// SPDX-License-Identifier: GPL-2.0-only /* tnum: tracked (or tristate) numbers * * A tnum tracks knowledge about the bits of a value. Each bit can be either * known (0 or 1), or unknown (x). Arithmetic operations on tnums will * propagate the unknown bits such that the tnum result represents all the * possible results for possible values of the operands. */ #include #include #define TNUM(_v, _m) (struct tnum){.value = _v, .mask = _m} /* A completely unknown value */ const struct tnum tnum_unknown = { .value = 0, .mask = -1 }; struct tnum tnum_const(u64 value) { return TNUM(value, 0); } struct tnum tnum_range(u64 min, u64 max) { u64 chi = min ^ max, delta; u8 bits = fls64(chi); /* special case, needed because 1ULL << 64 is undefined */ if (bits > 63) return tnum_unknown; /* e.g. if chi = 4, bits = 3, delta = (1<<3) - 1 = 7. * if chi = 0, bits = 0, delta = (1<<0) - 1 = 0, so we return * constant min (since min == max). */ delta = (1ULL << bits) - 1; return TNUM(min & ~delta, delta); } struct tnum tnum_lshift(struct tnum a, u8 shift) { return TNUM(a.value << shift, a.mask << shift); } struct tnum tnum_rshift(struct tnum a, u8 shift) { return TNUM(a.value >> shift, a.mask >> shift); } struct tnum tnum_arshift(struct tnum a, u8 min_shift, u8 insn_bitness) { /* if a.value is negative, arithmetic shifting by minimum shift * will have larger negative offset compared to more shifting. * If a.value is nonnegative, arithmetic shifting by minimum shift * will have larger positive offset compare to more shifting. */ if (insn_bitness == 32) return TNUM((u32)(((s32)a.value) >> min_shift), (u32)(((s32)a.mask) >> min_shift)); else return TNUM((s64)a.value >> min_shift, (s64)a.mask >> min_shift); } struct tnum tnum_add(struct tnum a, struct tnum b) { u64 sm, sv, sigma, chi, mu; sm = a.mask + b.mask; sv = a.value + b.value; sigma = sm + sv; chi = sigma ^ sv; mu = chi | a.mask | b.mask; return TNUM(sv & ~mu, mu); } struct tnum tnum_sub(struct tnum a, struct tnum b) { u64 dv, alpha, beta, chi, mu; dv = a.value - b.value; alpha = dv + a.mask; beta = dv - b.mask; chi = alpha ^ beta; mu = chi | a.mask | b.mask; return TNUM(dv & ~mu, mu); } struct tnum tnum_neg(struct tnum a) { return tnum_sub(TNUM(0, 0), a); } struct tnum tnum_and(struct tnum a, struct tnum b) { u64 alpha, beta, v; alpha = a.value | a.mask; beta = b.value | b.mask; v = a.value & b.value; return TNUM(v, alpha & beta & ~v); } struct tnum tnum_or(struct tnum a, struct tnum b) { u64 v, mu; v = a.value | b.value; mu = a.mask | b.mask; return TNUM(v, mu & ~v); } struct tnum tnum_xor(struct tnum a, struct tnum b) { u64 v, mu; v = a.value ^ b.value; mu = a.mask | b.mask; return TNUM(v & ~mu, mu); } /* Perform long multiplication, iterating through the bits in a using rshift: * - if LSB(a) is a known 0, keep current accumulator * - if LSB(a) is a known 1, add b to current accumulator * - if LSB(a) is unknown, take a union of the above cases. * * For example: * * acc_0: acc_1: * * 11 * -> 11 * -> 11 * -> union(0011, 1001) == x0x1 * x1 01 11 * ------ ------ ------ * 11 11 11 * xx 00 11 * ------ ------ ------ * ???? 0011 1001 */ struct tnum tnum_mul(struct tnum a, struct tnum b) { struct tnum acc = TNUM(0, 0); while (a.value || a.mask) { /* LSB of tnum a is a certain 1 */ if (a.value & 1) acc = tnum_add(acc, b); /* LSB of tnum a is uncertain */ else if (a.mask & 1) { /* acc = tnum_union(acc_0, acc_1), where acc_0 and * acc_1 are partial accumulators for cases * LSB(a) = certain 0 and LSB(a) = certain 1. * acc_0 = acc + 0 * b = acc. * acc_1 = acc + 1 * b = tnum_add(acc, b). */ acc = tnum_union(acc, tnum_add(acc, b)); } /* Note: no case for LSB is certain 0 */ a = tnum_rshift(a, 1); b = tnum_lshift(b, 1); } return acc; } bool tnum_overlap(struct tnum a, struct tnum b) { u64 mu; mu = ~a.mask & ~b.mask; return (a.value & mu) == (b.value & mu); } /* Note that if a and b disagree - i.e. one has a 'known 1' where the other has * a 'known 0' - this will return a 'known 1' for that bit. */ struct tnum tnum_intersect(struct tnum a, struct tnum b) { u64 v, mu; v = a.value | b.value; mu = a.mask & b.mask; return TNUM(v & ~mu, mu); } /* Returns a tnum with the uncertainty from both a and b, and in addition, new * uncertainty at any position that a and b disagree. This represents a * superset of the union of the concrete sets of both a and b. Despite the * overapproximation, it is optimal. */ struct tnum tnum_union(struct tnum a, struct tnum b) { u64 v = a.value & b.value; u64 mu = (a.value ^ b.value) | a.mask | b.mask; return TNUM(v & ~mu, mu); } struct tnum tnum_cast(struct tnum a, u8 size) { a.value &= (1ULL << (size * 8)) - 1; a.mask &= (1ULL << (size * 8)) - 1; return a; } bool tnum_is_aligned(struct tnum a, u64 size) { if (!size) return true; return !((a.value | a.mask) & (size - 1)); } bool tnum_in(struct tnum a, struct tnum b) { if (b.mask & ~a.mask) return false; b.value &= ~a.mask; return a.value == b.value; } int tnum_sbin(char *str, size_t size, struct tnum a) { size_t n; for (n = 64; n; n--) { if (n < size) { if (a.mask & 1) str[n - 1] = 'x'; else if (a.value & 1) str[n - 1] = '1'; else str[n - 1] = '0'; } a.mask >>= 1; a.value >>= 1; } str[min(size - 1, (size_t)64)] = 0; return 64; } struct tnum tnum_subreg(struct tnum a) { return tnum_cast(a, 4); } struct tnum tnum_clear_subreg(struct tnum a) { return tnum_lshift(tnum_rshift(a, 32), 32); } struct tnum tnum_with_subreg(struct tnum reg, struct tnum subreg) { return tnum_or(tnum_clear_subreg(reg), tnum_subreg(subreg)); } struct tnum tnum_const_subreg(struct tnum a, u32 value) { return tnum_with_subreg(a, tnum_const(value)); }