLINTNEXTLINE memset(c->num, 0, c->cap * sizeof(BcDig)); // Setup. BC_NUM_RDX_SET(c, BC_NUM_RDX_VAL(a)); c->scale = a->scale; c->len = a->len; // This is pulling the most significant limb of b in order to establish a // good "estimate" for the actual divisor. divisor = (BcBigDig) b->num[len - 1]; // The entire bit of code in this if statement is to tighten the estimate of // the divisor. The condition asks if b has any other non-zero limbs. if (len > 1 && bc_num_nonZeroDig(b->num, len - 1)) { // This takes a little bit of understanding. The "10*BC_BASE_DIGS/6+1" // results in either 16 for 64-bit 9-digit limbs or 7 for 32-bit 4-digit // limbs. Then it shifts a 1 by that many, which in both cases, puts the // result above *half* of the max value a limb can store. Basically, // this quickly calculates if the divisor is greater than half the max // of a limb. nonzero = (divisor > 1 << ((10 * BC_BASE_DIGS) / 6 + 1)); // If the divisor is *not* greater than half the limb... if (!nonzero) { // Extend the parameters by the number of missing digits in the // divisor. bc_num_divExtend(a, b, divisor); // Check bc_num_d(). In there, we grow a again and again. We do it // again here; we *always* want to be sure it is big enough. len = BC_MAX(a->len, b->len); bc_num_expand(a, len + 1); // Make a have a zero most significant limb to match the len. if (len + 1 > a->len) a->len = len + 1; // Grab the new divisor estimate, new because the shift has made it // different. len = b->len; end = a->len - len; divisor = (BcBigDig) b->num[len - 1]; nonzero = bc_num_nonZeroDig(b->num, len - 1); } } // If b has other nonzero limbs, we want the divisor to be one higher, so // that it is an upper bound. divisor += nonzero; // Make sure c can fit the new length. bc_num_expand(c, a->len); // NOLINTNEXTLINE memset(c->num, 0, BC_NUM_SIZE(c->cap)); assert(c->scale >= scale); rdx = BC_NUM_RDX_VAL(c) - BC_NUM_RDX(scale); BC_SIG_LOCK; bc_num_init(&cpb, len + 1); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // This is the actual division loop. for (i = end - 1; i < end && i >= rdx && BC_NUM_NONZERO(a); --i) { ssize_t cmp; BcDig* n; BcBigDig result; n = a->num + i; assert(n >= a->num); result = 0; cmp = bc_num_divCmp(n, b, len); // This is true if n is greater than b, which means that division can // proceed, so this inner loop is the part that implements one instance // of the division. while (cmp >= 0) { BcBigDig n1, dividend, quotient; // These should be named obviously enough. Just imagine that it's a // division of one limb. Because that's what it is. n1 = (BcBigDig) n[len]; dividend = n1 * BC_BASE_POW + (BcBigDig) n[len - 1]; quotient = (dividend / divisor); // If this is true, then we can just subtract. Remember: setting // quotient to 1 is not bad because we already know that n is // greater than b. if (quotient <= 1) { quotient = 1; bc_num_subArrays(n, b->num, len); } else { assert(quotient <= BC_BASE_POW); // We need to multiply and subtract for a quotient above 1. bc_num_mulArray(b, (BcBigDig) quotient, &cpb); bc_num_subArrays(n, cpb.num, cpb.len); } // The result is the *real* quotient, by the way, but it might take // multiple trips around this loop to get it. result += quotient; assert(result <= BC_BASE_POW); // And here's why it might take multiple trips: n might *still* be // greater than b. So we have to loop again. That's what this is // setting up for: the condition of the while loop. if (nonzero) cmp = bc_num_divCmp(n, b, len); else cmp = -1; } assert(result < BC_BASE_POW); // Store the actual limb quotient. c->num[i] = (BcDig) result; } err: BC_SIG_MAYLOCK; bc_num_free(&cpb); BC_LONGJMP_CONT; } /** * Implements division. This is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_d(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { size_t len, cpardx; BcNum cpa, cpb; if (BC_NUM_ZERO(b)) bc_err(BC_ERR_MATH_DIVIDE_BY_ZERO); if (BC_NUM_ZERO(a)) { bc_num_setToZero(c, scale); return; } if (BC_NUM_ONE(b)) { bc_num_copy(c, a); bc_num_retireMul(c, scale, BC_NUM_NEG(a), BC_NUM_NEG(b)); return; } // If this is true, we can use bc_num_divArray(), which would be faster. if (!BC_NUM_RDX_VAL(a) && !BC_NUM_RDX_VAL(b) && b->len == 1 && !scale) { BcBigDig rem; bc_num_divArray(a, (BcBigDig) b->num[0], c, &rem); bc_num_retireMul(c, scale, BC_NUM_NEG(a), BC_NUM_NEG(b)); return; } len = bc_num_divReq(a, b, scale); BC_SIG_LOCK; // Initialize copies of the parameters. We want the length of the first // operand copy to be as big as the result because of the way the division // is implemented. bc_num_init(&cpa, len); bc_num_copy(&cpa, a); bc_num_createCopy(&cpb, b); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; len = b->len; // Like the above comment, we want the copy of the first parameter to be // larger than the second parameter. if (len > cpa.len) { bc_num_expand(&cpa, bc_vm_growSize(len, 2)); bc_num_extend(&cpa, (len - cpa.len) * BC_BASE_DIGS); } cpardx = BC_NUM_RDX_VAL_NP(cpa); cpa.scale = cpardx * BC_BASE_DIGS; // This is just setting up the scale in preparation for the division. bc_num_extend(&cpa, b->scale); cpardx = BC_NUM_RDX_VAL_NP(cpa) - BC_NUM_RDX(b->scale); BC_NUM_RDX_SET_NP(cpa, cpardx); cpa.scale = cpardx * BC_BASE_DIGS; // Once again, just setting things up, this time to match scale. if (scale > cpa.scale) { bc_num_extend(&cpa, scale); cpardx = BC_NUM_RDX_VAL_NP(cpa); cpa.scale = cpardx * BC_BASE_DIGS; } // Grow if necessary. if (cpa.cap == cpa.len) bc_num_expand(&cpa, bc_vm_growSize(cpa.len, 1)); // We want an extra zero in front to make things simpler. cpa.num[cpa.len++] = 0; // Still setting things up. Why all of these things are needed is not // something that can be easily explained, but it has to do with making the // actual algorithm easier to understand because it can assume a lot of // things. Thus, you should view all of this setup code as establishing // assumptions for bc_num_d_long(), where the actual division happens. if (cpardx == cpa.len) cpa.len = bc_num_nonZeroLen(&cpa); if (BC_NUM_RDX_VAL_NP(cpb) == cpb.len) cpb.len = bc_num_nonZeroLen(&cpb); cpb.scale = 0; BC_NUM_RDX_SET_NP(cpb, 0); bc_num_d_long(&cpa, &cpb, c, scale); bc_num_retireMul(c, scale, BC_NUM_NEG(a), BC_NUM_NEG(b)); err: BC_SIG_MAYLOCK; bc_num_free(&cpb); bc_num_free(&cpa); BC_LONGJMP_CONT; } /** * Implements divmod. This is the actual modulus function; since modulus * requires a division anyway, this returns the quotient and modulus. Either can * be thrown out as desired. * @param a The first operand. * @param b The second operand. * @param c The return parameter for the quotient. * @param d The return parameter for the modulus. * @param scale The current scale. * @param ts The scale that the operation should be done to. Yes, it's not * necessarily the same as scale, per the bc spec. */ static void bc_num_r(BcNum* a, BcNum* b, BcNum* restrict c, BcNum* restrict d, size_t scale, size_t ts) { BcNum temp; bool neg; if (BC_NUM_ZERO(b)) bc_err(BC_ERR_MATH_DIVIDE_BY_ZERO); if (BC_NUM_ZERO(a)) { bc_num_setToZero(c, ts); bc_num_setToZero(d, ts); return; } BC_SIG_LOCK; bc_num_init(&temp, d->cap); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // Division. bc_num_d(a, b, c, scale); // We want an extra digit so we can safely truncate. if (scale) scale = ts + 1; assert(BC_NUM_RDX_VALID(c)); assert(BC_NUM_RDX_VALID(b)); // Implement the rest of the (a - (a / b) * b) formula. bc_num_m(c, b, &temp, scale); bc_num_sub(a, &temp, d, scale); // Extend if necessary. if (ts > d->scale && BC_NUM_NONZERO(d)) bc_num_extend(d, ts - d->scale); neg = BC_NUM_NEG(d); bc_num_retireMul(d, ts, BC_NUM_NEG(a), BC_NUM_NEG(b)); d->rdx = BC_NUM_NEG_VAL(d, BC_NUM_NONZERO(d) ? neg : false); err: BC_SIG_MAYLOCK; bc_num_free(&temp); BC_LONGJMP_CONT; } /** * Implements modulus/remainder. (Yes, I know they are different, but not in the * context of bc.) This is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_rem(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcNum c1; size_t ts; ts = bc_vm_growSize(scale, b->scale); ts = BC_MAX(ts, a->scale); BC_SIG_LOCK; // Need a temp for the quotient. bc_num_init(&c1, bc_num_mulReq(a, b, ts)); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; bc_num_r(a, b, &c1, c, scale, ts); err: BC_SIG_MAYLOCK; bc_num_free(&c1); BC_LONGJMP_CONT; } /** * Implements power (exponentiation). This is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_p(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcNum copy, btemp; BcBigDig exp; size_t powrdx, resrdx; bool neg; if (BC_ERR(bc_num_nonInt(b, &btemp))) bc_err(BC_ERR_MATH_NON_INTEGER); if (BC_NUM_ZERO(&btemp)) { bc_num_one(c); return; } if (BC_NUM_ZERO(a)) { if (BC_NUM_NEG_NP(btemp)) bc_err(BC_ERR_MATH_DIVIDE_BY_ZERO); bc_num_setToZero(c, scale); return; } if (BC_NUM_ONE(&btemp)) { if (!BC_NUM_NEG_NP(btemp)) bc_num_copy(c, a); else bc_num_inv(a, c, scale); return; } neg = BC_NUM_NEG_NP(btemp); BC_NUM_NEG_CLR_NP(btemp); exp = bc_num_bigdig(&btemp); BC_SIG_LOCK; bc_num_createCopy(©, a); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // If this is true, then we do not have to do a division, and we need to // set scale accordingly. if (!neg) { size_t max = BC_MAX(scale, a->scale), scalepow; scalepow = bc_num_mulOverflow(a->scale, exp); scale = BC_MIN(scalepow, max); } // This is only implementing the first exponentiation by squaring, until it // reaches the first time where the square is actually used. for (powrdx = a->scale; !(exp & 1); exp >>= 1) { powrdx <<= 1; assert(BC_NUM_RDX_VALID_NP(copy)); bc_num_mul(©, ©, ©, powrdx); } // Make c a copy of copy for the purpose of saving the squares that should // be saved. bc_num_copy(c, ©); resrdx = powrdx; // Now finish the exponentiation by squaring, this time saving the squares // as necessary. while (exp >>= 1) { powrdx <<= 1; assert(BC_NUM_RDX_VALID_NP(copy)); bc_num_mul(©, ©, ©, powrdx); // If this is true, we want to save that particular square. This does // that by multiplying c with copy. if (exp & 1) { resrdx += powrdx; assert(BC_NUM_RDX_VALID(c)); assert(BC_NUM_RDX_VALID_NP(copy)); bc_num_mul(c, ©, c, resrdx); } } // Invert if necessary. if (neg) bc_num_inv(c, c, scale); // Truncate if necessary. if (c->scale > scale) bc_num_truncate(c, c->scale - scale); bc_num_clean(c); err: BC_SIG_MAYLOCK; bc_num_free(©); BC_LONGJMP_CONT; } #if BC_ENABLE_EXTRA_MATH /** * Implements the places operator. This is a BcNumBinOp function. * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. */ static void bc_num_place(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcBigDig val; BC_UNUSED(scale); val = bc_num_intop(a, b, c); // Just truncate or extend as appropriate. if (val < c->scale) bc_num_truncate(c, c->scale - val); else if (val > c->scale) bc_num_extend(c, val - c->scale); } /** * Implements the left shift operator. This is a BcNumBinOp function. */ static void bc_num_left(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcBigDig val; BC_UNUSED(scale); val = bc_num_intop(a, b, c); bc_num_shiftLeft(c, (size_t) val); } /** * Implements the right shift operator. This is a BcNumBinOp function. */ static void bc_num_right(BcNum* a, BcNum* b, BcNum* restrict c, size_t scale) { BcBigDig val; BC_UNUSED(scale); val = bc_num_intop(a, b, c); if (BC_NUM_ZERO(c)) return; bc_num_shiftRight(c, (size_t) val); } #endif // BC_ENABLE_EXTRA_MATH /** * Prepares for, and calls, a binary operator function. This is probably the * most important function in the entire file because it establishes assumptions * that make the rest of the code so easy. Those assumptions include: * * - a is not the same pointer as c. * - b is not the same pointer as c. * - there is enough room in c for the result. * * Without these, this whole function would basically have to be duplicated for * *all* binary operators. * * @param a The first operand. * @param b The second operand. * @param c The return parameter. * @param scale The current scale. * @param req The number of limbs needed to fit the result. */ static void bc_num_binary(BcNum* a, BcNum* b, BcNum* c, size_t scale, BcNumBinOp op, size_t req) { BcNum* ptr_a; BcNum* ptr_b; BcNum num2; bool init = false; assert(a != NULL && b != NULL && c != NULL && op != NULL); assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); BC_SIG_LOCK; // Reallocate if c == a. if (c == a) { ptr_a = &num2; // NOLINTNEXTLINE memcpy(ptr_a, c, sizeof(BcNum)); init = true; } else { ptr_a = a; } // Also reallocate if c == b. if (c == b) { ptr_b = &num2; if (c != a) { // NOLINTNEXTLINE memcpy(ptr_b, c, sizeof(BcNum)); init = true; } } else { ptr_b = b; } // Actually reallocate. If we don't reallocate, we want to expand at the // very least. if (init) { bc_num_init(c, req); // Must prepare for cleanup. We want this here so that locals that got // set stay set since a longjmp() is not guaranteed to preserve locals. BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; } else { BC_SIG_UNLOCK; bc_num_expand(c, req); } // It is okay for a and b to be the same. If a binary operator function does // need them to be different, the binary operator function is responsible // for that. // Call the actual binary operator function. op(ptr_a, ptr_b, c, scale); assert(!BC_NUM_NEG(c) || BC_NUM_NONZERO(c)); assert(BC_NUM_RDX_VAL(c) <= c->len || !c->len); assert(BC_NUM_RDX_VALID(c)); assert(!c->len || c->num[c->len - 1] || BC_NUM_RDX_VAL(c) == c->len); err: // Cleanup only needed if we initialized c to a new number. if (init) { BC_SIG_MAYLOCK; bc_num_free(&num2); BC_LONGJMP_CONT; } } /** * Tests a number string for validity. This function has a history; I originally * wrote it because I did not trust my parser. Over time, however, I came to * trust it, so I was able to relegate this function to debug builds only, and I * used it in assert()'s. But then I created the library, and well, I can't * trust users, so I reused this for yelling at users. * @param val The string to check to see if it's a valid number string. * @return True if the string is a valid number string, false otherwise. */ bool bc_num_strValid(const char* restrict val) { bool radix = false; size_t i, len = strlen(val); // Notice that I don't check if there is a negative sign. That is not part // of a valid number, except in the library. The library-specific code takes // care of that part. // Nothing in the string is okay. if (!len) return true; // Loop through the characters. for (i = 0; i < len; ++i) { BcDig c = val[i]; // If we have found a radix point... if (c == '.') { // We don't allow two radices. if (radix) return false; radix = true; continue; } // We only allow digits and uppercase letters. if (!(isdigit(c) || isupper(c))) return false; } return true; } /** * Parses one character and returns the digit that corresponds to that * character according to the base. * @param c The character to parse. * @param base The base. * @return The character as a digit. */ static BcBigDig bc_num_parseChar(char c, size_t base) { assert(isupper(c) || isdigit(c)); // If a letter... if (isupper(c)) { // This returns the digit that directly corresponds with the letter. c = BC_NUM_NUM_LETTER(c); // If the digit is greater than the base, we clamp. c = ((size_t) c) >= base ? (char) base - 1 : c; } // Straight convert the digit to a number. else c -= '0'; return (BcBigDig) (uchar) c; } /** * Parses a string as a decimal number. This is separate because it's going to * be the most used, and it can be heavily optimized for decimal only. * @param n The number to parse into and return. Must be preallocated. * @param val The string to parse. */ static void bc_num_parseDecimal(BcNum* restrict n, const char* restrict val) { size_t len, i, temp, mod; const char* ptr; bool zero = true, rdx; // Eat leading zeroes. for (i = 0; val[i] == '0'; ++i) { continue; } val += i; assert(!val[0] || isalnum(val[0]) || val[0] == '.'); // All 0's. We can just return, since this procedure expects a virgin // (already 0) BcNum. if (!val[0]) return; // The length of the string is the length of the number, except it might be // one bigger because of a decimal point. len = strlen(val); // Find the location of the decimal point. ptr = strchr(val, '.'); rdx = (ptr != NULL); // We eat leading zeroes again. These leading zeroes are different because // they will come after the decimal point if they exist, and since that's // the case, they must be preserved. for (i = 0; i < len && (zero = (val[i] == '0' || val[i] == '.')); ++i) { continue; } // Set the scale of the number based on the location of the decimal point. // The casts to uintptr_t is to ensure that bc does not hit undefined // behavior when doing math on the values. n->scale = (size_t) (rdx * (((uintptr_t) (val + len)) - (((uintptr_t) ptr) + 1))); // Set rdx. BC_NUM_RDX_SET(n, BC_NUM_RDX(n->scale)); // Calculate length. First, the length of the integer, then the number of // digits in the last limb, then the length. i = len - (ptr == val ? 0 : i) - rdx; temp = BC_NUM_ROUND_POW(i); mod = n->scale % BC_BASE_DIGS; i = mod ? BC_BASE_DIGS - mod : 0; n->len = ((temp + i) / BC_BASE_DIGS); // Expand and zero. bc_num_expand(n, n->len); // NOLINTNEXTLINE memset(n->num, 0, BC_NUM_SIZE(n->len)); if (zero) { // I think I can set rdx directly to zero here because n should be a // new number with sign set to false. n->len = n->rdx = 0; } else { // There is actually stuff to parse if we make it here. Yay... BcBigDig exp, pow; assert(i <= BC_NUM_BIGDIG_MAX); // The exponent and power. exp = (BcBigDig) i; pow = bc_num_pow10[exp]; // Parse loop. We parse backwards because numbers are stored little // endian. for (i = len - 1; i < len; --i, ++exp) { char c = val[i]; // Skip the decimal point. if (c == '.') exp -= 1; else { // The index of the limb. size_t idx = exp / BC_BASE_DIGS; // Clamp for the base. if (isupper(c)) c = '9'; // Add the digit to the limb. n->num[idx] += (((BcBigDig) c) - '0') * pow; // Adjust the power and exponent. if ((exp + 1) % BC_BASE_DIGS == 0) pow = 1; else pow *= BC_BASE; } } } } /** * Parse a number in any base (besides decimal). * @param n The number to parse into and return. Must be preallocated. * @param val The string to parse. * @param base The base to parse as. */ static void bc_num_parseBase(BcNum* restrict n, const char* restrict val, BcBigDig base) { BcNum temp, mult1, mult2, result1, result2; BcNum* m1; BcNum* m2; BcNum* ptr; char c = 0; bool zero = true; BcBigDig v; size_t i, digs, len = strlen(val); // If zero, just return because the number should be virgin (already 0). for (i = 0; zero && i < len; ++i) { zero = (val[i] == '.' || val[i] == '0'); } if (zero) return; BC_SIG_LOCK; bc_num_init(&temp, BC_NUM_BIGDIG_LOG10); bc_num_init(&mult1, BC_NUM_BIGDIG_LOG10); BC_SETJMP_LOCKED(int_err); BC_SIG_UNLOCK; // We split parsing into parsing the integer and parsing the fractional // part. // Parse the integer part. This is the easy part because we just multiply // the number by the base, then add the digit. for (i = 0; i < len && (c = val[i]) && c != '.'; ++i) { // Convert the character to a digit. v = bc_num_parseChar(c, base); // Multiply the number. bc_num_mulArray(n, base, &mult1); // Convert the digit to a number and add. bc_num_bigdig2num(&temp, v); bc_num_add(&mult1, &temp, n, 0); } // If this condition is true, then we are done. We still need to do cleanup // though. if (i == len && !val[i]) goto int_err; // If we get here, we *must* be at the radix point. assert(val[i] == '.'); BC_SIG_LOCK; // Unset the jump to reset in for these new initializations. BC_UNSETJMP; bc_num_init(&mult2, BC_NUM_BIGDIG_LOG10); bc_num_init(&result1, BC_NUM_DEF_SIZE); bc_num_init(&result2, BC_NUM_DEF_SIZE); bc_num_one(&mult1); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // Pointers for easy switching. m1 = &mult1; m2 = &mult2; // Parse the fractional part. This is the hard part. for (i += 1, digs = 0; i < len && (c = val[i]); ++i, ++digs) { size_t rdx; // Convert the character to a digit. v = bc_num_parseChar(c, base); // We keep growing result2 according to the base because the more digits // after the radix, the more significant the digits close to the radix // should be. bc_num_mulArray(&result1, base, &result2); // Convert the digit to a number. bc_num_bigdig2num(&temp, v); // Add the digit into the fraction part. bc_num_add(&result2, &temp, &result1, 0); // Keep growing m1 and m2 for use after the loop. bc_num_mulArray(m1, base, m2); rdx = BC_NUM_RDX_VAL(m2); if (m2->len < rdx) m2->len = rdx; // Switch. ptr = m1; m1 = m2; m2 = ptr; } // This one cannot be a divide by 0 because mult starts out at 1, then is // multiplied by base, and base cannot be 0, so mult cannot be 0. And this // is the reason we keep growing m1 and m2; this division is what converts // the parsed fractional part from an integer to a fractional part. bc_num_div(&result1, m1, &result2, digs * 2); // Pretruncate. bc_num_truncate(&result2, digs); // The final add of the integer part to the fractional part. bc_num_add(n, &result2, n, digs); // Basic cleanup. if (BC_NUM_NONZERO(n)) { if (n->scale < digs) bc_num_extend(n, digs - n->scale); } else bc_num_zero(n); err: BC_SIG_MAYLOCK; bc_num_free(&result2); bc_num_free(&result1); bc_num_free(&mult2); int_err: BC_SIG_MAYLOCK; bc_num_free(&mult1); bc_num_free(&temp); BC_LONGJMP_CONT; } /** * Prints a backslash+newline combo if the number of characters needs it. This * is really a convenience function. */ static inline void bc_num_printNewline(void) { #if !BC_ENABLE_LIBRARY if (vm.nchars >= vm.line_len - 1 && vm.line_len) { bc_vm_putchar('\\', bc_flush_none); bc_vm_putchar('\n', bc_flush_err); } #endif // !BC_ENABLE_LIBRARY } /** * Prints a character after a backslash+newline, if needed. * @param c The character to print. * @param bslash Whether to print a backslash+newline. */ static void bc_num_putchar(int c, bool bslash) { if (c != '\n' && bslash) bc_num_printNewline(); bc_vm_putchar(c, bc_flush_save); } #if !BC_ENABLE_LIBRARY /** * Prints a character for a number's digit. This is for printing for dc's P * command. This function does not need to worry about radix points. This is a * BcNumDigitOp. * @param n The "digit" to print. * @param len The "length" of the digit, or number of characters that will * need to be printed for the digit. * @param rdx True if a decimal (radix) point should be printed. * @param bslash True if a backslash+newline should be printed if the character * limit for the line is reached, false otherwise. */ static void bc_num_printChar(size_t n, size_t len, bool rdx, bool bslash) { BC_UNUSED(rdx); BC_UNUSED(len); BC_UNUSED(bslash); assert(len == 1); bc_vm_putchar((uchar) n, bc_flush_save); } #endif // !BC_ENABLE_LIBRARY /** * Prints a series of characters for large bases. This is for printing in bases * above hexadecimal. This is a BcNumDigitOp. * @param n The "digit" to print. * @param len The "length" of the digit, or number of characters that will * need to be printed for the digit. * @param rdx True if a decimal (radix) point should be printed. * @param bslash True if a backslash+newline should be printed if the character * limit for the line is reached, false otherwise. */ static void bc_num_printDigits(size_t n, size_t len, bool rdx, bool bslash) { size_t exp, pow; // If needed, print the radix; otherwise, print a space to separate digits. bc_num_putchar(rdx ? '.' : ' ', true); // Calculate the exponent and power. for (exp = 0, pow = 1; exp < len - 1; ++exp, pow *= BC_BASE) { continue; } // Print each character individually. for (exp = 0; exp < len; pow /= BC_BASE, ++exp) { // The individual subdigit. size_t dig = n / pow; // Take the subdigit away. n -= dig * pow; // Print the subdigit. bc_num_putchar(((uchar) dig) + '0', bslash || exp != len - 1); } } /** * Prints a character for a number's digit. This is for printing in bases for * hexadecimal and below because they always print only one character at a time. * This is a BcNumDigitOp. * @param n The "digit" to print. * @param len The "length" of the digit, or number of characters that will * need to be printed for the digit. * @param rdx True if a decimal (radix) point should be printed. * @param bslash True if a backslash+newline should be printed if the character * limit for the line is reached, false otherwise. */ static void bc_num_printHex(size_t n, size_t len, bool rdx, bool bslash) { BC_UNUSED(len); BC_UNUSED(bslash); assert(len == 1); if (rdx) bc_num_putchar('.', true); bc_num_putchar(bc_num_hex_digits[n], bslash); } /** * Prints a decimal number. This is specially written for optimization since * this will be used the most and because bc's numbers are already in decimal. * @param n The number to print. * @param newline Whether to print backslash+newlines on long enough lines. */ static void bc_num_printDecimal(const BcNum* restrict n, bool newline) { size_t i, j, rdx = BC_NUM_RDX_VAL(n); bool zero = true; size_t buffer[BC_BASE_DIGS]; // Print loop. for (i = n->len - 1; i < n->len; --i) { BcDig n9 = n->num[i]; size_t temp; bool irdx = (i == rdx - 1); // Calculate the number of digits in the limb. zero = (zero & !irdx); temp = n->scale % BC_BASE_DIGS; temp = i || !temp ? 0 : BC_BASE_DIGS - temp; // NOLINTNEXTLINE memset(buffer, 0, BC_BASE_DIGS * sizeof(size_t)); // Fill the buffer with individual digits. for (j = 0; n9 && j < BC_BASE_DIGS; ++j) { buffer[j] = ((size_t) n9) % BC_BASE; n9 /= BC_BASE; } // Print the digits in the buffer. for (j = BC_BASE_DIGS - 1; j < BC_BASE_DIGS && j >= temp; --j) { // Figure out whether to print the decimal point. bool print_rdx = (irdx & (j == BC_BASE_DIGS - 1)); // The zero variable helps us skip leading zero digits in the limb. zero = (zero && buffer[j] == 0); if (!zero) { // While the first three arguments should be self-explanatory, // the last needs explaining. I don't want to print a newline // when the last digit to be printed could take the place of the // backslash rather than being pushed, as a single character, to // the next line. That's what that last argument does for bc. bc_num_printHex(buffer[j], 1, print_rdx, !newline || (j > temp || i != 0)); } } } } #if BC_ENABLE_EXTRA_MATH /** * Prints a number in scientific or engineering format. When doing this, we are * always printing in decimal. * @param n The number to print. * @param eng True if we are in engineering mode. * @param newline Whether to print backslash+newlines on long enough lines. */ static void bc_num_printExponent(const BcNum* restrict n, bool eng, bool newline) { size_t places, mod, nrdx = BC_NUM_RDX_VAL(n); bool neg = (n->len <= nrdx); BcNum temp, exp; BcDig digs[BC_NUM_BIGDIG_LOG10]; BC_SIG_LOCK; bc_num_createCopy(&temp, n); BC_SETJMP_LOCKED(exit); BC_SIG_UNLOCK; // We need to calculate the exponents, and they change based on whether the // number is all fractional or not, obviously. if (neg) { // Figure out how many limbs after the decimal point is zero. size_t i, idx = bc_num_nonZeroLen(n) - 1; places = 1; // Figure out how much in the last limb is zero. for (i = BC_BASE_DIGS - 1; i < BC_BASE_DIGS; --i) { if (bc_num_pow10[i] > (BcBigDig) n->num[idx]) places += 1; else break; } // Calculate the combination of zero limbs and zero digits in the last // limb. places += (nrdx - (idx + 1)) * BC_BASE_DIGS; mod = places % 3; // Calculate places if we are in engineering mode. if (eng && mod != 0) places += 3 - mod; // Shift the temp to the right place. bc_num_shiftLeft(&temp, places); } else { // This is the number of digits that we are supposed to put behind the // decimal point. places = bc_num_intDigits(n) - 1; // Calculate the true number based on whether engineering mode is // activated. mod = places % 3; if (eng && mod != 0) places -= 3 - (3 - mod); // Shift the temp to the right place. bc_num_shiftRight(&temp, places); } // Print the shifted number. bc_num_printDecimal(&temp, newline); // Print the e. bc_num_putchar('e', !newline); // Need to explicitly print a zero exponent. if (!places) { bc_num_printHex(0, 1, false, !newline); goto exit; } // Need to print sign for the exponent. if (neg) bc_num_putchar('-', true); // Create a temporary for the exponent... bc_num_setup(&exp, digs, BC_NUM_BIGDIG_LOG10); bc_num_bigdig2num(&exp, (BcBigDig) places); /// ..and print it. bc_num_printDecimal(&exp, newline); exit: BC_SIG_MAYLOCK; bc_num_free(&temp); BC_LONGJMP_CONT; } #endif // BC_ENABLE_EXTRA_MATH /** * Converts a number from limbs with base BC_BASE_POW to base @a pow, where * @a pow is obase^N. * @param n The number to convert. * @param rem BC_BASE_POW - @a pow. * @param pow The power of obase we will convert the number to. * @param idx The index of the number to start converting at. Doing the * conversion is O(n^2); we have to sweep through starting at the * least significant limb */ static void bc_num_printFixup(BcNum* restrict n, BcBigDig rem, BcBigDig pow, size_t idx) { size_t i, len = n->len - idx; BcBigDig acc; BcDig* a = n->num + idx; // Ignore if there's just one limb left. This is the part that requires the // extra loop after the one calling this function in bc_num_printPrepare(). if (len < 2) return; // Loop through the remaining limbs and convert. We start at the second limb // because we pull the value from the previous one as well. for (i = len - 1; i > 0; --i) { // Get the limb and add it to the previous, along with multiplying by // the remainder because that's the proper overflow. "acc" means // "accumulator," by the way. acc = ((BcBigDig) a[i]) * rem + ((BcBigDig) a[i - 1]); // Store a value in base pow in the previous limb. a[i - 1] = (BcDig) (acc % pow); // Divide by the base and accumulate the remaining value in the limb. acc /= pow; acc += (BcBigDig) a[i]; // If the accumulator is greater than the base... if (acc >= BC_BASE_POW) { // Do we need to grow? if (i == len - 1) { // Grow. len = bc_vm_growSize(len, 1); bc_num_expand(n, bc_vm_growSize(len, idx)); // Update the pointer because it may have moved. a = n->num + idx; // Zero out the last limb. a[len - 1] = 0; } // Overflow into the next limb since we are over the base. a[i + 1] += acc / BC_BASE_POW; acc %= BC_BASE_POW; } assert(acc < BC_BASE_POW); // Set the limb. a[i] = (BcDig) acc; } // We may have grown the number, so adjust the length. n->len = len + idx; } /** * Prepares a number for printing in a base that is not a divisor of * BC_BASE_POW. This basically converts the number from having limbs of base * BC_BASE_POW to limbs of pow, where pow is obase^N. * @param n The number to prepare for printing. * @param rem The remainder of BC_BASE_POW when divided by a power of the base. * @param pow The power of the base. */ static void bc_num_printPrepare(BcNum* restrict n, BcBigDig rem, BcBigDig pow) { size_t i; // Loop from the least significant limb to the most significant limb and // convert limbs in each pass. for (i = 0; i < n->len; ++i) { bc_num_printFixup(n, rem, pow, i); } // bc_num_printFixup() does not do everything it is supposed to, so we do // the last bit of cleanup here. That cleanup is to ensure that each limb // is less than pow and to expand the number to fit new limbs as necessary. for (i = 0; i < n->len; ++i) { assert(pow == ((BcBigDig) ((BcDig) pow))); // If the limb needs fixing... if (n->num[i] >= (BcDig) pow) { // Do we need to grow? if (i + 1 == n->len) { // Grow the number. n->len = bc_vm_growSize(n->len, 1); bc_num_expand(n, n->len); // Without this, we might use uninitialized data. n->num[i + 1] = 0; } assert(pow < BC_BASE_POW); // Overflow into the next limb. n->num[i + 1] += n->num[i] / ((BcDig) pow); n->num[i] %= (BcDig) pow; } } } static void bc_num_printNum(BcNum* restrict n, BcBigDig base, size_t len, BcNumDigitOp print, bool newline) { BcVec stack; BcNum intp, fracp1, fracp2, digit, flen1, flen2; BcNum* n1; BcNum* n2; BcNum* temp; BcBigDig dig = 0, acc, exp; BcBigDig* ptr; size_t i, j, nrdx, idigits; bool radix; BcDig digit_digs[BC_NUM_BIGDIG_LOG10 + 1]; assert(base > 1); // Easy case. Even with scale, we just print this. if (BC_NUM_ZERO(n)) { print(0, len, false, !newline); return; } // This function uses an algorithm that Stefan Esser came // up with to print the integer part of a number. What it does is convert // intp into a number of the specified base, but it does it directly, // instead of just doing a series of divisions and printing the remainders // in reverse order. // // Let me explain in a bit more detail: // // The algorithm takes the current least significant limb (after intp has // been converted to an integer) and the next to least significant limb, and // it converts the least significant limb into one of the specified base, // putting any overflow into the next to least significant limb. It iterates // through the whole number, from least significant to most significant, // doing this conversion. At the end of that iteration, the least // significant limb is converted, but the others are not, so it iterates // again, starting at the next to least significant limb. It keeps doing // that conversion, skipping one more limb than the last time, until all // limbs have been converted. Then it prints them in reverse order. // // That is the gist of the algorithm. It leaves out several things, such as // the fact that limbs are not always converted into the specified base, but // into something close, basically a power of the specified base. In // Stefan's words, "You could consider BcDigs to be of base 10^BC_BASE_DIGS // in the normal case and obase^N for the largest value of N that satisfies // obase^N <= 10^BC_BASE_DIGS. [This means that] the result is not in base // "obase", but in base "obase^N", which happens to be printable as a number // of base "obase" without consideration for neighbouring BcDigs." This fact // is what necessitates the existence of the loop later in this function. // // The conversion happens in bc_num_printPrepare() where the outer loop // happens and bc_num_printFixup() where the inner loop, or actual // conversion, happens. In other words, bc_num_printPrepare() is where the // loop that starts at the least significant limb and goes to the most // significant limb. Then, on every iteration of its loop, it calls // bc_num_printFixup(), which has the inner loop of actually converting // the limbs it passes into limbs of base obase^N rather than base // BC_BASE_POW. nrdx = BC_NUM_RDX_VAL(n); BC_SIG_LOCK; // The stack is what allows us to reverse the digits for printing. bc_vec_init(&stack, sizeof(BcBigDig), BC_DTOR_NONE); bc_num_init(&fracp1, nrdx); // intp will be the "integer part" of the number, so copy it. bc_num_createCopy(&intp, n); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // Make intp an integer. bc_num_truncate(&intp, intp.scale); // Get the fractional part out. bc_num_sub(n, &intp, &fracp1, 0); // If the base is not the same as the last base used for printing, we need // to update the cached exponent and power. Yes, we cache the values of the // exponent and power. That is to prevent us from calculating them every // time because printing will probably happen multiple times on the same // base. if (base != vm.last_base) { vm.last_pow = 1; vm.last_exp = 0; // Calculate the exponent and power. while (vm.last_pow * base <= BC_BASE_POW) { vm.last_pow *= base; vm.last_exp += 1; } // Also, the remainder and base itself. vm.last_rem = BC_BASE_POW - vm.last_pow; vm.last_base = base; } exp = vm.last_exp; // If vm.last_rem is 0, then the base we are printing in is a divisor of // BC_BASE_POW, which is the easy case because it means that BC_BASE_POW is // a power of obase, and no conversion is needed. If it *is* 0, then we have // the hard case, and we have to prepare the number for the base. if (vm.last_rem != 0) bc_num_printPrepare(&intp, vm.last_rem, vm.last_pow); // After the conversion comes the surprisingly easy part. From here on out, // this is basically naive code that I wrote, adjusted for the larger bases. // Fill the stack of digits for the integer part. for (i = 0; i < intp.len; ++i) { // Get the limb. acc = (BcBigDig) intp.num[i]; // Turn the limb into digits of base obase. for (j = 0; j < exp && (i < intp.len - 1 || acc != 0); ++j) { // This condition is true if we are not at the last digit. if (j != exp - 1) { dig = acc % base; acc /= base; } else { dig = acc; acc = 0; } assert(dig < base); // Push the digit onto the stack. bc_vec_push(&stack, &dig); } assert(acc == 0); } // Go through the stack backwards and print each digit. for (i = 0; i < stack.len; ++i) { ptr = bc_vec_item_rev(&stack, i); assert(ptr != NULL); // While the first three arguments should be self-explanatory, the last // needs explaining. I don't want to print a newline when the last digit // to be printed could take the place of the backslash rather than being // pushed, as a single character, to the next line. That's what that // last argument does for bc. print(*ptr, len, false, !newline || (n->scale != 0 || i == stack.len - 1)); } // We are done if there is no fractional part. if (!n->scale) goto err; BC_SIG_LOCK; // Reset the jump because some locals are changing. BC_UNSETJMP; bc_num_init(&fracp2, nrdx); bc_num_setup(&digit, digit_digs, sizeof(digit_digs) / sizeof(BcDig)); bc_num_init(&flen1, BC_NUM_BIGDIG_LOG10); bc_num_init(&flen2, BC_NUM_BIGDIG_LOG10); BC_SETJMP_LOCKED(frac_err); BC_SIG_UNLOCK; bc_num_one(&flen1); radix = true; // Pointers for easy switching. n1 = &flen1; n2 = &flen2; fracp2.scale = n->scale; BC_NUM_RDX_SET_NP(fracp2, BC_NUM_RDX(fracp2.scale)); // As long as we have not reached the scale of the number, keep printing. while ((idigits = bc_num_intDigits(n1)) <= n->scale) { // These numbers will keep growing. bc_num_expand(&fracp2, fracp1.len + 1); bc_num_mulArray(&fracp1, base, &fracp2); nrdx = BC_NUM_RDX_VAL_NP(fracp2); // Ensure an invariant. if (fracp2.len < nrdx) fracp2.len = nrdx; // fracp is guaranteed to be non-negative and small enough. dig = bc_num_bigdig2(&fracp2); // Convert the digit to a number and subtract it from the number. bc_num_bigdig2num(&digit, dig); bc_num_sub(&fracp2, &digit, &fracp1, 0); // While the first three arguments should be self-explanatory, the last // needs explaining. I don't want to print a newline when the last digit // to be printed could take the place of the backslash rather than being // pushed, as a single character, to the next line. That's what that // last argument does for bc. print(dig, len, radix, !newline || idigits != n->scale); // Update the multipliers. bc_num_mulArray(n1, base, n2); radix = false; // Switch. temp = n1; n1 = n2; n2 = temp; } frac_err: BC_SIG_MAYLOCK; bc_num_free(&flen2); bc_num_free(&flen1); bc_num_free(&fracp2); err: BC_SIG_MAYLOCK; bc_num_free(&fracp1); bc_num_free(&intp); bc_vec_free(&stack); BC_LONGJMP_CONT; } /** * Prints a number in the specified base, or rather, figures out which function * to call to print the number in the specified base and calls it. * @param n The number to print. * @param base The base to print in. * @param newline Whether to print backslash+newlines on long enough lines. */ static void bc_num_printBase(BcNum* restrict n, BcBigDig base, bool newline) { size_t width; BcNumDigitOp print; bool neg = BC_NUM_NEG(n); // Clear the sign because it makes the actual printing easier when we have // to do math. BC_NUM_NEG_CLR(n); // Bases at hexadecimal and below are printed as one character, larger bases // are printed as a series of digits separated by spaces. if (base <= BC_NUM_MAX_POSIX_IBASE) { width = 1; print = bc_num_printHex; } else { assert(base <= BC_BASE_POW); width = bc_num_log10(base - 1); print = bc_num_printDigits; } // Print. bc_num_printNum(n, base, width, print, newline); // Reset the sign. n->rdx = BC_NUM_NEG_VAL(n, neg); } #if !BC_ENABLE_LIBRARY void bc_num_stream(BcNum* restrict n) { bc_num_printNum(n, BC_NUM_STREAM_BASE, 1, bc_num_printChar, false); } #endif // !BC_ENABLE_LIBRARY void bc_num_setup(BcNum* restrict n, BcDig* restrict num, size_t cap) { assert(n != NULL); n->num = num; n->cap = cap; bc_num_zero(n); } void bc_num_init(BcNum* restrict n, size_t req) { BcDig* num; BC_SIG_ASSERT_LOCKED; assert(n != NULL); // BC_NUM_DEF_SIZE is set to be about the smallest allocation size that // malloc() returns in practice, so just use it. req = req >= BC_NUM_DEF_SIZE ? req : BC_NUM_DEF_SIZE; // If we can't use a temp, allocate. if (req != BC_NUM_DEF_SIZE || (num = bc_vm_takeTemp()) == NULL) { num = bc_vm_malloc(BC_NUM_SIZE(req)); } bc_num_setup(n, num, req); } void bc_num_clear(BcNum* restrict n) { n->num = NULL; n->cap = 0; } void bc_num_free(void* num) { BcNum* n = (BcNum*) num; BC_SIG_ASSERT_LOCKED; assert(n != NULL); if (n->cap == BC_NUM_DEF_SIZE) bc_vm_addTemp(n->num); else free(n->num); } void bc_num_copy(BcNum* d, const BcNum* s) { assert(d != NULL && s != NULL); if (d == s) return; bc_num_expand(d, s->len); d->len = s->len; // I can just copy directly here because the sign *and* rdx will be // properly preserved. d->rdx = s->rdx; d->scale = s->scale; // NOLINTNEXTLINE memcpy(d->num, s->num, BC_NUM_SIZE(d->len)); } void bc_num_createCopy(BcNum* d, const BcNum* s) { BC_SIG_ASSERT_LOCKED; bc_num_init(d, s->len); bc_num_copy(d, s); } void bc_num_createFromBigdig(BcNum* restrict n, BcBigDig val) { BC_SIG_ASSERT_LOCKED; bc_num_init(n, BC_NUM_BIGDIG_LOG10); bc_num_bigdig2num(n, val); } size_t bc_num_scale(const BcNum* restrict n) { return n->scale; } size_t bc_num_len(const BcNum* restrict n) { size_t len = n->len; // Always return at least 1. if (BC_NUM_ZERO(n)) return n->scale ? n->scale : 1; // If this is true, there is no integer portion of the number. if (BC_NUM_RDX_VAL(n) == len) { // We have to take into account the fact that some of the digits right // after the decimal could be zero. If that is the case, we need to // ignore them until we hit the first non-zero digit. size_t zero, scale; // The number of limbs with non-zero digits. len = bc_num_nonZeroLen(n); // Get the number of digits in the last limb. scale = n->scale % BC_BASE_DIGS; scale = scale ? scale : BC_BASE_DIGS; // Get the number of zero digits. zero = bc_num_zeroDigits(n->num + len - 1); // Calculate the true length. len = len * BC_BASE_DIGS - zero - (BC_BASE_DIGS - scale); } // Otherwise, count the number of int digits and return that plus the scale. else len = bc_num_intDigits(n) + n->scale; return len; } void bc_num_parse(BcNum* restrict n, const char* restrict val, BcBigDig base) { assert(n != NULL && val != NULL && base); assert(base >= BC_NUM_MIN_BASE && base <= vm.maxes[BC_PROG_GLOBALS_IBASE]); assert(bc_num_strValid(val)); // A one character number is *always* parsed as though the base was the // maximum allowed ibase, per the bc spec. if (!val[1]) { BcBigDig dig = bc_num_parseChar(val[0], BC_NUM_MAX_LBASE); bc_num_bigdig2num(n, dig); } else if (base == BC_BASE) bc_num_parseDecimal(n, val); else bc_num_parseBase(n, val, base); assert(BC_NUM_RDX_VALID(n)); } void bc_num_print(BcNum* restrict n, BcBigDig base, bool newline) { assert(n != NULL); assert(BC_ENABLE_EXTRA_MATH || base >= BC_NUM_MIN_BASE); // We may need a newline, just to start. bc_num_printNewline(); if (BC_NUM_NONZERO(n)) { // Print the sign. if (BC_NUM_NEG(n)) bc_num_putchar('-', true); // Print the leading zero if necessary. if (BC_Z && BC_NUM_RDX_VAL(n) == n->len) { bc_num_printHex(0, 1, false, !newline); } } // Short-circuit 0. if (BC_NUM_ZERO(n)) bc_num_printHex(0, 1, false, !newline); else if (base == BC_BASE) bc_num_printDecimal(n, newline); #if BC_ENABLE_EXTRA_MATH else if (base == 0 || base == 1) { bc_num_printExponent(n, base != 0, newline); } #endif // BC_ENABLE_EXTRA_MATH else bc_num_printBase(n, base, newline); if (newline) bc_num_putchar('\n', false); } BcBigDig bc_num_bigdig2(const BcNum* restrict n) { // This function returns no errors because it's guaranteed to succeed if // its preconditions are met. Those preconditions include both n needs to // be non-NULL, n being non-negative, and n being less than vm.max. If all // of that is true, then we can just convert without worrying about negative // errors or overflow. BcBigDig r = 0; size_t nrdx = BC_NUM_RDX_VAL(n); assert(n != NULL); assert(!BC_NUM_NEG(n)); assert(bc_num_cmp(n, &vm.max) < 0); assert(n->len - nrdx <= 3); // There is a small speed win from unrolling the loop here, and since it // only adds 53 bytes, I decided that it was worth it. switch (n->len - nrdx) { case 3: { r = (BcBigDig) n->num[nrdx + 2]; // Fallthrough. BC_FALLTHROUGH } case 2: { r = r * BC_BASE_POW + (BcBigDig) n->num[nrdx + 1]; // Fallthrough. BC_FALLTHROUGH } case 1: { r = r * BC_BASE_POW + (BcBigDig) n->num[nrdx]; } } return r; } BcBigDig bc_num_bigdig(const BcNum* restrict n) { assert(n != NULL); // This error checking is extremely important, and if you do not have a // guarantee that converting a number will always succeed in a particular // case, you *must* call this function to get these error checks. This // includes all instances of numbers inputted by the user or calculated by // the user. Otherwise, you can call the faster bc_num_bigdig2(). if (BC_ERR(BC_NUM_NEG(n))) bc_err(BC_ERR_MATH_NEGATIVE); if (BC_ERR(bc_num_cmp(n, &vm.max) >= 0)) bc_err(BC_ERR_MATH_OVERFLOW); return bc_num_bigdig2(n); } void bc_num_bigdig2num(BcNum* restrict n, BcBigDig val) { BcDig* ptr; size_t i; assert(n != NULL); bc_num_zero(n); // Already 0. if (!val) return; // Expand first. This is the only way this function can fail, and it's a // fatal error. bc_num_expand(n, BC_NUM_BIGDIG_LOG10); // The conversion is easy because numbers are laid out in little-endian // order. for (ptr = n->num, i = 0; val; ++i, val /= BC_BASE_POW) { ptr[i] = val % BC_BASE_POW; } n->len = i; } #if BC_ENABLE_EXTRA_MATH void bc_num_rng(const BcNum* restrict n, BcRNG* rng) { BcNum temp, temp2, intn, frac; BcRand state1, state2, inc1, inc2; size_t nrdx = BC_NUM_RDX_VAL(n); // This function holds the secret of how I interpret a seed number for the // PRNG. Well, it's actually in the development manual // (manuals/development.md#pseudo-random-number-generator), so look there // before you try to understand this. BC_SIG_LOCK; bc_num_init(&temp, n->len); bc_num_init(&temp2, n->len); bc_num_init(&frac, nrdx); bc_num_init(&intn, bc_num_int(n)); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; assert(BC_NUM_RDX_VALID_NP(vm.max)); // NOLINTNEXTLINE memcpy(frac.num, n->num, BC_NUM_SIZE(nrdx)); frac.len = nrdx; BC_NUM_RDX_SET_NP(frac, nrdx); frac.scale = n->scale; assert(BC_NUM_RDX_VALID_NP(frac)); assert(BC_NUM_RDX_VALID_NP(vm.max2)); // Multiply the fraction and truncate so that it's an integer. The // truncation is what clamps it, by the way. bc_num_mul(&frac, &vm.max2, &temp, 0); bc_num_truncate(&temp, temp.scale); bc_num_copy(&frac, &temp); // Get the integer. // NOLINTNEXTLINE memcpy(intn.num, n->num + nrdx, BC_NUM_SIZE(bc_num_int(n))); intn.len = bc_num_int(n); // This assert is here because it has to be true. It is also here to justify // some optimizations. assert(BC_NUM_NONZERO(&vm.max)); // If there *was* a fractional part... if (BC_NUM_NONZERO(&frac)) { // This divmod splits frac into the two state parts. bc_num_divmod(&frac, &vm.max, &temp, &temp2, 0); // frac is guaranteed to be smaller than vm.max * vm.max (pow). // This means that when dividing frac by vm.max, as above, the // quotient and remainder are both guaranteed to be less than vm.max, // which means we can use bc_num_bigdig2() here and not worry about // overflow. state1 = (BcRand) bc_num_bigdig2(&temp2); state2 = (BcRand) bc_num_bigdig2(&temp); } else state1 = state2 = 0; // If there *was* an integer part... if (BC_NUM_NONZERO(&intn)) { // This divmod splits intn into the two inc parts. bc_num_divmod(&intn, &vm.max, &temp, &temp2, 0); // Because temp2 is the mod of vm.max, from above, it is guaranteed // to be small enough to use bc_num_bigdig2(). inc1 = (BcRand) bc_num_bigdig2(&temp2); // Clamp the second inc part. if (bc_num_cmp(&temp, &vm.max) >= 0) { bc_num_copy(&temp2, &temp); bc_num_mod(&temp2, &vm.max, &temp, 0); } // The if statement above ensures that temp is less than vm.max, which // means that we can use bc_num_bigdig2() here. inc2 = (BcRand) bc_num_bigdig2(&temp); } else inc1 = inc2 = 0; bc_rand_seed(rng, state1, state2, inc1, inc2); err: BC_SIG_MAYLOCK; bc_num_free(&intn); bc_num_free(&frac); bc_num_free(&temp2); bc_num_free(&temp); BC_LONGJMP_CONT; } void bc_num_createFromRNG(BcNum* restrict n, BcRNG* rng) { BcRand s1, s2, i1, i2; BcNum conv, temp1, temp2, temp3; BcDig temp1_num[BC_RAND_NUM_SIZE], temp2_num[BC_RAND_NUM_SIZE]; BcDig conv_num[BC_NUM_BIGDIG_LOG10]; BC_SIG_LOCK; bc_num_init(&temp3, 2 * BC_RAND_NUM_SIZE); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; bc_num_setup(&temp1, temp1_num, sizeof(temp1_num) / sizeof(BcDig)); bc_num_setup(&temp2, temp2_num, sizeof(temp2_num) / sizeof(BcDig)); bc_num_setup(&conv, conv_num, sizeof(conv_num) / sizeof(BcDig)); // This assert is here because it has to be true. It is also here to justify // the assumption that vm.max is not zero. assert(BC_NUM_NONZERO(&vm.max)); // Because this is true, we can just ignore math errors that would happen // otherwise. assert(BC_NUM_NONZERO(&vm.max2)); bc_rand_getRands(rng, &s1, &s2, &i1, &i2); // Put the second piece of state into a number. bc_num_bigdig2num(&conv, (BcBigDig) s2); assert(BC_NUM_RDX_VALID_NP(conv)); // Multiply by max to make room for the first piece of state. bc_num_mul(&conv, &vm.max, &temp1, 0); // Add in the first piece of state. bc_num_bigdig2num(&conv, (BcBigDig) s1); bc_num_add(&conv, &temp1, &temp2, 0); // Divide to make it an entirely fractional part. bc_num_div(&temp2, &vm.max2, &temp3, BC_RAND_STATE_BITS); // Now start on the increment parts. It's the same process without the // divide, so put the second piece of increment into a number. bc_num_bigdig2num(&conv, (BcBigDig) i2); assert(BC_NUM_RDX_VALID_NP(conv)); // Multiply by max to make room for the first piece of increment. bc_num_mul(&conv, &vm.max, &temp1, 0); // Add in the first piece of increment. bc_num_bigdig2num(&conv, (BcBigDig) i1); bc_num_add(&conv, &temp1, &temp2, 0); // Now add the two together. bc_num_add(&temp2, &temp3, n, 0); assert(BC_NUM_RDX_VALID(n)); err: BC_SIG_MAYLOCK; bc_num_free(&temp3); BC_LONGJMP_CONT; } void bc_num_irand(BcNum* restrict a, BcNum* restrict b, BcRNG* restrict rng) { BcNum atemp; size_t i, len; assert(a != b); if (BC_ERR(BC_NUM_NEG(a))) bc_err(BC_ERR_MATH_NEGATIVE); // If either of these are true, then the numbers are integers. if (BC_NUM_ZERO(a) || BC_NUM_ONE(a)) return; if (BC_ERR(bc_num_nonInt(a, &atemp))) bc_err(BC_ERR_MATH_NON_INTEGER); assert(atemp.len); len = atemp.len - 1; // Just generate a random number for each limb. for (i = 0; i < len; ++i) { b->num[i] = (BcDig) bc_rand_bounded(rng, BC_BASE_POW); } // Do the last digit explicitly because the bound must be right. But only // do it if the limb does not equal 1. If it does, we have already hit the // limit. if (atemp.num[i] != 1) { b->num[i] = (BcDig) bc_rand_bounded(rng, (BcRand) atemp.num[i]); b->len = atemp.len; } // We want 1 less len in the case where we skip the last limb. else b->len = len; bc_num_clean(b); assert(BC_NUM_RDX_VALID(b)); } #endif // BC_ENABLE_EXTRA_MATH size_t bc_num_addReq(const BcNum* a, const BcNum* b, size_t scale) { size_t aint, bint, ardx, brdx; // Addition and subtraction require the max of the length of the two numbers // plus 1. BC_UNUSED(scale); ardx = BC_NUM_RDX_VAL(a); aint = bc_num_int(a); assert(aint <= a->len && ardx <= a->len); brdx = BC_NUM_RDX_VAL(b); bint = bc_num_int(b); assert(bint <= b->len && brdx <= b->len); ardx = BC_MAX(ardx, brdx); aint = BC_MAX(aint, bint); return bc_vm_growSize(bc_vm_growSize(ardx, aint), 1); } size_t bc_num_mulReq(const BcNum* a, const BcNum* b, size_t scale) { size_t max, rdx; // Multiplication requires the sum of the lengths of the numbers. rdx = bc_vm_growSize(BC_NUM_RDX_VAL(a), BC_NUM_RDX_VAL(b)); max = BC_NUM_RDX(scale); max = bc_vm_growSize(BC_MAX(max, rdx), 1); rdx = bc_vm_growSize(bc_vm_growSize(bc_num_int(a), bc_num_int(b)), max); return rdx; } size_t bc_num_divReq(const BcNum* a, const BcNum* b, size_t scale) { size_t max, rdx; // Division requires the length of the dividend plus the scale. rdx = bc_vm_growSize(BC_NUM_RDX_VAL(a), BC_NUM_RDX_VAL(b)); max = BC_NUM_RDX(scale); max = bc_vm_growSize(BC_MAX(max, rdx), 1); rdx = bc_vm_growSize(bc_num_int(a), max); return rdx; } size_t bc_num_powReq(const BcNum* a, const BcNum* b, size_t scale) { BC_UNUSED(scale); return bc_vm_growSize(bc_vm_growSize(a->len, b->len), 1); } #if BC_ENABLE_EXTRA_MATH size_t bc_num_placesReq(const BcNum* a, const BcNum* b, size_t scale) { BC_UNUSED(scale); return a->len + b->len - BC_NUM_RDX_VAL(a) - BC_NUM_RDX_VAL(b); } #endif // BC_ENABLE_EXTRA_MATH void bc_num_add(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, false, bc_num_as, bc_num_addReq(a, b, scale)); } void bc_num_sub(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, true, bc_num_as, bc_num_addReq(a, b, scale)); } void bc_num_mul(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_m, bc_num_mulReq(a, b, scale)); } void bc_num_div(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_d, bc_num_divReq(a, b, scale)); } void bc_num_mod(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_rem, bc_num_divReq(a, b, scale)); } void bc_num_pow(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_p, bc_num_powReq(a, b, scale)); } #if BC_ENABLE_EXTRA_MATH void bc_num_places(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_place, bc_num_placesReq(a, b, scale)); } void bc_num_lshift(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_left, bc_num_placesReq(a, b, scale)); } void bc_num_rshift(BcNum* a, BcNum* b, BcNum* c, size_t scale) { assert(BC_NUM_RDX_VALID(a)); assert(BC_NUM_RDX_VALID(b)); bc_num_binary(a, b, c, scale, bc_num_right, bc_num_placesReq(a, b, scale)); } #endif // BC_ENABLE_EXTRA_MATH void bc_num_sqrt(BcNum* restrict a, BcNum* restrict b, size_t scale) { BcNum num1, num2, half, f, fprime; BcNum* x0; BcNum* x1; BcNum* temp; size_t pow, len, rdx, req, resscale; BcDig half_digs[1]; assert(a != NULL && b != NULL && a != b); if (BC_ERR(BC_NUM_NEG(a))) bc_err(BC_ERR_MATH_NEGATIVE); // We want to calculate to a's scale if it is bigger so that the result will // truncate properly. if (a->scale > scale) scale = a->scale; // Set parameters for the result. len = bc_vm_growSize(bc_num_intDigits(a), 1); rdx = BC_NUM_RDX(scale); // Square root needs half of the length of the parameter. req = bc_vm_growSize(BC_MAX(rdx, BC_NUM_RDX_VAL(a)), len >> 1); BC_SIG_LOCK; // Unlike the binary operators, this function is the only single parameter // function and is expected to initialize the result. This means that it // expects that b is *NOT* preallocated. We allocate it here. bc_num_init(b, bc_vm_growSize(req, 1)); BC_SIG_UNLOCK; assert(a != NULL && b != NULL && a != b); assert(a->num != NULL && b->num != NULL); // Easy case. if (BC_NUM_ZERO(a)) { bc_num_setToZero(b, scale); return; } // Another easy case. if (BC_NUM_ONE(a)) { bc_num_one(b); bc_num_extend(b, scale); return; } // Set the parameters again. rdx = BC_NUM_RDX(scale); rdx = BC_MAX(rdx, BC_NUM_RDX_VAL(a)); len = bc_vm_growSize(a->len, rdx); BC_SIG_LOCK; bc_num_init(&num1, len); bc_num_init(&num2, len); bc_num_setup(&half, half_digs, sizeof(half_digs) / sizeof(BcDig)); // There is a division by two in the formula. We setup a number that's 1/2 // so that we can use multiplication instead of heavy division. bc_num_one(&half); half.num[0] = BC_BASE_POW / 2; half.len = 1; BC_NUM_RDX_SET_NP(half, 1); half.scale = 1; bc_num_init(&f, len); bc_num_init(&fprime, len); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; // Pointers for easy switching. x0 = &num1; x1 = &num2; // Start with 1. bc_num_one(x0); // The power of the operand is needed for the estimate. pow = bc_num_intDigits(a); // The code in this if statement calculates the initial estimate. First, if // a is less than 0, then 0 is a good estimate. Otherwise, we want something // in the same ballpark. That ballpark is pow. if (pow) { // An odd number is served by starting with 2^((pow-1)/2), and an even // number is served by starting with 6^((pow-2)/2). Why? Because math. if (pow & 1) x0->num[0] = 2; else x0->num[0] = 6; pow -= 2 - (pow & 1); bc_num_shiftLeft(x0, pow / 2); } // I can set the rdx here directly because neg should be false. x0->scale = x0->rdx = 0; resscale = (scale + BC_BASE_DIGS) + 2; // This is the calculation loop. This compare goes to 0 eventually as the // difference between the two numbers gets smaller than resscale. while (bc_num_cmp(x1, x0)) { assert(BC_NUM_NONZERO(x0)); // This loop directly corresponds to the iteration in Newton's method. // If you know the formula, this loop makes sense. Go study the formula. bc_num_div(a, x0, &f, resscale); bc_num_add(x0, &f, &fprime, resscale); assert(BC_NUM_RDX_VALID_NP(fprime)); assert(BC_NUM_RDX_VALID_NP(half)); bc_num_mul(&fprime, &half, x1, resscale); // Switch. temp = x0; x0 = x1; x1 = temp; } // Copy to the result and truncate. bc_num_copy(b, x0); if (b->scale > scale) bc_num_truncate(b, b->scale - scale); assert(!BC_NUM_NEG(b) || BC_NUM_NONZERO(b)); assert(BC_NUM_RDX_VALID(b)); assert(BC_NUM_RDX_VAL(b) <= b->len || !b->len); assert(!b->len || b->num[b->len - 1] || BC_NUM_RDX_VAL(b) == b->len); err: BC_SIG_MAYLOCK; bc_num_free(&fprime); bc_num_free(&f); bc_num_free(&num2); bc_num_free(&num1); BC_LONGJMP_CONT; } void bc_num_divmod(BcNum* a, BcNum* b, BcNum* c, BcNum* d, size_t scale) { size_t ts, len; BcNum *ptr_a, num2; bool init = false; // The bulk of this function is just doing what bc_num_binary() does for the // binary operators. However, it assumes that only c and a can be equal. // Set up the parameters. ts = BC_MAX(scale + b->scale, a->scale); len = bc_num_mulReq(a, b, ts); assert(a != NULL && b != NULL && c != NULL && d != NULL); assert(c != d && a != d && b != d && b != c); // Initialize or expand as necessary. if (c == a) { // NOLINTNEXTLINE memcpy(&num2, c, sizeof(BcNum)); ptr_a = &num2; BC_SIG_LOCK; bc_num_init(c, len); init = true; BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; } else { ptr_a = a; bc_num_expand(c, len); } // Do the quick version if possible. if (BC_NUM_NONZERO(a) && !BC_NUM_RDX_VAL(a) && !BC_NUM_RDX_VAL(b) && b->len == 1 && !scale) { BcBigDig rem; bc_num_divArray(ptr_a, (BcBigDig) b->num[0], c, &rem); assert(rem < BC_BASE_POW); d->num[0] = (BcDig) rem; d->len = (rem != 0); } // Do the slow method. else bc_num_r(ptr_a, b, c, d, scale, ts); assert(!BC_NUM_NEG(c) || BC_NUM_NONZERO(c)); assert(BC_NUM_RDX_VALID(c)); assert(BC_NUM_RDX_VAL(c) <= c->len || !c->len); assert(!c->len || c->num[c->len - 1] || BC_NUM_RDX_VAL(c) == c->len); assert(!BC_NUM_NEG(d) || BC_NUM_NONZERO(d)); assert(BC_NUM_RDX_VALID(d)); assert(BC_NUM_RDX_VAL(d) <= d->len || !d->len); assert(!d->len || d->num[d->len - 1] || BC_NUM_RDX_VAL(d) == d->len); err: // Only cleanup if we initialized. if (init) { BC_SIG_MAYLOCK; bc_num_free(&num2); BC_LONGJMP_CONT; } } void bc_num_modexp(BcNum* a, BcNum* b, BcNum* c, BcNum* restrict d) { BcNum base, exp, two, temp, atemp, btemp, ctemp; BcDig two_digs[2]; assert(a != NULL && b != NULL && c != NULL && d != NULL); assert(a != d && b != d && c != d); if (BC_ERR(BC_NUM_ZERO(c))) bc_err(BC_ERR_MATH_DIVIDE_BY_ZERO); if (BC_ERR(BC_NUM_NEG(b))) bc_err(BC_ERR_MATH_NEGATIVE); #ifndef NDEBUG // This is entirely for quieting a useless scan-build error. btemp.len = 0; ctemp.len = 0; #endif // NDEBUG // Eliminate fractional parts that are zero or error if they are not zero. if (BC_ERR(bc_num_nonInt(a, &atemp) || bc_num_nonInt(b, &btemp) || bc_num_nonInt(c, &ctemp))) { bc_err(BC_ERR_MATH_NON_INTEGER); } bc_num_expand(d, ctemp.len); BC_SIG_LOCK; bc_num_init(&base, ctemp.len); bc_num_setup(&two, two_digs, sizeof(two_digs) / sizeof(BcDig)); bc_num_init(&temp, btemp.len + 1); bc_num_createCopy(&exp, &btemp); BC_SETJMP_LOCKED(err); BC_SIG_UNLOCK; bc_num_one(&two); two.num[0] = 2; bc_num_one(d); // We already checked for 0. bc_num_rem(&atemp, &ctemp, &base, 0); // If you know the algorithm I used, the memory-efficient method, then this // loop should be self-explanatory because it is the calculation loop. while (BC_NUM_NONZERO(&exp)) { // Num two cannot be 0, so no errors. bc_num_divmod(&exp, &two, &exp, &temp, 0); if (BC_NUM_ONE(&temp) && !BC_NUM_NEG_NP(temp)) { assert(BC_NUM_RDX_VALID(d)); assert(BC_NUM_RDX_VALID_NP(base)); bc_num_mul(d, &base, &temp, 0); // We already checked for 0. bc_num_rem(&temp, &ctemp, d, 0); } assert(BC_NUM_RDX_VALID_NP(base)); bc_num_mul(&base, &base, &temp, 0); // We already checked for 0. bc_num_rem(&temp, &ctemp, &base, 0); } err: BC_SIG_MAYLOCK; bc_num_free(&exp); bc_num_free(&temp); bc_num_free(&base); BC_LONGJMP_CONT; assert(!BC_NUM_NEG(d) || d->len); assert(BC_NUM_RDX_VALID(d)); assert(!d->len || d->num[d->len - 1] || BC_NUM_RDX_VAL(d) == d->len); } #if BC_DEBUG_CODE void bc_num_printDebug(const BcNum* n, const char* name, bool emptyline) { bc_file_puts(&vm.fout, bc_flush_none, name); bc_file_puts(&vm.fout, bc_flush_none, ": "); bc_num_printDecimal(n, true); bc_file_putchar(&vm.fout, bc_flush_err, '\n'); if (emptyline) bc_file_putchar(&vm.fout, bc_flush_err, '\n'); vm.nchars = 0; } void bc_num_printDigs(const BcDig* n, size_t len, bool emptyline) { size_t i; for (i = len - 1; i < len; --i) { bc_file_printf(&vm.fout, " %lu", (unsigned long) n[i]); } bc_file_putchar(&vm.fout, bc_flush_err, '\n'); if (emptyline) bc_file_putchar(&vm.fout, bc_flush_err, '\n'); vm.nchars = 0; } void bc_num_printWithDigs(const BcNum* n, const char* name, bool emptyline) { bc_file_puts(&vm.fout, bc_flush_none, name); bc_file_printf(&vm.fout, " len: %zu, rdx: %zu, scale: %zu\n", name, n->len, BC_NUM_RDX_VAL(n), n->scale); bc_num_printDigs(n->num, n->len, emptyline); } void bc_num_dump(const char* varname, const BcNum* n) { ulong i, scale = n->scale; bc_file_printf(&vm.ferr, "\n%s = %s", varname, n->len ? (BC_NUM_NEG(n) ? "-" : "+") : "0 "); for (i = n->len - 1; i < n->len; --i) { if (i + 1 == BC_NUM_RDX_VAL(n)) { bc_file_puts(&vm.ferr, bc_flush_none, ". "); } if (scale / BC_BASE_DIGS != BC_NUM_RDX_VAL(n) - i - 1) { bc_file_printf(&vm.ferr, "%lu ", (unsigned long) n->num[i]); } else { int mod = scale % BC_BASE_DIGS; int d = BC_BASE_DIGS - mod; BcDig div; if (mod != 0) { div = n->num[i] / ((BcDig) bc_num_pow10[(ulong) d]); bc_file_printf(&vm.ferr, "%lu", (unsigned long) div); } div = n->num[i] % ((BcDig) bc_num_pow10[(ulong) d]); bc_file_printf(&vm.ferr, " ' %lu ", (unsigned long) div); } } bc_file_printf(&vm.ferr, "(%zu | %zu.%zu / %zu) %lu\n", n->scale, n->len, BC_NUM_RDX_VAL(n), n->cap, (unsigned long) (void*) n->num); bc_file_flush(&vm.ferr, bc_flush_err); } #endif // BC_DEBUG_CODE