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-rw-r--r--src/crypto/slow-hash.c252
-rw-r--r--src/crypto/variant2_int_sqrt.h163
2 files changed, 361 insertions, 54 deletions
diff --git a/src/crypto/slow-hash.c b/src/crypto/slow-hash.c
index 9d4fc0dfa..a4d2b58de 100644
--- a/src/crypto/slow-hash.c
+++ b/src/crypto/slow-hash.c
@@ -38,6 +38,7 @@
#include "common/int-util.h"
#include "hash-ops.h"
#include "oaes_lib.h"
+#include "variant2_int_sqrt.h"
#define MEMORY (1 << 21) // 2MB scratchpad
#define ITER (1 << 20)
@@ -50,7 +51,7 @@ extern int aesb_single_round(const uint8_t *in, uint8_t*out, const uint8_t *expa
extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *expandedKey);
#define VARIANT1_1(p) \
- do if (variant > 0) \
+ do if (variant == 1) \
{ \
const uint8_t tmp = ((const uint8_t*)(p))[11]; \
static const uint32_t table = 0x75310; \
@@ -59,7 +60,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
} while(0)
#define VARIANT1_2(p) \
- do if (variant > 0) \
+ do if (variant == 1) \
{ \
xor64(p, tweak1_2); \
} while(0)
@@ -67,7 +68,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
#define VARIANT1_CHECK() \
do if (length < 43) \
{ \
- fprintf(stderr, "Cryptonight variants need at least 43 bytes of data"); \
+ fprintf(stderr, "Cryptonight variant 1 needs at least 43 bytes of data"); \
_exit(1); \
} while(0)
@@ -75,7 +76,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
#define VARIANT1_PORTABLE_INIT() \
uint8_t tweak1_2[8]; \
- do if (variant > 0) \
+ do if (variant == 1) \
{ \
VARIANT1_CHECK(); \
memcpy(&tweak1_2, &state.hs.b[192], sizeof(tweak1_2)); \
@@ -83,11 +84,119 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
} while(0)
#define VARIANT1_INIT64() \
- if (variant > 0) \
+ if (variant == 1) \
{ \
VARIANT1_CHECK(); \
} \
- const uint64_t tweak1_2 = variant > 0 ? (state.hs.w[24] ^ (*((const uint64_t*)NONCE_POINTER))) : 0
+ const uint64_t tweak1_2 = (variant == 1) ? (state.hs.w[24] ^ (*((const uint64_t*)NONCE_POINTER))) : 0
+
+#define VARIANT2_INIT64() \
+ uint64_t division_result = 0; \
+ uint64_t sqrt_result = 0; \
+ do if (variant >= 2) \
+ { \
+ U64(b)[2] = state.hs.w[8] ^ state.hs.w[10]; \
+ U64(b)[3] = state.hs.w[9] ^ state.hs.w[11]; \
+ division_result = state.hs.w[12]; \
+ sqrt_result = state.hs.w[13]; \
+ } while (0)
+
+#define VARIANT2_PORTABLE_INIT() \
+ uint64_t division_result = 0; \
+ uint64_t sqrt_result = 0; \
+ do if (variant >= 2) \
+ { \
+ memcpy(b + AES_BLOCK_SIZE, state.hs.b + 64, AES_BLOCK_SIZE); \
+ xor64(b + AES_BLOCK_SIZE, state.hs.b + 80); \
+ xor64(b + AES_BLOCK_SIZE + 8, state.hs.b + 88); \
+ division_result = state.hs.w[12]; \
+ sqrt_result = state.hs.w[13]; \
+ } while (0)
+
+#define VARIANT2_SHUFFLE_ADD_SSE2(base_ptr, offset) \
+ do if (variant >= 2) \
+ { \
+ const __m128i chunk1 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x10))); \
+ const __m128i chunk2 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x20))); \
+ const __m128i chunk3 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x30))); \
+ _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x10)), _mm_add_epi64(chunk3, _b1)); \
+ _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x20)), _mm_add_epi64(chunk1, _b)); \
+ _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x30)), _mm_add_epi64(chunk2, _a)); \
+ } while (0)
+
+#define VARIANT2_SHUFFLE_ADD_NEON(base_ptr, offset) \
+ do if (variant >= 2) \
+ { \
+ const uint64x2_t chunk1 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x10))); \
+ const uint64x2_t chunk2 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x20))); \
+ const uint64x2_t chunk3 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x30))); \
+ vst1q_u64(U64((base_ptr) + ((offset) ^ 0x10)), vaddq_u64(chunk3, vreinterpretq_u64_u8(_b1))); \
+ vst1q_u64(U64((base_ptr) + ((offset) ^ 0x20)), vaddq_u64(chunk1, vreinterpretq_u64_u8(_b))); \
+ vst1q_u64(U64((base_ptr) + ((offset) ^ 0x30)), vaddq_u64(chunk2, vreinterpretq_u64_u8(_a))); \
+ } while (0)
+
+#define VARIANT2_PORTABLE_SHUFFLE_ADD(base_ptr, offset) \
+ do if (variant >= 2) \
+ { \
+ uint64_t* chunk1 = U64((base_ptr) + ((offset) ^ 0x10)); \
+ uint64_t* chunk2 = U64((base_ptr) + ((offset) ^ 0x20)); \
+ uint64_t* chunk3 = U64((base_ptr) + ((offset) ^ 0x30)); \
+ \
+ const uint64_t chunk1_old[2] = { chunk1[0], chunk1[1] }; \
+ \
+ uint64_t b1[2]; \
+ memcpy(b1, b + 16, 16); \
+ chunk1[0] = chunk3[0] + b1[0]; \
+ chunk1[1] = chunk3[1] + b1[1]; \
+ \
+ uint64_t a0[2]; \
+ memcpy(a0, a, 16); \
+ chunk3[0] = chunk2[0] + a0[0]; \
+ chunk3[1] = chunk2[1] + a0[1]; \
+ \
+ uint64_t b0[2]; \
+ memcpy(b0, b, 16); \
+ chunk2[0] = chunk1_old[0] + b0[0]; \
+ chunk2[1] = chunk1_old[1] + b0[1]; \
+ } while (0)
+
+#define VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr) \
+ ((uint64_t*)(b))[0] ^= division_result ^ (sqrt_result << 32); \
+ { \
+ const uint64_t dividend = ((uint64_t*)(ptr))[1]; \
+ const uint32_t divisor = (((uint64_t*)(ptr))[0] + (uint32_t)(sqrt_result << 1)) | 0x80000001UL; \
+ division_result = ((uint32_t)(dividend / divisor)) + \
+ (((uint64_t)(dividend % divisor)) << 32); \
+ } \
+ const uint64_t sqrt_input = ((uint64_t*)(ptr))[0] + division_result
+
+#define VARIANT2_INTEGER_MATH_SSE2(b, ptr) \
+ do if (variant >= 2) \
+ { \
+ VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+ VARIANT2_INTEGER_MATH_SQRT_STEP_SSE2(); \
+ VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result); \
+ } while(0)
+
+#if defined DBL_MANT_DIG && (DBL_MANT_DIG >= 50)
+ // double precision floating point type has enough bits of precision on current platform
+ #define VARIANT2_PORTABLE_INTEGER_MATH(b, ptr) \
+ do if (variant >= 2) \
+ { \
+ VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+ VARIANT2_INTEGER_MATH_SQRT_STEP_FP64(); \
+ VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result); \
+ } while (0)
+#else
+ // double precision floating point type is not good enough on current platform
+ // fall back to the reference code (integer only)
+ #define VARIANT2_PORTABLE_INTEGER_MATH(b, ptr) \
+ do if (variant >= 2) \
+ { \
+ VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+ VARIANT2_INTEGER_MATH_SQRT_STEP_REF(); \
+ } while (0)
+#endif
#if !defined NO_AES && (defined(__x86_64__) || (defined(_MSC_VER) && defined(_WIN64)))
// Optimised code below, uses x86-specific intrinsics, SSE2, AES-NI
@@ -164,19 +273,22 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
* This code is based upon an optimized implementation by dga.
*/
#define post_aes() \
+ VARIANT2_SHUFFLE_ADD_SSE2(hp_state, j); \
_mm_store_si128(R128(c), _c); \
- _b = _mm_xor_si128(_b, _c); \
- _mm_store_si128(R128(&hp_state[j]), _b); \
+ _mm_store_si128(R128(&hp_state[j]), _mm_xor_si128(_b, _c)); \
VARIANT1_1(&hp_state[j]); \
j = state_index(c); \
p = U64(&hp_state[j]); \
b[0] = p[0]; b[1] = p[1]; \
+ VARIANT2_INTEGER_MATH_SSE2(b, c); \
__mul(); \
+ VARIANT2_SHUFFLE_ADD_SSE2(hp_state, j); \
a[0] += hi; a[1] += lo; \
p = U64(&hp_state[j]); \
p[0] = a[0]; p[1] = a[1]; \
a[0] ^= b[0]; a[1] ^= b[1]; \
VARIANT1_2(p + 1); \
+ _b1 = _b; \
_b = _c; \
#if defined(_MSC_VER)
@@ -570,10 +682,10 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
uint8_t text[INIT_SIZE_BYTE];
RDATA_ALIGN16 uint64_t a[2];
- RDATA_ALIGN16 uint64_t b[2];
+ RDATA_ALIGN16 uint64_t b[4];
RDATA_ALIGN16 uint64_t c[2];
union cn_slow_hash_state state;
- __m128i _a, _b, _c;
+ __m128i _a, _b, _b1, _c;
uint64_t hi, lo;
size_t i, j;
@@ -599,6 +711,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
memcpy(text, state.init, INIT_SIZE_BYTE);
VARIANT1_INIT64();
+ VARIANT2_INIT64();
/* CryptoNight Step 2: Iteratively encrypt the results from Keccak to fill
* the 2MB large random access buffer.
@@ -637,6 +750,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
*/
_b = _mm_load_si128(R128(b));
+ _b1 = _mm_load_si128(R128(b) + 1);
// Two independent versions, one with AES, one without, to ensure that
// the useAes test is only performed once, not every iteration.
if(useAes)
@@ -761,19 +875,22 @@ union cn_slow_hash_state
_a = vld1q_u8((const uint8_t *)a); \
#define post_aes() \
+ VARIANT2_SHUFFLE_ADD_NEON(hp_state, j); \
vst1q_u8((uint8_t *)c, _c); \
- _b = veorq_u8(_b, _c); \
- vst1q_u8(&hp_state[j], _b); \
+ vst1q_u8(&hp_state[j], veorq_u8(_b, _c)); \
VARIANT1_1(&hp_state[j]); \
j = state_index(c); \
p = U64(&hp_state[j]); \
b[0] = p[0]; b[1] = p[1]; \
+ VARIANT2_PORTABLE_INTEGER_MATH(b, c); \
__mul(); \
+ VARIANT2_SHUFFLE_ADD_NEON(hp_state, j); \
a[0] += hi; a[1] += lo; \
p = U64(&hp_state[j]); \
p[0] = a[0]; p[1] = a[1]; \
a[0] ^= b[0]; a[1] ^= b[1]; \
VARIANT1_2(p + 1); \
+ _b1 = _b; \
_b = _c; \
@@ -912,10 +1029,10 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
uint8_t text[INIT_SIZE_BYTE];
RDATA_ALIGN16 uint64_t a[2];
- RDATA_ALIGN16 uint64_t b[2];
+ RDATA_ALIGN16 uint64_t b[4];
RDATA_ALIGN16 uint64_t c[2];
union cn_slow_hash_state state;
- uint8x16_t _a, _b, _c, zero = {0};
+ uint8x16_t _a, _b, _b1, _c, zero = {0};
uint64_t hi, lo;
size_t i, j;
@@ -936,6 +1053,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
memcpy(text, state.init, INIT_SIZE_BYTE);
VARIANT1_INIT64();
+ VARIANT2_INIT64();
/* CryptoNight Step 2: Iteratively encrypt the results from Keccak to fill
* the 2MB large random access buffer.
@@ -959,7 +1077,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
*/
_b = vld1q_u8((const uint8_t *)b);
-
+ _b1 = vld1q_u8(((const uint8_t *)b) + AES_BLOCK_SIZE);
for(i = 0; i < ITER / 2; i++)
{
@@ -1075,6 +1193,11 @@ __asm__ __volatile__(
#endif /* !aarch64 */
#endif // NO_OPTIMIZED_MULTIPLY_ON_ARM
+STATIC INLINE void copy_block(uint8_t* dst, const uint8_t* src)
+{
+ memcpy(dst, src, AES_BLOCK_SIZE);
+}
+
STATIC INLINE void sum_half_blocks(uint8_t* a, const uint8_t* b)
{
uint64_t a0, a1, b0, b1;
@@ -1109,7 +1232,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
{
uint8_t text[INIT_SIZE_BYTE];
uint8_t a[AES_BLOCK_SIZE];
- uint8_t b[AES_BLOCK_SIZE];
+ uint8_t b[AES_BLOCK_SIZE * 2];
+ uint8_t c[AES_BLOCK_SIZE];
+ uint8_t c1[AES_BLOCK_SIZE];
uint8_t d[AES_BLOCK_SIZE];
uint8_t aes_key[AES_KEY_SIZE];
RDATA_ALIGN16 uint8_t expandedKey[256];
@@ -1138,11 +1263,12 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
}
memcpy(text, state.init, INIT_SIZE_BYTE);
- VARIANT1_INIT64();
-
aes_ctx = (oaes_ctx *) oaes_alloc();
oaes_key_import_data(aes_ctx, state.hs.b, AES_KEY_SIZE);
+ VARIANT1_INIT64();
+ VARIANT2_INIT64();
+
// use aligned data
memcpy(expandedKey, aes_ctx->key->exp_data, aes_ctx->key->exp_data_len);
for(i = 0; i < MEMORY / INIT_SIZE_BYTE; i++)
@@ -1163,23 +1289,33 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
#define state_index(x) ((*(uint32_t *) x) & MASK)
// Iteration 1
- p = &long_state[state_index(a)];
+ j = state_index(a);
+ p = &long_state[j];
aesb_single_round(p, p, a);
+ copy_block(c1, p);
- xor_blocks(b, p);
- swap_blocks(b, p);
- swap_blocks(a, b);
+ VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+ xor_blocks(p, b);
VARIANT1_1(p);
// Iteration 2
- p = &long_state[state_index(a)];
-
- mul(a, p, d);
- sum_half_blocks(b, d);
- swap_blocks(b, p);
- xor_blocks(b, p);
- swap_blocks(a, b);
- VARIANT1_2(U64(p) + 1);
+ j = state_index(c1);
+ p = &long_state[j];
+ copy_block(c, p);
+
+ VARIANT2_PORTABLE_INTEGER_MATH(c, c1);
+ mul(c1, c, d);
+ VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+ sum_half_blocks(a, d);
+ swap_blocks(a, c);
+ xor_blocks(a, c);
+ VARIANT1_2(U64(c) + 1);
+ copy_block(p, c);
+
+ if (variant >= 2) {
+ copy_block(b + AES_BLOCK_SIZE, b);
+ }
+ copy_block(b, c1);
}
memcpy(text, state.init, INIT_SIZE_BYTE);
@@ -1298,8 +1434,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
union cn_slow_hash_state state;
uint8_t text[INIT_SIZE_BYTE];
uint8_t a[AES_BLOCK_SIZE];
- uint8_t b[AES_BLOCK_SIZE];
- uint8_t c[AES_BLOCK_SIZE];
+ uint8_t b[AES_BLOCK_SIZE * 2];
+ uint8_t c1[AES_BLOCK_SIZE];
+ uint8_t c2[AES_BLOCK_SIZE];
uint8_t d[AES_BLOCK_SIZE];
size_t i, j;
uint8_t aes_key[AES_KEY_SIZE];
@@ -1315,6 +1452,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
aes_ctx = (oaes_ctx *) oaes_alloc();
VARIANT1_PORTABLE_INIT();
+ VARIANT2_PORTABLE_INIT();
oaes_key_import_data(aes_ctx, aes_key, AES_KEY_SIZE);
for (i = 0; i < MEMORY / INIT_SIZE_BYTE; i++) {
@@ -1324,9 +1462,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
memcpy(&long_state[i * INIT_SIZE_BYTE], text, INIT_SIZE_BYTE);
}
- for (i = 0; i < 16; i++) {
- a[i] = state.k[ i] ^ state.k[32 + i];
- b[i] = state.k[16 + i] ^ state.k[48 + i];
+ for (i = 0; i < AES_BLOCK_SIZE; i++) {
+ a[i] = state.k[ i] ^ state.k[AES_BLOCK_SIZE * 2 + i];
+ b[i] = state.k[AES_BLOCK_SIZE + i] ^ state.k[AES_BLOCK_SIZE * 3 + i];
}
for (i = 0; i < ITER / 2; i++) {
@@ -1335,26 +1473,32 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
* next address <-+
*/
/* Iteration 1 */
- j = e2i(a, MEMORY / AES_BLOCK_SIZE);
- copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
- aesb_single_round(c, c, a);
- xor_blocks(b, c);
- swap_blocks(b, c);
- copy_block(&long_state[j * AES_BLOCK_SIZE], c);
- assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
- swap_blocks(a, b);
- VARIANT1_1(&long_state[j * AES_BLOCK_SIZE]);
+ j = e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
+ copy_block(c1, &long_state[j]);
+ aesb_single_round(c1, c1, a);
+ VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+ copy_block(&long_state[j], c1);
+ xor_blocks(&long_state[j], b);
+ assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE);
+ VARIANT1_1(&long_state[j]);
/* Iteration 2 */
- j = e2i(a, MEMORY / AES_BLOCK_SIZE);
- copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
- mul(a, c, d);
- sum_half_blocks(b, d);
- swap_blocks(b, c);
- xor_blocks(b, c);
- VARIANT1_2(c + 8);
- copy_block(&long_state[j * AES_BLOCK_SIZE], c);
- assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
- swap_blocks(a, b);
+ j = e2i(c1, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
+ copy_block(c2, &long_state[j]);
+ VARIANT2_PORTABLE_INTEGER_MATH(c2, c1);
+ mul(c1, c2, d);
+ VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+ swap_blocks(a, c1);
+ sum_half_blocks(c1, d);
+ swap_blocks(c1, c2);
+ xor_blocks(c1, c2);
+ VARIANT1_2(c2 + 8);
+ copy_block(&long_state[j], c2);
+ assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE);
+ if (variant >= 2) {
+ copy_block(b + AES_BLOCK_SIZE, b);
+ }
+ copy_block(b, a);
+ copy_block(a, c1);
}
memcpy(text, state.init, INIT_SIZE_BYTE);
diff --git a/src/crypto/variant2_int_sqrt.h b/src/crypto/variant2_int_sqrt.h
new file mode 100644
index 000000000..b405bb798
--- /dev/null
+++ b/src/crypto/variant2_int_sqrt.h
@@ -0,0 +1,163 @@
+#ifndef VARIANT2_INT_SQRT_H
+#define VARIANT2_INT_SQRT_H
+
+#include <math.h>
+#include <float.h>
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_SSE2() \
+ do { \
+ const __m128i exp_double_bias = _mm_set_epi64x(0, 1023ULL << 52); \
+ __m128d x = _mm_castsi128_pd(_mm_add_epi64(_mm_cvtsi64_si128(sqrt_input >> 12), exp_double_bias)); \
+ x = _mm_sqrt_sd(_mm_setzero_pd(), x); \
+ sqrt_result = (uint64_t)(_mm_cvtsi128_si64(_mm_sub_epi64(_mm_castpd_si128(x), exp_double_bias))) >> 19; \
+ } while(0)
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_FP64() \
+ do { \
+ sqrt_result = sqrt(sqrt_input + 18446744073709551616.0) * 2.0 - 8589934592.0; \
+ } while(0)
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_REF() \
+ sqrt_result = integer_square_root_v2(sqrt_input)
+
+// Reference implementation of the integer square root for Cryptonight variant 2
+// Computes integer part of "sqrt(2^64 + n) * 2 - 2^33"
+//
+// In other words, given 64-bit unsigned integer n:
+// 1) Write it as x = 1.NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN000... in binary (1 <= x < 2, all 64 bits of n are used)
+// 2) Calculate sqrt(x) = 1.0RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR... (1 <= sqrt(x) < sqrt(2), so it will always start with "1.0" in binary)
+// 3) Take 32 bits that come after "1.0" and return them as a 32-bit unsigned integer, discard all remaining bits
+//
+// Some sample inputs and outputs:
+//
+// Input | Output | Exact value of "sqrt(2^64 + n) * 2 - 2^33"
+// -----------------|------------|-------------------------------------------
+// 0 | 0 | 0
+// 2^32 | 0 | 0.99999999994179233909330885695244...
+// 2^32 + 1 | 1 | 1.0000000001746229827200734316305...
+// 2^50 | 262140 | 262140.00012206565608606978175873...
+// 2^55 + 20963331 | 8384515 | 8384515.9999999997673963974959744...
+// 2^55 + 20963332 | 8384516 | 8384516
+// 2^62 + 26599786 | 1013904242 | 1013904242.9999999999479374853545...
+// 2^62 + 26599787 | 1013904243 | 1013904243.0000000001561875439364...
+// 2^64 - 1 | 3558067407 | 3558067407.9041987696409179931096...
+
+// The reference implementation as it is now uses only unsigned int64 arithmetic, so it can't have undefined behavior
+// It was tested once for all edge cases and confirmed correct
+static inline uint32_t integer_square_root_v2(uint64_t n)
+{
+ uint64_t r = 1ULL << 63;
+
+ for (uint64_t bit = 1ULL << 60; bit; bit >>= 2)
+ {
+ const bool b = (n < r + bit);
+ const uint64_t n_next = n - (r + bit);
+ const uint64_t r_next = r + bit * 2;
+ n = b ? n : n_next;
+ r = b ? r : r_next;
+ r >>= 1;
+ }
+
+ return r * 2 + ((n > r) ? 1 : 0);
+}
+
+/*
+VARIANT2_INTEGER_MATH_SQRT_FIXUP checks that "r" is an integer part of "sqrt(2^64 + sqrt_input) * 2 - 2^33" and adds or subtracts 1 if needed
+It's hard to understand how it works, so here is a full calculation of formulas used in VARIANT2_INTEGER_MATH_SQRT_FIXUP
+
+The following inequalities must hold for r if it's an integer part of "sqrt(2^64 + sqrt_input) * 2 - 2^33":
+1) r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+2) r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+
+We need to check them using only unsigned integer arithmetic to avoid rounding errors and undefined behavior
+
+First inequality: r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+-----------------------------------------------------------------------------------
+r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+r + 2^33 <= sqrt(2^64 + sqrt_input) * 2
+r/2 + 2^32 <= sqrt(2^64 + sqrt_input)
+(r/2 + 2^32)^2 <= 2^64 + sqrt_input
+
+Rewrite r as r = s * 2 + b (s = trunc(r/2), b is 0 or 1)
+
+((s*2+b)/2 + 2^32)^2 <= 2^64 + sqrt_input
+(s*2+b)^2/4 + 2*2^32*(s*2+b)/2 + 2^64 <= 2^64 + sqrt_input
+(s*2+b)^2/4 + 2*2^32*(s*2+b)/2 <= sqrt_input
+(s*2+b)^2/4 + 2^32*r <= sqrt_input
+(s^2*4+2*s*2*b+b^2)/4 + 2^32*r <= sqrt_input
+s^2+s*b+b^2/4 + 2^32*r <= sqrt_input
+s*(s+b) + b^2/4 + 2^32*r <= sqrt_input
+
+Let r2 = s*(s+b) + r*2^32
+r2 + b^2/4 <= sqrt_input
+
+If this inequality doesn't hold, then we must decrement r: IF "r2 + b^2/4 > sqrt_input" THEN r = r - 1
+
+b can be 0 or 1
+If b is 0 then we need to compare "r2 > sqrt_input"
+If b is 1 then b^2/4 = 0.25, so we need to compare "r2 + 0.25 > sqrt_input"
+Since both r2 and sqrt_input are integers, we can safely replace it with "r2 + 1 > sqrt_input"
+-----------------------------------------------------------------------------------
+Both cases can be merged to a single expression "r2 + b > sqrt_input"
+-----------------------------------------------------------------------------------
+There will be no overflow when calculating "r2 + b", so it's safe to compare with sqrt_input:
+r2 + b = s*(s+b) + r*2^32 + b
+The largest value s, b and r can have is s = 1779033703, b = 1, r = 3558067407 when sqrt_input = 2^64 - 1
+r2 + b <= 1779033703*1779033704 + 3558067407*2^32 + 1 = 18446744068217447385 < 2^64
+
+Second inequality: r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+-----------------------------------------------------------------------------------
+r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+r + 1 + 2^33 > sqrt(2^64 + sqrt_input) * 2
+((r+1)/2 + 2^32)^2 > 2^64 + sqrt_input
+
+Rewrite r as r = s * 2 + b (s = trunc(r/2), b is 0 or 1)
+
+((s*2+b+1)/2 + 2^32)^2 > 2^64 + sqrt_input
+(s*2+b+1)^2/4 + 2*(s*2+b+1)/2*2^32 + 2^64 > 2^64 + sqrt_input
+(s*2+b+1)^2/4 + (s*2+b+1)*2^32 > sqrt_input
+(s*2+b+1)^2/4 + (r+1)*2^32 > sqrt_input
+(s*2+(b+1))^2/4 + r*2^32 + 2^32 > sqrt_input
+(s^2*4+2*s*2*(b+1)+(b+1)^2)/4 + r*2^32 + 2^32 > sqrt_input
+s^2+s*(b+1)+(b+1)^2/4 + r*2^32 + 2^32 > sqrt_input
+s*(s+b) + s + (b+1)^2/4 + r*2^32 + 2^32 > sqrt_input
+
+Let r2 = s*(s+b) + r*2^32
+
+r2 + s + (b+1)^2/4 + 2^32 > sqrt_input
+r2 + 2^32 + (b+1)^2/4 > sqrt_input - s
+
+If this inequality doesn't hold, then we must decrement r: IF "r2 + 2^32 + (b+1)^2/4 <= sqrt_input - s" THEN r = r - 1
+b can be 0 or 1
+If b is 0 then we need to compare "r2 + 2^32 + 1/4 <= sqrt_input - s" which is equal to "r2 + 2^32 < sqrt_input - s" because all numbers here are integers
+If b is 1 then (b+1)^2/4 = 1, so we need to compare "r2 + 2^32 + 1 <= sqrt_input - s" which is also equal to "r2 + 2^32 < sqrt_input - s"
+-----------------------------------------------------------------------------------
+Both cases can be merged to a single expression "r2 + 2^32 < sqrt_input - s"
+-----------------------------------------------------------------------------------
+There will be no overflow when calculating "r2 + 2^32":
+r2 + 2^32 = s*(s+b) + r*2^32 + 2^32 = s*(s+b) + (r+1)*2^32
+The largest value s, b and r can have is s = 1779033703, b = 1, r = 3558067407 when sqrt_input = 2^64 - 1
+r2 + b <= 1779033703*1779033704 + 3558067408*2^32 = 18446744072512414680 < 2^64
+
+There will be no integer overflow when calculating "sqrt_input - s", i.e. "sqrt_input >= s" at all times:
+s = trunc(r/2) = trunc(sqrt(2^64 + sqrt_input) - 2^32) < sqrt(2^64 + sqrt_input) - 2^32 + 1
+sqrt_input > sqrt(2^64 + sqrt_input) - 2^32 + 1
+sqrt_input + 2^32 - 1 > sqrt(2^64 + sqrt_input)
+(sqrt_input + 2^32 - 1)^2 > sqrt_input + 2^64
+sqrt_input^2 + 2*sqrt_input*(2^32 - 1) + (2^32-1)^2 > sqrt_input + 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 2) + (2^32-1)^2 > sqrt_input + 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) + (2^32-1)^2 > 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) + 2^64-2^33+1 > 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) - 2^33 + 1 > 0
+This inequality is true if sqrt_input > 1 and it's easy to check that s = 0 if sqrt_input is 0 or 1, so there will be no integer overflow
+*/
+
+#define VARIANT2_INTEGER_MATH_SQRT_FIXUP(r) \
+ do { \
+ const uint64_t s = r >> 1; \
+ const uint64_t b = r & 1; \
+ const uint64_t r2 = (uint64_t)(s) * (s + b) + (r << 32); \
+ r += ((r2 + b > sqrt_input) ? -1 : 0) + ((r2 + (1ULL << 32) < sqrt_input - s) ? 1 : 0); \
+ } while(0)
+
+#endif
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