/* Deterministic conversion between `double` and its decimal text form. * * The bulk of this file is vendored verbatim from Wuffs so that it stays cheap * to diff against upstream when picking up fixes: * * https://github.com/google/wuffs * internal/cgen/base/floatconv-submodule-data.c * internal/cgen/base/floatconv-submodule-code.c * * Copyright 2020 The Wuffs Authors. * SPDX-License-Identifier: Apache-2.0 OR MIT * * Parsing is Eisel-Lemire with an exact high-precision-decimal fallback; * rendering runs the same decimal machinery backwards. Both are correctly * rounded, locale independent and (with the one exception guarded below) use * integer arithmetic only, which is what makes them deterministic. * * Deviations from upstream are tagged `[pocketpy]`: * 1. the f16/f32 entry points are dropped, as is the wuffs_base__ machinery * they need; what little remains is reimplemented in the shim below; * 2. every entry point is `static`, since pocketpy exposes its own API at * the bottom of this file; * 3. the shortest-round-trip renderer switches to exponent notation on * CPython's threshold rather than C's "%g" one; * 4. the sole floating-point fast path is compiled out on targets with * excess intermediate precision. * * Do not run clang-format over the vendored region; `scripts/format.py` skips * this file on purpose. */ #include "pocketpy/common/floatconv.h" #include #include #include #include /* ---------------- [pocketpy] wuffs_base__ shim ---------------- * * Just enough of Wuffs' base module for the vendored code to compile. These * are copied from internal/cgen/base/fundamental-public.h and * internal/cgen/base/strconv-public.h. */ #define WUFFS_BASE__MAYBE_STATIC static typedef struct wuffs_base__slice_u8__struct { uint8_t* ptr; size_t len; } wuffs_base__slice_u8; typedef struct wuffs_base__status__struct { const char* repr; } wuffs_base__status; typedef struct wuffs_base__result_f64__struct { wuffs_base__status status; double value; } wuffs_base__result_f64; static const char wuffs_base__error__bad_argument[] = "#base: bad argument"; static const char wuffs_base__error__bad_receiver[] = "#base: bad receiver"; static inline wuffs_base__status // wuffs_base__make_status(const char* repr) { wuffs_base__status z; z.repr = repr; return z; } static inline int32_t // wuffs_base__i32__max(int32_t x, int32_t y) { return x > y ? x : y; } static inline uint32_t // wuffs_base__u32__min(uint32_t x, uint32_t y) { return x < y ? x : y; } #if (defined(__GNUC__) || defined(__clang__)) && (__SIZEOF_LONG__ == 8) static inline uint32_t // wuffs_base__count_leading_zeroes_u64(uint64_t u) { return u ? ((uint32_t)(__builtin_clzl(u))) : 64u; } #else static inline uint32_t // wuffs_base__count_leading_zeroes_u64(uint64_t u) { if (u == 0) { return 64; } uint32_t n = 0; if ((u >> 32) == 0) { n |= 32; u <<= 32; } if ((u >> 48) == 0) { n |= 16; u <<= 16; } if ((u >> 56) == 0) { n |= 8; u <<= 8; } if ((u >> 60) == 0) { n |= 4; u <<= 4; } if ((u >> 62) == 0) { n |= 2; u <<= 2; } if ((u >> 63) == 0) { n |= 1; u <<= 1; } return n; } #endif typedef struct wuffs_base__multiply_u64__output__struct { uint64_t lo; uint64_t hi; } wuffs_base__multiply_u64__output; static inline wuffs_base__multiply_u64__output // wuffs_base__multiply_u64(uint64_t x, uint64_t y) { #if defined(__SIZEOF_INT128__) __uint128_t z = ((__uint128_t)x) * ((__uint128_t)y); wuffs_base__multiply_u64__output o; o.lo = ((uint64_t)(z)); o.hi = ((uint64_t)(z >> 64)); return o; #else uint64_t x0 = x & 0xFFFFFFFF; uint64_t x1 = x >> 32; uint64_t y0 = y & 0xFFFFFFFF; uint64_t y1 = y >> 32; uint64_t w0 = x0 * y0; uint64_t t = (x1 * y0) + (w0 >> 32); uint64_t w1 = t & 0xFFFFFFFF; uint64_t w2 = t >> 32; w1 += x0 * y1; wuffs_base__multiply_u64__output o; o.lo = x * y; o.hi = (x1 * y1) + w2 + (w1 >> 32); return o; #endif } static inline void // wuffs_base__poke_u24le__no_bounds_check(uint8_t* p, uint32_t x) { p[0] = (uint8_t)(x >> 0); p[1] = (uint8_t)(x >> 8); p[2] = (uint8_t)(x >> 16); } static inline void // wuffs_base__poke_u32le__no_bounds_check(uint8_t* p, uint32_t x) { p[0] = (uint8_t)(x >> 0); p[1] = (uint8_t)(x >> 8); p[2] = (uint8_t)(x >> 16); p[3] = (uint8_t)(x >> 24); } static inline uint64_t // wuffs_base__ieee_754_bit_representation__from_f64_to_u64(double f) { uint64_t u = 0; if (sizeof(uint64_t) == sizeof(double)) { memcpy(&u, &f, sizeof(uint64_t)); } return u; } static inline double // wuffs_base__ieee_754_bit_representation__from_u64_to_f64(uint64_t u) { double f = 0; if (sizeof(uint64_t) == sizeof(double)) { memcpy(&f, &u, sizeof(uint64_t)); } return f; } // Options for wuffs_base__parse_number_f64. #define WUFFS_BASE__PARSE_NUMBER_XXX__DEFAULT_OPTIONS ((uint32_t)0x00000000) #define WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_MULTIPLE_LEADING_ZEROES \ ((uint32_t)0x00000001) #define WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES ((uint32_t)0x00000002) #define WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA \ ((uint32_t)0x00000010) #define WUFFS_BASE__PARSE_NUMBER_FXX__REJECT_INF_AND_NAN ((uint32_t)0x00000020) // Options for wuffs_base__render_number_f64. #define WUFFS_BASE__RENDER_NUMBER_XXX__DEFAULT_OPTIONS ((uint32_t)0x00000000) #define WUFFS_BASE__RENDER_NUMBER_XXX__ALIGN_RIGHT ((uint32_t)0x00000100) #define WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN ((uint32_t)0x00000200) #define WUFFS_BASE__RENDER_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA \ ((uint32_t)0x00001000) #define WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_ABSENT ((uint32_t)0x00002000) #define WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_PRESENT ((uint32_t)0x00004000) #define WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION \ ((uint32_t)0x00008000) /* [pocketpy] Deviation 3. * * Wuffs' "%g" notation follows C and switches to an exponent once the decimal * point moves past 6 digits, so 1e6 would render as "1e+06". CPython's repr() * instead switches once `decimal_point > 16`, which is what * `wuffs_private_impl__high_prec_dec__render_*` calls an `e_threshold` of 16. * See `format_float_short` in CPython's Python/pystrtod.c. */ #define PK_FLOATCONV_REPR_E_THRESHOLD 16 /* [pocketpy] Deviation 4. * * Everything below is integer arithmetic except for one `d *= power_of_10` * fast path in wuffs_base__parse_number_f64. That multiply is exact and * correctly rounded on an IEEE-754 target, but on a target that evaluates * doubles in a wider format -- 32-bit x86 using the x87 stack is the one that * still matters -- it rounds twice and can land one ulp away from what the * integer path computes. Compile it out there and let Eisel-Lemire handle * those inputs instead; the answer is the same, just a few ns slower. */ #if !defined(FLT_EVAL_METHOD) || (FLT_EVAL_METHOD == 0) #define PK_FLOATCONV_HAS_EXACT_DOUBLE_ARITHMETIC 1 #else #define PK_FLOATCONV_HAS_EXACT_DOUBLE_ARITHMETIC 0 #endif /* ---------------- end of the [pocketpy] shim ---------------- */ // ---------------- IEEE 754 Floating Point // The etc__hpd_left_shift and etc__powers_of_5 tables were printed by // script/print-hpd-left-shift.go. That script has an optional -comments flag, // whose output is not copied here, which prints further detail. // // These tables are used in // wuffs_private_impl__high_prec_dec__lshift_num_new_digits. // wuffs_private_impl__hpd_left_shift[i] encodes the number of new digits // created after multiplying a positive integer by (1 << i): the additional // length in the decimal representation. For example, shifting "234" by 3 // (equivalent to multiplying by 8) will produce "1872". Going from a 3-length // string to a 4-length string means that 1 new digit was added (and existing // digits may have changed). // // Shifting by i can add either N or N-1 new digits, depending on whether the // original positive integer compares >= or < to the i'th power of 5 (as 10 // equals 2 * 5). Comparison is lexicographic, not numerical. // // For example, shifting by 4 (i.e. multiplying by 16) can add 1 or 2 new // digits, depending on a lexicographic comparison to (5 ** 4), i.e. "625": // - ("1" << 4) is "16", which adds 1 new digit. // - ("5678" << 4) is "90848", which adds 1 new digit. // - ("624" << 4) is "9984", which adds 1 new digit. // - ("62498" << 4) is "999968", which adds 1 new digit. // - ("625" << 4) is "10000", which adds 2 new digits. // - ("625001" << 4) is "10000016", which adds 2 new digits. // - ("7008" << 4) is "112128", which adds 2 new digits. // - ("99" << 4) is "1584", which adds 2 new digits. // // Thus, when i is 4, N is 2 and (5 ** i) is "625". This etc__hpd_left_shift // array encodes this as: // - etc__hpd_left_shift[4] is 0x1006 = (2 << 11) | 0x0006. // - etc__hpd_left_shift[5] is 0x1009 = (? << 11) | 0x0009. // where the ? isn't relevant for i == 4. // // The high 5 bits of etc__hpd_left_shift[i] is N, the higher of the two // possible number of new digits. The low 11 bits are an offset into the // etc__powers_of_5 array (of length 0x051C, so offsets fit in 11 bits). When i // is 4, its offset and the next one is 6 and 9, and etc__powers_of_5[6 .. 9] // is the string "\x06\x02\x05", so the relevant power of 5 is "625". // // Thanks to Ken Thompson for the original idea. static const uint16_t wuffs_private_impl__hpd_left_shift[65] = { 0x0000, 0x0800, 0x0801, 0x0803, 0x1006, 0x1009, 0x100D, 0x1812, 0x1817, 0x181D, 0x2024, 0x202B, 0x2033, 0x203C, 0x2846, 0x2850, 0x285B, 0x3067, 0x3073, 0x3080, 0x388E, 0x389C, 0x38AB, 0x38BB, 0x40CC, 0x40DD, 0x40EF, 0x4902, 0x4915, 0x4929, 0x513E, 0x5153, 0x5169, 0x5180, 0x5998, 0x59B0, 0x59C9, 0x61E3, 0x61FD, 0x6218, 0x6A34, 0x6A50, 0x6A6D, 0x6A8B, 0x72AA, 0x72C9, 0x72E9, 0x7B0A, 0x7B2B, 0x7B4D, 0x8370, 0x8393, 0x83B7, 0x83DC, 0x8C02, 0x8C28, 0x8C4F, 0x9477, 0x949F, 0x94C8, 0x9CF2, 0x051C, 0x051C, 0x051C, 0x051C, }; // wuffs_private_impl__powers_of_5 contains the powers of 5, concatenated // together: "5", "25", "125", "625", "3125", etc. static const uint8_t wuffs_private_impl__powers_of_5[0x051C] = { 5, 2, 5, 1, 2, 5, 6, 2, 5, 3, 1, 2, 5, 1, 5, 6, 2, 5, 7, 8, 1, 2, 5, 3, 9, 0, 6, 2, 5, 1, 9, 5, 3, 1, 2, 5, 9, 7, 6, 5, 6, 2, 5, 4, 8, 8, 2, 8, 1, 2, 5, 2, 4, 4, 1, 4, 0, 6, 2, 5, 1, 2, 2, 0, 7, 0, 3, 1, 2, 5, 6, 1, 0, 3, 5, 1, 5, 6, 2, 5, 3, 0, 5, 1, 7, 5, 7, 8, 1, 2, 5, 1, 5, 2, 5, 8, 7, 8, 9, 0, 6, 2, 5, 7, 6, 2, 9, 3, 9, 4, 5, 3, 1, 2, 5, 3, 8, 1, 4, 6, 9, 7, 2, 6, 5, 6, 2, 5, 1, 9, 0, 7, 3, 4, 8, 6, 3, 2, 8, 1, 2, 5, 9, 5, 3, 6, 7, 4, 3, 1, 6, 4, 0, 6, 2, 5, 4, 7, 6, 8, 3, 7, 1, 5, 8, 2, 0, 3, 1, 2, 5, 2, 3, 8, 4, 1, 8, 5, 7, 9, 1, 0, 1, 5, 6, 2, 5, 1, 1, 9, 2, 0, 9, 2, 8, 9, 5, 5, 0, 7, 8, 1, 2, 5, 5, 9, 6, 0, 4, 6, 4, 4, 7, 7, 5, 3, 9, 0, 6, 2, 5, 2, 9, 8, 0, 2, 3, 2, 2, 3, 8, 7, 6, 9, 5, 3, 1, 2, 5, 1, 4, 9, 0, 1, 1, 6, 1, 1, 9, 3, 8, 4, 7, 6, 5, 6, 2, 5, 7, 4, 5, 0, 5, 8, 0, 5, 9, 6, 9, 2, 3, 8, 2, 8, 1, 2, 5, 3, 7, 2, 5, 2, 9, 0, 2, 9, 8, 4, 6, 1, 9, 1, 4, 0, 6, 2, 5, 1, 8, 6, 2, 6, 4, 5, 1, 4, 9, 2, 3, 0, 9, 5, 7, 0, 3, 1, 2, 5, 9, 3, 1, 3, 2, 2, 5, 7, 4, 6, 1, 5, 4, 7, 8, 5, 1, 5, 6, 2, 5, 4, 6, 5, 6, 6, 1, 2, 8, 7, 3, 0, 7, 7, 3, 9, 2, 5, 7, 8, 1, 2, 5, 2, 3, 2, 8, 3, 0, 6, 4, 3, 6, 5, 3, 8, 6, 9, 6, 2, 8, 9, 0, 6, 2, 5, 1, 1, 6, 4, 1, 5, 3, 2, 1, 8, 2, 6, 9, 3, 4, 8, 1, 4, 4, 5, 3, 1, 2, 5, 5, 8, 2, 0, 7, 6, 6, 0, 9, 1, 3, 4, 6, 7, 4, 0, 7, 2, 2, 6, 5, 6, 2, 5, 2, 9, 1, 0, 3, 8, 3, 0, 4, 5, 6, 7, 3, 3, 7, 0, 3, 6, 1, 3, 2, 8, 1, 2, 5, 1, 4, 5, 5, 1, 9, 1, 5, 2, 2, 8, 3, 6, 6, 8, 5, 1, 8, 0, 6, 6, 4, 0, 6, 2, 5, 7, 2, 7, 5, 9, 5, 7, 6, 1, 4, 1, 8, 3, 4, 2, 5, 9, 0, 3, 3, 2, 0, 3, 1, 2, 5, 3, 6, 3, 7, 9, 7, 8, 8, 0, 7, 0, 9, 1, 7, 1, 2, 9, 5, 1, 6, 6, 0, 1, 5, 6, 2, 5, 1, 8, 1, 8, 9, 8, 9, 4, 0, 3, 5, 4, 5, 8, 5, 6, 4, 7, 5, 8, 3, 0, 0, 7, 8, 1, 2, 5, 9, 0, 9, 4, 9, 4, 7, 0, 1, 7, 7, 2, 9, 2, 8, 2, 3, 7, 9, 1, 5, 0, 3, 9, 0, 6, 2, 5, 4, 5, 4, 7, 4, 7, 3, 5, 0, 8, 8, 6, 4, 6, 4, 1, 1, 8, 9, 5, 7, 5, 1, 9, 5, 3, 1, 2, 5, 2, 2, 7, 3, 7, 3, 6, 7, 5, 4, 4, 3, 2, 3, 2, 0, 5, 9, 4, 7, 8, 7, 5, 9, 7, 6, 5, 6, 2, 5, 1, 1, 3, 6, 8, 6, 8, 3, 7, 7, 2, 1, 6, 1, 6, 0, 2, 9, 7, 3, 9, 3, 7, 9, 8, 8, 2, 8, 1, 2, 5, 5, 6, 8, 4, 3, 4, 1, 8, 8, 6, 0, 8, 0, 8, 0, 1, 4, 8, 6, 9, 6, 8, 9, 9, 4, 1, 4, 0, 6, 2, 5, 2, 8, 4, 2, 1, 7, 0, 9, 4, 3, 0, 4, 0, 4, 0, 0, 7, 4, 3, 4, 8, 4, 4, 9, 7, 0, 7, 0, 3, 1, 2, 5, 1, 4, 2, 1, 0, 8, 5, 4, 7, 1, 5, 2, 0, 2, 0, 0, 3, 7, 1, 7, 4, 2, 2, 4, 8, 5, 3, 5, 1, 5, 6, 2, 5, 7, 1, 0, 5, 4, 2, 7, 3, 5, 7, 6, 0, 1, 0, 0, 1, 8, 5, 8, 7, 1, 1, 2, 4, 2, 6, 7, 5, 7, 8, 1, 2, 5, 3, 5, 5, 2, 7, 1, 3, 6, 7, 8, 8, 0, 0, 5, 0, 0, 9, 2, 9, 3, 5, 5, 6, 2, 1, 3, 3, 7, 8, 9, 0, 6, 2, 5, 1, 7, 7, 6, 3, 5, 6, 8, 3, 9, 4, 0, 0, 2, 5, 0, 4, 6, 4, 6, 7, 7, 8, 1, 0, 6, 6, 8, 9, 4, 5, 3, 1, 2, 5, 8, 8, 8, 1, 7, 8, 4, 1, 9, 7, 0, 0, 1, 2, 5, 2, 3, 2, 3, 3, 8, 9, 0, 5, 3, 3, 4, 4, 7, 2, 6, 5, 6, 2, 5, 4, 4, 4, 0, 8, 9, 2, 0, 9, 8, 5, 0, 0, 6, 2, 6, 1, 6, 1, 6, 9, 4, 5, 2, 6, 6, 7, 2, 3, 6, 3, 2, 8, 1, 2, 5, 2, 2, 2, 0, 4, 4, 6, 0, 4, 9, 2, 5, 0, 3, 1, 3, 0, 8, 0, 8, 4, 7, 2, 6, 3, 3, 3, 6, 1, 8, 1, 6, 4, 0, 6, 2, 5, 1, 1, 1, 0, 2, 2, 3, 0, 2, 4, 6, 2, 5, 1, 5, 6, 5, 4, 0, 4, 2, 3, 6, 3, 1, 6, 6, 8, 0, 9, 0, 8, 2, 0, 3, 1, 2, 5, 5, 5, 5, 1, 1, 1, 5, 1, 2, 3, 1, 2, 5, 7, 8, 2, 7, 0, 2, 1, 1, 8, 1, 5, 8, 3, 4, 0, 4, 5, 4, 1, 0, 1, 5, 6, 2, 5, 2, 7, 7, 5, 5, 5, 7, 5, 6, 1, 5, 6, 2, 8, 9, 1, 3, 5, 1, 0, 5, 9, 0, 7, 9, 1, 7, 0, 2, 2, 7, 0, 5, 0, 7, 8, 1, 2, 5, 1, 3, 8, 7, 7, 7, 8, 7, 8, 0, 7, 8, 1, 4, 4, 5, 6, 7, 5, 5, 2, 9, 5, 3, 9, 5, 8, 5, 1, 1, 3, 5, 2, 5, 3, 9, 0, 6, 2, 5, 6, 9, 3, 8, 8, 9, 3, 9, 0, 3, 9, 0, 7, 2, 2, 8, 3, 7, 7, 6, 4, 7, 6, 9, 7, 9, 2, 5, 5, 6, 7, 6, 2, 6, 9, 5, 3, 1, 2, 5, 3, 4, 6, 9, 4, 4, 6, 9, 5, 1, 9, 5, 3, 6, 1, 4, 1, 8, 8, 8, 2, 3, 8, 4, 8, 9, 6, 2, 7, 8, 3, 8, 1, 3, 4, 7, 6, 5, 6, 2, 5, 1, 7, 3, 4, 7, 2, 3, 4, 7, 5, 9, 7, 6, 8, 0, 7, 0, 9, 4, 4, 1, 1, 9, 2, 4, 4, 8, 1, 3, 9, 1, 9, 0, 6, 7, 3, 8, 2, 8, 1, 2, 5, 8, 6, 7, 3, 6, 1, 7, 3, 7, 9, 8, 8, 4, 0, 3, 5, 4, 7, 2, 0, 5, 9, 6, 2, 2, 4, 0, 6, 9, 5, 9, 5, 3, 3, 6, 9, 1, 4, 0, 6, 2, 5, }; // -------- // wuffs_private_impl__powers_of_10 contains truncated approximations to the // powers of 10, ranging from 1e-307 to 1e+288 inclusive, as 596 pairs of // uint64_t values (a 128-bit mantissa). // // There's also an implicit third column (implied by a linear formula involving // the base-10 exponent) that is the base-2 exponent, biased by a magic // constant. That constant (1214 or 0x04BE) equals 1023 + 191. 1023 is the bias // for IEEE 754 double-precision floating point. 191 is ((3 * 64) - 1) and // wuffs_private_impl__parse_number_f64_eisel_lemire works with // multiples-of-64-bit mantissas. // // For example, the third row holds the approximation to 1e-305: // 0xE0B62E29_29ABA83C_331ACDAB_FE94DE87 * (2 ** (0x0049 - 0x04BE)) // // Similarly, 1e+4 is approximated by: // 0x9C400000_00000000_00000000_00000000 * (2 ** (0x044C - 0x04BE)) // // Similarly, 1e+68 is approximated by: // 0xED63A231_D4C4FB27_4CA7AAA8_63EE4BDD * (2 ** (0x0520 - 0x04BE)) // // This table was generated by by script/print-mpb-powers-of-10.go static const uint64_t wuffs_private_impl__powers_of_10[596][2] = { {0xA5D3B6D479F8E056, 0x8FD0C16206306BAB}, // 1e-307 {0x8F48A4899877186C, 0xB3C4F1BA87BC8696}, // 1e-306 {0x331ACDABFE94DE87, 0xE0B62E2929ABA83C}, // 1e-305 {0x9FF0C08B7F1D0B14, 0x8C71DCD9BA0B4925}, // 1e-304 {0x07ECF0AE5EE44DD9, 0xAF8E5410288E1B6F}, // 1e-303 {0xC9E82CD9F69D6150, 0xDB71E91432B1A24A}, // 1e-302 {0xBE311C083A225CD2, 0x892731AC9FAF056E}, // 1e-301 {0x6DBD630A48AAF406, 0xAB70FE17C79AC6CA}, // 1e-300 {0x092CBBCCDAD5B108, 0xD64D3D9DB981787D}, // 1e-299 {0x25BBF56008C58EA5, 0x85F0468293F0EB4E}, // 1e-298 {0xAF2AF2B80AF6F24E, 0xA76C582338ED2621}, // 1e-297 {0x1AF5AF660DB4AEE1, 0xD1476E2C07286FAA}, // 1e-296 {0x50D98D9FC890ED4D, 0x82CCA4DB847945CA}, // 1e-295 {0xE50FF107BAB528A0, 0xA37FCE126597973C}, // 1e-294 {0x1E53ED49A96272C8, 0xCC5FC196FEFD7D0C}, // 1e-293 {0x25E8E89C13BB0F7A, 0xFF77B1FCBEBCDC4F}, // 1e-292 {0x77B191618C54E9AC, 0x9FAACF3DF73609B1}, // 1e-291 {0xD59DF5B9EF6A2417, 0xC795830D75038C1D}, // 1e-290 {0x4B0573286B44AD1D, 0xF97AE3D0D2446F25}, // 1e-289 {0x4EE367F9430AEC32, 0x9BECCE62836AC577}, // 1e-288 {0x229C41F793CDA73F, 0xC2E801FB244576D5}, // 1e-287 {0x6B43527578C1110F, 0xF3A20279ED56D48A}, // 1e-286 {0x830A13896B78AAA9, 0x9845418C345644D6}, // 1e-285 {0x23CC986BC656D553, 0xBE5691EF416BD60C}, // 1e-284 {0x2CBFBE86B7EC8AA8, 0xEDEC366B11C6CB8F}, // 1e-283 {0x7BF7D71432F3D6A9, 0x94B3A202EB1C3F39}, // 1e-282 {0xDAF5CCD93FB0CC53, 0xB9E08A83A5E34F07}, // 1e-281 {0xD1B3400F8F9CFF68, 0xE858AD248F5C22C9}, // 1e-280 {0x23100809B9C21FA1, 0x91376C36D99995BE}, // 1e-279 {0xABD40A0C2832A78A, 0xB58547448FFFFB2D}, // 1e-278 {0x16C90C8F323F516C, 0xE2E69915B3FFF9F9}, // 1e-277 {0xAE3DA7D97F6792E3, 0x8DD01FAD907FFC3B}, // 1e-276 {0x99CD11CFDF41779C, 0xB1442798F49FFB4A}, // 1e-275 {0x40405643D711D583, 0xDD95317F31C7FA1D}, // 1e-274 {0x482835EA666B2572, 0x8A7D3EEF7F1CFC52}, // 1e-273 {0xDA3243650005EECF, 0xAD1C8EAB5EE43B66}, // 1e-272 {0x90BED43E40076A82, 0xD863B256369D4A40}, // 1e-271 {0x5A7744A6E804A291, 0x873E4F75E2224E68}, // 1e-270 {0x711515D0A205CB36, 0xA90DE3535AAAE202}, // 1e-269 {0x0D5A5B44CA873E03, 0xD3515C2831559A83}, // 1e-268 {0xE858790AFE9486C2, 0x8412D9991ED58091}, // 1e-267 {0x626E974DBE39A872, 0xA5178FFF668AE0B6}, // 1e-266 {0xFB0A3D212DC8128F, 0xCE5D73FF402D98E3}, // 1e-265 {0x7CE66634BC9D0B99, 0x80FA687F881C7F8E}, // 1e-264 {0x1C1FFFC1EBC44E80, 0xA139029F6A239F72}, // 1e-263 {0xA327FFB266B56220, 0xC987434744AC874E}, // 1e-262 {0x4BF1FF9F0062BAA8, 0xFBE9141915D7A922}, // 1e-261 {0x6F773FC3603DB4A9, 0x9D71AC8FADA6C9B5}, // 1e-260 {0xCB550FB4384D21D3, 0xC4CE17B399107C22}, // 1e-259 {0x7E2A53A146606A48, 0xF6019DA07F549B2B}, // 1e-258 {0x2EDA7444CBFC426D, 0x99C102844F94E0FB}, // 1e-257 {0xFA911155FEFB5308, 0xC0314325637A1939}, // 1e-256 {0x793555AB7EBA27CA, 0xF03D93EEBC589F88}, // 1e-255 {0x4BC1558B2F3458DE, 0x96267C7535B763B5}, // 1e-254 {0x9EB1AAEDFB016F16, 0xBBB01B9283253CA2}, // 1e-253 {0x465E15A979C1CADC, 0xEA9C227723EE8BCB}, // 1e-252 {0x0BFACD89EC191EC9, 0x92A1958A7675175F}, // 1e-251 {0xCEF980EC671F667B, 0xB749FAED14125D36}, // 1e-250 {0x82B7E12780E7401A, 0xE51C79A85916F484}, // 1e-249 {0xD1B2ECB8B0908810, 0x8F31CC0937AE58D2}, // 1e-248 {0x861FA7E6DCB4AA15, 0xB2FE3F0B8599EF07}, // 1e-247 {0x67A791E093E1D49A, 0xDFBDCECE67006AC9}, // 1e-246 {0xE0C8BB2C5C6D24E0, 0x8BD6A141006042BD}, // 1e-245 {0x58FAE9F773886E18, 0xAECC49914078536D}, // 1e-244 {0xAF39A475506A899E, 0xDA7F5BF590966848}, // 1e-243 {0x6D8406C952429603, 0x888F99797A5E012D}, // 1e-242 {0xC8E5087BA6D33B83, 0xAAB37FD7D8F58178}, // 1e-241 {0xFB1E4A9A90880A64, 0xD5605FCDCF32E1D6}, // 1e-240 {0x5CF2EEA09A55067F, 0x855C3BE0A17FCD26}, // 1e-239 {0xF42FAA48C0EA481E, 0xA6B34AD8C9DFC06F}, // 1e-238 {0xF13B94DAF124DA26, 0xD0601D8EFC57B08B}, // 1e-237 {0x76C53D08D6B70858, 0x823C12795DB6CE57}, // 1e-236 {0x54768C4B0C64CA6E, 0xA2CB1717B52481ED}, // 1e-235 {0xA9942F5DCF7DFD09, 0xCB7DDCDDA26DA268}, // 1e-234 {0xD3F93B35435D7C4C, 0xFE5D54150B090B02}, // 1e-233 {0xC47BC5014A1A6DAF, 0x9EFA548D26E5A6E1}, // 1e-232 {0x359AB6419CA1091B, 0xC6B8E9B0709F109A}, // 1e-231 {0xC30163D203C94B62, 0xF867241C8CC6D4C0}, // 1e-230 {0x79E0DE63425DCF1D, 0x9B407691D7FC44F8}, // 1e-229 {0x985915FC12F542E4, 0xC21094364DFB5636}, // 1e-228 {0x3E6F5B7B17B2939D, 0xF294B943E17A2BC4}, // 1e-227 {0xA705992CEECF9C42, 0x979CF3CA6CEC5B5A}, // 1e-226 {0x50C6FF782A838353, 0xBD8430BD08277231}, // 1e-225 {0xA4F8BF5635246428, 0xECE53CEC4A314EBD}, // 1e-224 {0x871B7795E136BE99, 0x940F4613AE5ED136}, // 1e-223 {0x28E2557B59846E3F, 0xB913179899F68584}, // 1e-222 {0x331AEADA2FE589CF, 0xE757DD7EC07426E5}, // 1e-221 {0x3FF0D2C85DEF7621, 0x9096EA6F3848984F}, // 1e-220 {0x0FED077A756B53A9, 0xB4BCA50B065ABE63}, // 1e-219 {0xD3E8495912C62894, 0xE1EBCE4DC7F16DFB}, // 1e-218 {0x64712DD7ABBBD95C, 0x8D3360F09CF6E4BD}, // 1e-217 {0xBD8D794D96AACFB3, 0xB080392CC4349DEC}, // 1e-216 {0xECF0D7A0FC5583A0, 0xDCA04777F541C567}, // 1e-215 {0xF41686C49DB57244, 0x89E42CAAF9491B60}, // 1e-214 {0x311C2875C522CED5, 0xAC5D37D5B79B6239}, // 1e-213 {0x7D633293366B828B, 0xD77485CB25823AC7}, // 1e-212 {0xAE5DFF9C02033197, 0x86A8D39EF77164BC}, // 1e-211 {0xD9F57F830283FDFC, 0xA8530886B54DBDEB}, // 1e-210 {0xD072DF63C324FD7B, 0xD267CAA862A12D66}, // 1e-209 {0x4247CB9E59F71E6D, 0x8380DEA93DA4BC60}, // 1e-208 {0x52D9BE85F074E608, 0xA46116538D0DEB78}, // 1e-207 {0x67902E276C921F8B, 0xCD795BE870516656}, // 1e-206 {0x00BA1CD8A3DB53B6, 0x806BD9714632DFF6}, // 1e-205 {0x80E8A40ECCD228A4, 0xA086CFCD97BF97F3}, // 1e-204 {0x6122CD128006B2CD, 0xC8A883C0FDAF7DF0}, // 1e-203 {0x796B805720085F81, 0xFAD2A4B13D1B5D6C}, // 1e-202 {0xCBE3303674053BB0, 0x9CC3A6EEC6311A63}, // 1e-201 {0xBEDBFC4411068A9C, 0xC3F490AA77BD60FC}, // 1e-200 {0xEE92FB5515482D44, 0xF4F1B4D515ACB93B}, // 1e-199 {0x751BDD152D4D1C4A, 0x991711052D8BF3C5}, // 1e-198 {0xD262D45A78A0635D, 0xBF5CD54678EEF0B6}, // 1e-197 {0x86FB897116C87C34, 0xEF340A98172AACE4}, // 1e-196 {0xD45D35E6AE3D4DA0, 0x9580869F0E7AAC0E}, // 1e-195 {0x8974836059CCA109, 0xBAE0A846D2195712}, // 1e-194 {0x2BD1A438703FC94B, 0xE998D258869FACD7}, // 1e-193 {0x7B6306A34627DDCF, 0x91FF83775423CC06}, // 1e-192 {0x1A3BC84C17B1D542, 0xB67F6455292CBF08}, // 1e-191 {0x20CABA5F1D9E4A93, 0xE41F3D6A7377EECA}, // 1e-190 {0x547EB47B7282EE9C, 0x8E938662882AF53E}, // 1e-189 {0xE99E619A4F23AA43, 0xB23867FB2A35B28D}, // 1e-188 {0x6405FA00E2EC94D4, 0xDEC681F9F4C31F31}, // 1e-187 {0xDE83BC408DD3DD04, 0x8B3C113C38F9F37E}, // 1e-186 {0x9624AB50B148D445, 0xAE0B158B4738705E}, // 1e-185 {0x3BADD624DD9B0957, 0xD98DDAEE19068C76}, // 1e-184 {0xE54CA5D70A80E5D6, 0x87F8A8D4CFA417C9}, // 1e-183 {0x5E9FCF4CCD211F4C, 0xA9F6D30A038D1DBC}, // 1e-182 {0x7647C3200069671F, 0xD47487CC8470652B}, // 1e-181 {0x29ECD9F40041E073, 0x84C8D4DFD2C63F3B}, // 1e-180 {0xF468107100525890, 0xA5FB0A17C777CF09}, // 1e-179 {0x7182148D4066EEB4, 0xCF79CC9DB955C2CC}, // 1e-178 {0xC6F14CD848405530, 0x81AC1FE293D599BF}, // 1e-177 {0xB8ADA00E5A506A7C, 0xA21727DB38CB002F}, // 1e-176 {0xA6D90811F0E4851C, 0xCA9CF1D206FDC03B}, // 1e-175 {0x908F4A166D1DA663, 0xFD442E4688BD304A}, // 1e-174 {0x9A598E4E043287FE, 0x9E4A9CEC15763E2E}, // 1e-173 {0x40EFF1E1853F29FD, 0xC5DD44271AD3CDBA}, // 1e-172 {0xD12BEE59E68EF47C, 0xF7549530E188C128}, // 1e-171 {0x82BB74F8301958CE, 0x9A94DD3E8CF578B9}, // 1e-170 {0xE36A52363C1FAF01, 0xC13A148E3032D6E7}, // 1e-169 {0xDC44E6C3CB279AC1, 0xF18899B1BC3F8CA1}, // 1e-168 {0x29AB103A5EF8C0B9, 0x96F5600F15A7B7E5}, // 1e-167 {0x7415D448F6B6F0E7, 0xBCB2B812DB11A5DE}, // 1e-166 {0x111B495B3464AD21, 0xEBDF661791D60F56}, // 1e-165 {0xCAB10DD900BEEC34, 0x936B9FCEBB25C995}, // 1e-164 {0x3D5D514F40EEA742, 0xB84687C269EF3BFB}, // 1e-163 {0x0CB4A5A3112A5112, 0xE65829B3046B0AFA}, // 1e-162 {0x47F0E785EABA72AB, 0x8FF71A0FE2C2E6DC}, // 1e-161 {0x59ED216765690F56, 0xB3F4E093DB73A093}, // 1e-160 {0x306869C13EC3532C, 0xE0F218B8D25088B8}, // 1e-159 {0x1E414218C73A13FB, 0x8C974F7383725573}, // 1e-158 {0xE5D1929EF90898FA, 0xAFBD2350644EEACF}, // 1e-157 {0xDF45F746B74ABF39, 0xDBAC6C247D62A583}, // 1e-156 {0x6B8BBA8C328EB783, 0x894BC396CE5DA772}, // 1e-155 {0x066EA92F3F326564, 0xAB9EB47C81F5114F}, // 1e-154 {0xC80A537B0EFEFEBD, 0xD686619BA27255A2}, // 1e-153 {0xBD06742CE95F5F36, 0x8613FD0145877585}, // 1e-152 {0x2C48113823B73704, 0xA798FC4196E952E7}, // 1e-151 {0xF75A15862CA504C5, 0xD17F3B51FCA3A7A0}, // 1e-150 {0x9A984D73DBE722FB, 0x82EF85133DE648C4}, // 1e-149 {0xC13E60D0D2E0EBBA, 0xA3AB66580D5FDAF5}, // 1e-148 {0x318DF905079926A8, 0xCC963FEE10B7D1B3}, // 1e-147 {0xFDF17746497F7052, 0xFFBBCFE994E5C61F}, // 1e-146 {0xFEB6EA8BEDEFA633, 0x9FD561F1FD0F9BD3}, // 1e-145 {0xFE64A52EE96B8FC0, 0xC7CABA6E7C5382C8}, // 1e-144 {0x3DFDCE7AA3C673B0, 0xF9BD690A1B68637B}, // 1e-143 {0x06BEA10CA65C084E, 0x9C1661A651213E2D}, // 1e-142 {0x486E494FCFF30A62, 0xC31BFA0FE5698DB8}, // 1e-141 {0x5A89DBA3C3EFCCFA, 0xF3E2F893DEC3F126}, // 1e-140 {0xF89629465A75E01C, 0x986DDB5C6B3A76B7}, // 1e-139 {0xF6BBB397F1135823, 0xBE89523386091465}, // 1e-138 {0x746AA07DED582E2C, 0xEE2BA6C0678B597F}, // 1e-137 {0xA8C2A44EB4571CDC, 0x94DB483840B717EF}, // 1e-136 {0x92F34D62616CE413, 0xBA121A4650E4DDEB}, // 1e-135 {0x77B020BAF9C81D17, 0xE896A0D7E51E1566}, // 1e-134 {0x0ACE1474DC1D122E, 0x915E2486EF32CD60}, // 1e-133 {0x0D819992132456BA, 0xB5B5ADA8AAFF80B8}, // 1e-132 {0x10E1FFF697ED6C69, 0xE3231912D5BF60E6}, // 1e-131 {0xCA8D3FFA1EF463C1, 0x8DF5EFABC5979C8F}, // 1e-130 {0xBD308FF8A6B17CB2, 0xB1736B96B6FD83B3}, // 1e-129 {0xAC7CB3F6D05DDBDE, 0xDDD0467C64BCE4A0}, // 1e-128 {0x6BCDF07A423AA96B, 0x8AA22C0DBEF60EE4}, // 1e-127 {0x86C16C98D2C953C6, 0xAD4AB7112EB3929D}, // 1e-126 {0xE871C7BF077BA8B7, 0xD89D64D57A607744}, // 1e-125 {0x11471CD764AD4972, 0x87625F056C7C4A8B}, // 1e-124 {0xD598E40D3DD89BCF, 0xA93AF6C6C79B5D2D}, // 1e-123 {0x4AFF1D108D4EC2C3, 0xD389B47879823479}, // 1e-122 {0xCEDF722A585139BA, 0x843610CB4BF160CB}, // 1e-121 {0xC2974EB4EE658828, 0xA54394FE1EEDB8FE}, // 1e-120 {0x733D226229FEEA32, 0xCE947A3DA6A9273E}, // 1e-119 {0x0806357D5A3F525F, 0x811CCC668829B887}, // 1e-118 {0xCA07C2DCB0CF26F7, 0xA163FF802A3426A8}, // 1e-117 {0xFC89B393DD02F0B5, 0xC9BCFF6034C13052}, // 1e-116 {0xBBAC2078D443ACE2, 0xFC2C3F3841F17C67}, // 1e-115 {0xD54B944B84AA4C0D, 0x9D9BA7832936EDC0}, // 1e-114 {0x0A9E795E65D4DF11, 0xC5029163F384A931}, // 1e-113 {0x4D4617B5FF4A16D5, 0xF64335BCF065D37D}, // 1e-112 {0x504BCED1BF8E4E45, 0x99EA0196163FA42E}, // 1e-111 {0xE45EC2862F71E1D6, 0xC06481FB9BCF8D39}, // 1e-110 {0x5D767327BB4E5A4C, 0xF07DA27A82C37088}, // 1e-109 {0x3A6A07F8D510F86F, 0x964E858C91BA2655}, // 1e-108 {0x890489F70A55368B, 0xBBE226EFB628AFEA}, // 1e-107 {0x2B45AC74CCEA842E, 0xEADAB0ABA3B2DBE5}, // 1e-106 {0x3B0B8BC90012929D, 0x92C8AE6B464FC96F}, // 1e-105 {0x09CE6EBB40173744, 0xB77ADA0617E3BBCB}, // 1e-104 {0xCC420A6A101D0515, 0xE55990879DDCAABD}, // 1e-103 {0x9FA946824A12232D, 0x8F57FA54C2A9EAB6}, // 1e-102 {0x47939822DC96ABF9, 0xB32DF8E9F3546564}, // 1e-101 {0x59787E2B93BC56F7, 0xDFF9772470297EBD}, // 1e-100 {0x57EB4EDB3C55B65A, 0x8BFBEA76C619EF36}, // 1e-99 {0xEDE622920B6B23F1, 0xAEFAE51477A06B03}, // 1e-98 {0xE95FAB368E45ECED, 0xDAB99E59958885C4}, // 1e-97 {0x11DBCB0218EBB414, 0x88B402F7FD75539B}, // 1e-96 {0xD652BDC29F26A119, 0xAAE103B5FCD2A881}, // 1e-95 {0x4BE76D3346F0495F, 0xD59944A37C0752A2}, // 1e-94 {0x6F70A4400C562DDB, 0x857FCAE62D8493A5}, // 1e-93 {0xCB4CCD500F6BB952, 0xA6DFBD9FB8E5B88E}, // 1e-92 {0x7E2000A41346A7A7, 0xD097AD07A71F26B2}, // 1e-91 {0x8ED400668C0C28C8, 0x825ECC24C873782F}, // 1e-90 {0x728900802F0F32FA, 0xA2F67F2DFA90563B}, // 1e-89 {0x4F2B40A03AD2FFB9, 0xCBB41EF979346BCA}, // 1e-88 {0xE2F610C84987BFA8, 0xFEA126B7D78186BC}, // 1e-87 {0x0DD9CA7D2DF4D7C9, 0x9F24B832E6B0F436}, // 1e-86 {0x91503D1C79720DBB, 0xC6EDE63FA05D3143}, // 1e-85 {0x75A44C6397CE912A, 0xF8A95FCF88747D94}, // 1e-84 {0xC986AFBE3EE11ABA, 0x9B69DBE1B548CE7C}, // 1e-83 {0xFBE85BADCE996168, 0xC24452DA229B021B}, // 1e-82 {0xFAE27299423FB9C3, 0xF2D56790AB41C2A2}, // 1e-81 {0xDCCD879FC967D41A, 0x97C560BA6B0919A5}, // 1e-80 {0x5400E987BBC1C920, 0xBDB6B8E905CB600F}, // 1e-79 {0x290123E9AAB23B68, 0xED246723473E3813}, // 1e-78 {0xF9A0B6720AAF6521, 0x9436C0760C86E30B}, // 1e-77 {0xF808E40E8D5B3E69, 0xB94470938FA89BCE}, // 1e-76 {0xB60B1D1230B20E04, 0xE7958CB87392C2C2}, // 1e-75 {0xB1C6F22B5E6F48C2, 0x90BD77F3483BB9B9}, // 1e-74 {0x1E38AEB6360B1AF3, 0xB4ECD5F01A4AA828}, // 1e-73 {0x25C6DA63C38DE1B0, 0xE2280B6C20DD5232}, // 1e-72 {0x579C487E5A38AD0E, 0x8D590723948A535F}, // 1e-71 {0x2D835A9DF0C6D851, 0xB0AF48EC79ACE837}, // 1e-70 {0xF8E431456CF88E65, 0xDCDB1B2798182244}, // 1e-69 {0x1B8E9ECB641B58FF, 0x8A08F0F8BF0F156B}, // 1e-68 {0xE272467E3D222F3F, 0xAC8B2D36EED2DAC5}, // 1e-67 {0x5B0ED81DCC6ABB0F, 0xD7ADF884AA879177}, // 1e-66 {0x98E947129FC2B4E9, 0x86CCBB52EA94BAEA}, // 1e-65 {0x3F2398D747B36224, 0xA87FEA27A539E9A5}, // 1e-64 {0x8EEC7F0D19A03AAD, 0xD29FE4B18E88640E}, // 1e-63 {0x1953CF68300424AC, 0x83A3EEEEF9153E89}, // 1e-62 {0x5FA8C3423C052DD7, 0xA48CEAAAB75A8E2B}, // 1e-61 {0x3792F412CB06794D, 0xCDB02555653131B6}, // 1e-60 {0xE2BBD88BBEE40BD0, 0x808E17555F3EBF11}, // 1e-59 {0x5B6ACEAEAE9D0EC4, 0xA0B19D2AB70E6ED6}, // 1e-58 {0xF245825A5A445275, 0xC8DE047564D20A8B}, // 1e-57 {0xEED6E2F0F0D56712, 0xFB158592BE068D2E}, // 1e-56 {0x55464DD69685606B, 0x9CED737BB6C4183D}, // 1e-55 {0xAA97E14C3C26B886, 0xC428D05AA4751E4C}, // 1e-54 {0xD53DD99F4B3066A8, 0xF53304714D9265DF}, // 1e-53 {0xE546A8038EFE4029, 0x993FE2C6D07B7FAB}, // 1e-52 {0xDE98520472BDD033, 0xBF8FDB78849A5F96}, // 1e-51 {0x963E66858F6D4440, 0xEF73D256A5C0F77C}, // 1e-50 {0xDDE7001379A44AA8, 0x95A8637627989AAD}, // 1e-49 {0x5560C018580D5D52, 0xBB127C53B17EC159}, // 1e-48 {0xAAB8F01E6E10B4A6, 0xE9D71B689DDE71AF}, // 1e-47 {0xCAB3961304CA70E8, 0x9226712162AB070D}, // 1e-46 {0x3D607B97C5FD0D22, 0xB6B00D69BB55C8D1}, // 1e-45 {0x8CB89A7DB77C506A, 0xE45C10C42A2B3B05}, // 1e-44 {0x77F3608E92ADB242, 0x8EB98A7A9A5B04E3}, // 1e-43 {0x55F038B237591ED3, 0xB267ED1940F1C61C}, // 1e-42 {0x6B6C46DEC52F6688, 0xDF01E85F912E37A3}, // 1e-41 {0x2323AC4B3B3DA015, 0x8B61313BBABCE2C6}, // 1e-40 {0xABEC975E0A0D081A, 0xAE397D8AA96C1B77}, // 1e-39 {0x96E7BD358C904A21, 0xD9C7DCED53C72255}, // 1e-38 {0x7E50D64177DA2E54, 0x881CEA14545C7575}, // 1e-37 {0xDDE50BD1D5D0B9E9, 0xAA242499697392D2}, // 1e-36 {0x955E4EC64B44E864, 0xD4AD2DBFC3D07787}, // 1e-35 {0xBD5AF13BEF0B113E, 0x84EC3C97DA624AB4}, // 1e-34 {0xECB1AD8AEACDD58E, 0xA6274BBDD0FADD61}, // 1e-33 {0x67DE18EDA5814AF2, 0xCFB11EAD453994BA}, // 1e-32 {0x80EACF948770CED7, 0x81CEB32C4B43FCF4}, // 1e-31 {0xA1258379A94D028D, 0xA2425FF75E14FC31}, // 1e-30 {0x096EE45813A04330, 0xCAD2F7F5359A3B3E}, // 1e-29 {0x8BCA9D6E188853FC, 0xFD87B5F28300CA0D}, // 1e-28 {0x775EA264CF55347D, 0x9E74D1B791E07E48}, // 1e-27 {0x95364AFE032A819D, 0xC612062576589DDA}, // 1e-26 {0x3A83DDBD83F52204, 0xF79687AED3EEC551}, // 1e-25 {0xC4926A9672793542, 0x9ABE14CD44753B52}, // 1e-24 {0x75B7053C0F178293, 0xC16D9A0095928A27}, // 1e-23 {0x5324C68B12DD6338, 0xF1C90080BAF72CB1}, // 1e-22 {0xD3F6FC16EBCA5E03, 0x971DA05074DA7BEE}, // 1e-21 {0x88F4BB1CA6BCF584, 0xBCE5086492111AEA}, // 1e-20 {0x2B31E9E3D06C32E5, 0xEC1E4A7DB69561A5}, // 1e-19 {0x3AFF322E62439FCF, 0x9392EE8E921D5D07}, // 1e-18 {0x09BEFEB9FAD487C2, 0xB877AA3236A4B449}, // 1e-17 {0x4C2EBE687989A9B3, 0xE69594BEC44DE15B}, // 1e-16 {0x0F9D37014BF60A10, 0x901D7CF73AB0ACD9}, // 1e-15 {0x538484C19EF38C94, 0xB424DC35095CD80F}, // 1e-14 {0x2865A5F206B06FB9, 0xE12E13424BB40E13}, // 1e-13 {0xF93F87B7442E45D3, 0x8CBCCC096F5088CB}, // 1e-12 {0xF78F69A51539D748, 0xAFEBFF0BCB24AAFE}, // 1e-11 {0xB573440E5A884D1B, 0xDBE6FECEBDEDD5BE}, // 1e-10 {0x31680A88F8953030, 0x89705F4136B4A597}, // 1e-9 {0xFDC20D2B36BA7C3D, 0xABCC77118461CEFC}, // 1e-8 {0x3D32907604691B4C, 0xD6BF94D5E57A42BC}, // 1e-7 {0xA63F9A49C2C1B10F, 0x8637BD05AF6C69B5}, // 1e-6 {0x0FCF80DC33721D53, 0xA7C5AC471B478423}, // 1e-5 {0xD3C36113404EA4A8, 0xD1B71758E219652B}, // 1e-4 {0x645A1CAC083126E9, 0x83126E978D4FDF3B}, // 1e-3 {0x3D70A3D70A3D70A3, 0xA3D70A3D70A3D70A}, // 1e-2 {0xCCCCCCCCCCCCCCCC, 0xCCCCCCCCCCCCCCCC}, // 1e-1 {0x0000000000000000, 0x8000000000000000}, // 1e0 {0x0000000000000000, 0xA000000000000000}, // 1e1 {0x0000000000000000, 0xC800000000000000}, // 1e2 {0x0000000000000000, 0xFA00000000000000}, // 1e3 {0x0000000000000000, 0x9C40000000000000}, // 1e4 {0x0000000000000000, 0xC350000000000000}, // 1e5 {0x0000000000000000, 0xF424000000000000}, // 1e6 {0x0000000000000000, 0x9896800000000000}, // 1e7 {0x0000000000000000, 0xBEBC200000000000}, // 1e8 {0x0000000000000000, 0xEE6B280000000000}, // 1e9 {0x0000000000000000, 0x9502F90000000000}, // 1e10 {0x0000000000000000, 0xBA43B74000000000}, // 1e11 {0x0000000000000000, 0xE8D4A51000000000}, // 1e12 {0x0000000000000000, 0x9184E72A00000000}, // 1e13 {0x0000000000000000, 0xB5E620F480000000}, // 1e14 {0x0000000000000000, 0xE35FA931A0000000}, // 1e15 {0x0000000000000000, 0x8E1BC9BF04000000}, // 1e16 {0x0000000000000000, 0xB1A2BC2EC5000000}, // 1e17 {0x0000000000000000, 0xDE0B6B3A76400000}, // 1e18 {0x0000000000000000, 0x8AC7230489E80000}, // 1e19 {0x0000000000000000, 0xAD78EBC5AC620000}, // 1e20 {0x0000000000000000, 0xD8D726B7177A8000}, // 1e21 {0x0000000000000000, 0x878678326EAC9000}, // 1e22 {0x0000000000000000, 0xA968163F0A57B400}, // 1e23 {0x0000000000000000, 0xD3C21BCECCEDA100}, // 1e24 {0x0000000000000000, 0x84595161401484A0}, // 1e25 {0x0000000000000000, 0xA56FA5B99019A5C8}, // 1e26 {0x0000000000000000, 0xCECB8F27F4200F3A}, // 1e27 {0x4000000000000000, 0x813F3978F8940984}, // 1e28 {0x5000000000000000, 0xA18F07D736B90BE5}, // 1e29 {0xA400000000000000, 0xC9F2C9CD04674EDE}, // 1e30 {0x4D00000000000000, 0xFC6F7C4045812296}, // 1e31 {0xF020000000000000, 0x9DC5ADA82B70B59D}, // 1e32 {0x6C28000000000000, 0xC5371912364CE305}, // 1e33 {0xC732000000000000, 0xF684DF56C3E01BC6}, // 1e34 {0x3C7F400000000000, 0x9A130B963A6C115C}, // 1e35 {0x4B9F100000000000, 0xC097CE7BC90715B3}, // 1e36 {0x1E86D40000000000, 0xF0BDC21ABB48DB20}, // 1e37 {0x1314448000000000, 0x96769950B50D88F4}, // 1e38 {0x17D955A000000000, 0xBC143FA4E250EB31}, // 1e39 {0x5DCFAB0800000000, 0xEB194F8E1AE525FD}, // 1e40 {0x5AA1CAE500000000, 0x92EFD1B8D0CF37BE}, // 1e41 {0xF14A3D9E40000000, 0xB7ABC627050305AD}, // 1e42 {0x6D9CCD05D0000000, 0xE596B7B0C643C719}, // 1e43 {0xE4820023A2000000, 0x8F7E32CE7BEA5C6F}, // 1e44 {0xDDA2802C8A800000, 0xB35DBF821AE4F38B}, // 1e45 {0xD50B2037AD200000, 0xE0352F62A19E306E}, // 1e46 {0x4526F422CC340000, 0x8C213D9DA502DE45}, // 1e47 {0x9670B12B7F410000, 0xAF298D050E4395D6}, // 1e48 {0x3C0CDD765F114000, 0xDAF3F04651D47B4C}, // 1e49 {0xA5880A69FB6AC800, 0x88D8762BF324CD0F}, // 1e50 {0x8EEA0D047A457A00, 0xAB0E93B6EFEE0053}, // 1e51 {0x72A4904598D6D880, 0xD5D238A4ABE98068}, // 1e52 {0x47A6DA2B7F864750, 0x85A36366EB71F041}, // 1e53 {0x999090B65F67D924, 0xA70C3C40A64E6C51}, // 1e54 {0xFFF4B4E3F741CF6D, 0xD0CF4B50CFE20765}, // 1e55 {0xBFF8F10E7A8921A4, 0x82818F1281ED449F}, // 1e56 {0xAFF72D52192B6A0D, 0xA321F2D7226895C7}, // 1e57 {0x9BF4F8A69F764490, 0xCBEA6F8CEB02BB39}, // 1e58 {0x02F236D04753D5B4, 0xFEE50B7025C36A08}, // 1e59 {0x01D762422C946590, 0x9F4F2726179A2245}, // 1e60 {0x424D3AD2B7B97EF5, 0xC722F0EF9D80AAD6}, // 1e61 {0xD2E0898765A7DEB2, 0xF8EBAD2B84E0D58B}, // 1e62 {0x63CC55F49F88EB2F, 0x9B934C3B330C8577}, // 1e63 {0x3CBF6B71C76B25FB, 0xC2781F49FFCFA6D5}, // 1e64 {0x8BEF464E3945EF7A, 0xF316271C7FC3908A}, // 1e65 {0x97758BF0E3CBB5AC, 0x97EDD871CFDA3A56}, // 1e66 {0x3D52EEED1CBEA317, 0xBDE94E8E43D0C8EC}, // 1e67 {0x4CA7AAA863EE4BDD, 0xED63A231D4C4FB27}, // 1e68 {0x8FE8CAA93E74EF6A, 0x945E455F24FB1CF8}, // 1e69 {0xB3E2FD538E122B44, 0xB975D6B6EE39E436}, // 1e70 {0x60DBBCA87196B616, 0xE7D34C64A9C85D44}, // 1e71 {0xBC8955E946FE31CD, 0x90E40FBEEA1D3A4A}, // 1e72 {0x6BABAB6398BDBE41, 0xB51D13AEA4A488DD}, // 1e73 {0xC696963C7EED2DD1, 0xE264589A4DCDAB14}, // 1e74 {0xFC1E1DE5CF543CA2, 0x8D7EB76070A08AEC}, // 1e75 {0x3B25A55F43294BCB, 0xB0DE65388CC8ADA8}, // 1e76 {0x49EF0EB713F39EBE, 0xDD15FE86AFFAD912}, // 1e77 {0x6E3569326C784337, 0x8A2DBF142DFCC7AB}, // 1e78 {0x49C2C37F07965404, 0xACB92ED9397BF996}, // 1e79 {0xDC33745EC97BE906, 0xD7E77A8F87DAF7FB}, // 1e80 {0x69A028BB3DED71A3, 0x86F0AC99B4E8DAFD}, // 1e81 {0xC40832EA0D68CE0C, 0xA8ACD7C0222311BC}, // 1e82 {0xF50A3FA490C30190, 0xD2D80DB02AABD62B}, // 1e83 {0x792667C6DA79E0FA, 0x83C7088E1AAB65DB}, // 1e84 {0x577001B891185938, 0xA4B8CAB1A1563F52}, // 1e85 {0xED4C0226B55E6F86, 0xCDE6FD5E09ABCF26}, // 1e86 {0x544F8158315B05B4, 0x80B05E5AC60B6178}, // 1e87 {0x696361AE3DB1C721, 0xA0DC75F1778E39D6}, // 1e88 {0x03BC3A19CD1E38E9, 0xC913936DD571C84C}, // 1e89 {0x04AB48A04065C723, 0xFB5878494ACE3A5F}, // 1e90 {0x62EB0D64283F9C76, 0x9D174B2DCEC0E47B}, // 1e91 {0x3BA5D0BD324F8394, 0xC45D1DF942711D9A}, // 1e92 {0xCA8F44EC7EE36479, 0xF5746577930D6500}, // 1e93 {0x7E998B13CF4E1ECB, 0x9968BF6ABBE85F20}, // 1e94 {0x9E3FEDD8C321A67E, 0xBFC2EF456AE276E8}, // 1e95 {0xC5CFE94EF3EA101E, 0xEFB3AB16C59B14A2}, // 1e96 {0xBBA1F1D158724A12, 0x95D04AEE3B80ECE5}, // 1e97 {0x2A8A6E45AE8EDC97, 0xBB445DA9CA61281F}, // 1e98 {0xF52D09D71A3293BD, 0xEA1575143CF97226}, // 1e99 {0x593C2626705F9C56, 0x924D692CA61BE758}, // 1e100 {0x6F8B2FB00C77836C, 0xB6E0C377CFA2E12E}, // 1e101 {0x0B6DFB9C0F956447, 0xE498F455C38B997A}, // 1e102 {0x4724BD4189BD5EAC, 0x8EDF98B59A373FEC}, // 1e103 {0x58EDEC91EC2CB657, 0xB2977EE300C50FE7}, // 1e104 {0x2F2967B66737E3ED, 0xDF3D5E9BC0F653E1}, // 1e105 {0xBD79E0D20082EE74, 0x8B865B215899F46C}, // 1e106 {0xECD8590680A3AA11, 0xAE67F1E9AEC07187}, // 1e107 {0xE80E6F4820CC9495, 0xDA01EE641A708DE9}, // 1e108 {0x3109058D147FDCDD, 0x884134FE908658B2}, // 1e109 {0xBD4B46F0599FD415, 0xAA51823E34A7EEDE}, // 1e110 {0x6C9E18AC7007C91A, 0xD4E5E2CDC1D1EA96}, // 1e111 {0x03E2CF6BC604DDB0, 0x850FADC09923329E}, // 1e112 {0x84DB8346B786151C, 0xA6539930BF6BFF45}, // 1e113 {0xE612641865679A63, 0xCFE87F7CEF46FF16}, // 1e114 {0x4FCB7E8F3F60C07E, 0x81F14FAE158C5F6E}, // 1e115 {0xE3BE5E330F38F09D, 0xA26DA3999AEF7749}, // 1e116 {0x5CADF5BFD3072CC5, 0xCB090C8001AB551C}, // 1e117 {0x73D9732FC7C8F7F6, 0xFDCB4FA002162A63}, // 1e118 {0x2867E7FDDCDD9AFA, 0x9E9F11C4014DDA7E}, // 1e119 {0xB281E1FD541501B8, 0xC646D63501A1511D}, // 1e120 {0x1F225A7CA91A4226, 0xF7D88BC24209A565}, // 1e121 {0x3375788DE9B06958, 0x9AE757596946075F}, // 1e122 {0x0052D6B1641C83AE, 0xC1A12D2FC3978937}, // 1e123 {0xC0678C5DBD23A49A, 0xF209787BB47D6B84}, // 1e124 {0xF840B7BA963646E0, 0x9745EB4D50CE6332}, // 1e125 {0xB650E5A93BC3D898, 0xBD176620A501FBFF}, // 1e126 {0xA3E51F138AB4CEBE, 0xEC5D3FA8CE427AFF}, // 1e127 {0xC66F336C36B10137, 0x93BA47C980E98CDF}, // 1e128 {0xB80B0047445D4184, 0xB8A8D9BBE123F017}, // 1e129 {0xA60DC059157491E5, 0xE6D3102AD96CEC1D}, // 1e130 {0x87C89837AD68DB2F, 0x9043EA1AC7E41392}, // 1e131 {0x29BABE4598C311FB, 0xB454E4A179DD1877}, // 1e132 {0xF4296DD6FEF3D67A, 0xE16A1DC9D8545E94}, // 1e133 {0x1899E4A65F58660C, 0x8CE2529E2734BB1D}, // 1e134 {0x5EC05DCFF72E7F8F, 0xB01AE745B101E9E4}, // 1e135 {0x76707543F4FA1F73, 0xDC21A1171D42645D}, // 1e136 {0x6A06494A791C53A8, 0x899504AE72497EBA}, // 1e137 {0x0487DB9D17636892, 0xABFA45DA0EDBDE69}, // 1e138 {0x45A9D2845D3C42B6, 0xD6F8D7509292D603}, // 1e139 {0x0B8A2392BA45A9B2, 0x865B86925B9BC5C2}, // 1e140 {0x8E6CAC7768D7141E, 0xA7F26836F282B732}, // 1e141 {0x3207D795430CD926, 0xD1EF0244AF2364FF}, // 1e142 {0x7F44E6BD49E807B8, 0x8335616AED761F1F}, // 1e143 {0x5F16206C9C6209A6, 0xA402B9C5A8D3A6E7}, // 1e144 {0x36DBA887C37A8C0F, 0xCD036837130890A1}, // 1e145 {0xC2494954DA2C9789, 0x802221226BE55A64}, // 1e146 {0xF2DB9BAA10B7BD6C, 0xA02AA96B06DEB0FD}, // 1e147 {0x6F92829494E5ACC7, 0xC83553C5C8965D3D}, // 1e148 {0xCB772339BA1F17F9, 0xFA42A8B73ABBF48C}, // 1e149 {0xFF2A760414536EFB, 0x9C69A97284B578D7}, // 1e150 {0xFEF5138519684ABA, 0xC38413CF25E2D70D}, // 1e151 {0x7EB258665FC25D69, 0xF46518C2EF5B8CD1}, // 1e152 {0xEF2F773FFBD97A61, 0x98BF2F79D5993802}, // 1e153 {0xAAFB550FFACFD8FA, 0xBEEEFB584AFF8603}, // 1e154 {0x95BA2A53F983CF38, 0xEEAABA2E5DBF6784}, // 1e155 {0xDD945A747BF26183, 0x952AB45CFA97A0B2}, // 1e156 {0x94F971119AEEF9E4, 0xBA756174393D88DF}, // 1e157 {0x7A37CD5601AAB85D, 0xE912B9D1478CEB17}, // 1e158 {0xAC62E055C10AB33A, 0x91ABB422CCB812EE}, // 1e159 {0x577B986B314D6009, 0xB616A12B7FE617AA}, // 1e160 {0xED5A7E85FDA0B80B, 0xE39C49765FDF9D94}, // 1e161 {0x14588F13BE847307, 0x8E41ADE9FBEBC27D}, // 1e162 {0x596EB2D8AE258FC8, 0xB1D219647AE6B31C}, // 1e163 {0x6FCA5F8ED9AEF3BB, 0xDE469FBD99A05FE3}, // 1e164 {0x25DE7BB9480D5854, 0x8AEC23D680043BEE}, // 1e165 {0xAF561AA79A10AE6A, 0xADA72CCC20054AE9}, // 1e166 {0x1B2BA1518094DA04, 0xD910F7FF28069DA4}, // 1e167 {0x90FB44D2F05D0842, 0x87AA9AFF79042286}, // 1e168 {0x353A1607AC744A53, 0xA99541BF57452B28}, // 1e169 {0x42889B8997915CE8, 0xD3FA922F2D1675F2}, // 1e170 {0x69956135FEBADA11, 0x847C9B5D7C2E09B7}, // 1e171 {0x43FAB9837E699095, 0xA59BC234DB398C25}, // 1e172 {0x94F967E45E03F4BB, 0xCF02B2C21207EF2E}, // 1e173 {0x1D1BE0EEBAC278F5, 0x8161AFB94B44F57D}, // 1e174 {0x6462D92A69731732, 0xA1BA1BA79E1632DC}, // 1e175 {0x7D7B8F7503CFDCFE, 0xCA28A291859BBF93}, // 1e176 {0x5CDA735244C3D43E, 0xFCB2CB35E702AF78}, // 1e177 {0x3A0888136AFA64A7, 0x9DEFBF01B061ADAB}, // 1e178 {0x088AAA1845B8FDD0, 0xC56BAEC21C7A1916}, // 1e179 {0x8AAD549E57273D45, 0xF6C69A72A3989F5B}, // 1e180 {0x36AC54E2F678864B, 0x9A3C2087A63F6399}, // 1e181 {0x84576A1BB416A7DD, 0xC0CB28A98FCF3C7F}, // 1e182 {0x656D44A2A11C51D5, 0xF0FDF2D3F3C30B9F}, // 1e183 {0x9F644AE5A4B1B325, 0x969EB7C47859E743}, // 1e184 {0x873D5D9F0DDE1FEE, 0xBC4665B596706114}, // 1e185 {0xA90CB506D155A7EA, 0xEB57FF22FC0C7959}, // 1e186 {0x09A7F12442D588F2, 0x9316FF75DD87CBD8}, // 1e187 {0x0C11ED6D538AEB2F, 0xB7DCBF5354E9BECE}, // 1e188 {0x8F1668C8A86DA5FA, 0xE5D3EF282A242E81}, // 1e189 {0xF96E017D694487BC, 0x8FA475791A569D10}, // 1e190 {0x37C981DCC395A9AC, 0xB38D92D760EC4455}, // 1e191 {0x85BBE253F47B1417, 0xE070F78D3927556A}, // 1e192 {0x93956D7478CCEC8E, 0x8C469AB843B89562}, // 1e193 {0x387AC8D1970027B2, 0xAF58416654A6BABB}, // 1e194 {0x06997B05FCC0319E, 0xDB2E51BFE9D0696A}, // 1e195 {0x441FECE3BDF81F03, 0x88FCF317F22241E2}, // 1e196 {0xD527E81CAD7626C3, 0xAB3C2FDDEEAAD25A}, // 1e197 {0x8A71E223D8D3B074, 0xD60B3BD56A5586F1}, // 1e198 {0xF6872D5667844E49, 0x85C7056562757456}, // 1e199 {0xB428F8AC016561DB, 0xA738C6BEBB12D16C}, // 1e200 {0xE13336D701BEBA52, 0xD106F86E69D785C7}, // 1e201 {0xECC0024661173473, 0x82A45B450226B39C}, // 1e202 {0x27F002D7F95D0190, 0xA34D721642B06084}, // 1e203 {0x31EC038DF7B441F4, 0xCC20CE9BD35C78A5}, // 1e204 {0x7E67047175A15271, 0xFF290242C83396CE}, // 1e205 {0x0F0062C6E984D386, 0x9F79A169BD203E41}, // 1e206 {0x52C07B78A3E60868, 0xC75809C42C684DD1}, // 1e207 {0xA7709A56CCDF8A82, 0xF92E0C3537826145}, // 1e208 {0x88A66076400BB691, 0x9BBCC7A142B17CCB}, // 1e209 {0x6ACFF893D00EA435, 0xC2ABF989935DDBFE}, // 1e210 {0x0583F6B8C4124D43, 0xF356F7EBF83552FE}, // 1e211 {0xC3727A337A8B704A, 0x98165AF37B2153DE}, // 1e212 {0x744F18C0592E4C5C, 0xBE1BF1B059E9A8D6}, // 1e213 {0x1162DEF06F79DF73, 0xEDA2EE1C7064130C}, // 1e214 {0x8ADDCB5645AC2BA8, 0x9485D4D1C63E8BE7}, // 1e215 {0x6D953E2BD7173692, 0xB9A74A0637CE2EE1}, // 1e216 {0xC8FA8DB6CCDD0437, 0xE8111C87C5C1BA99}, // 1e217 {0x1D9C9892400A22A2, 0x910AB1D4DB9914A0}, // 1e218 {0x2503BEB6D00CAB4B, 0xB54D5E4A127F59C8}, // 1e219 {0x2E44AE64840FD61D, 0xE2A0B5DC971F303A}, // 1e220 {0x5CEAECFED289E5D2, 0x8DA471A9DE737E24}, // 1e221 {0x7425A83E872C5F47, 0xB10D8E1456105DAD}, // 1e222 {0xD12F124E28F77719, 0xDD50F1996B947518}, // 1e223 {0x82BD6B70D99AAA6F, 0x8A5296FFE33CC92F}, // 1e224 {0x636CC64D1001550B, 0xACE73CBFDC0BFB7B}, // 1e225 {0x3C47F7E05401AA4E, 0xD8210BEFD30EFA5A}, // 1e226 {0x65ACFAEC34810A71, 0x8714A775E3E95C78}, // 1e227 {0x7F1839A741A14D0D, 0xA8D9D1535CE3B396}, // 1e228 {0x1EDE48111209A050, 0xD31045A8341CA07C}, // 1e229 {0x934AED0AAB460432, 0x83EA2B892091E44D}, // 1e230 {0xF81DA84D5617853F, 0xA4E4B66B68B65D60}, // 1e231 {0x36251260AB9D668E, 0xCE1DE40642E3F4B9}, // 1e232 {0xC1D72B7C6B426019, 0x80D2AE83E9CE78F3}, // 1e233 {0xB24CF65B8612F81F, 0xA1075A24E4421730}, // 1e234 {0xDEE033F26797B627, 0xC94930AE1D529CFC}, // 1e235 {0x169840EF017DA3B1, 0xFB9B7CD9A4A7443C}, // 1e236 {0x8E1F289560EE864E, 0x9D412E0806E88AA5}, // 1e237 {0xF1A6F2BAB92A27E2, 0xC491798A08A2AD4E}, // 1e238 {0xAE10AF696774B1DB, 0xF5B5D7EC8ACB58A2}, // 1e239 {0xACCA6DA1E0A8EF29, 0x9991A6F3D6BF1765}, // 1e240 {0x17FD090A58D32AF3, 0xBFF610B0CC6EDD3F}, // 1e241 {0xDDFC4B4CEF07F5B0, 0xEFF394DCFF8A948E}, // 1e242 {0x4ABDAF101564F98E, 0x95F83D0A1FB69CD9}, // 1e243 {0x9D6D1AD41ABE37F1, 0xBB764C4CA7A4440F}, // 1e244 {0x84C86189216DC5ED, 0xEA53DF5FD18D5513}, // 1e245 {0x32FD3CF5B4E49BB4, 0x92746B9BE2F8552C}, // 1e246 {0x3FBC8C33221DC2A1, 0xB7118682DBB66A77}, // 1e247 {0x0FABAF3FEAA5334A, 0xE4D5E82392A40515}, // 1e248 {0x29CB4D87F2A7400E, 0x8F05B1163BA6832D}, // 1e249 {0x743E20E9EF511012, 0xB2C71D5BCA9023F8}, // 1e250 {0x914DA9246B255416, 0xDF78E4B2BD342CF6}, // 1e251 {0x1AD089B6C2F7548E, 0x8BAB8EEFB6409C1A}, // 1e252 {0xA184AC2473B529B1, 0xAE9672ABA3D0C320}, // 1e253 {0xC9E5D72D90A2741E, 0xDA3C0F568CC4F3E8}, // 1e254 {0x7E2FA67C7A658892, 0x8865899617FB1871}, // 1e255 {0xDDBB901B98FEEAB7, 0xAA7EEBFB9DF9DE8D}, // 1e256 {0x552A74227F3EA565, 0xD51EA6FA85785631}, // 1e257 {0xD53A88958F87275F, 0x8533285C936B35DE}, // 1e258 {0x8A892ABAF368F137, 0xA67FF273B8460356}, // 1e259 {0x2D2B7569B0432D85, 0xD01FEF10A657842C}, // 1e260 {0x9C3B29620E29FC73, 0x8213F56A67F6B29B}, // 1e261 {0x8349F3BA91B47B8F, 0xA298F2C501F45F42}, // 1e262 {0x241C70A936219A73, 0xCB3F2F7642717713}, // 1e263 {0xED238CD383AA0110, 0xFE0EFB53D30DD4D7}, // 1e264 {0xF4363804324A40AA, 0x9EC95D1463E8A506}, // 1e265 {0xB143C6053EDCD0D5, 0xC67BB4597CE2CE48}, // 1e266 {0xDD94B7868E94050A, 0xF81AA16FDC1B81DA}, // 1e267 {0xCA7CF2B4191C8326, 0x9B10A4E5E9913128}, // 1e268 {0xFD1C2F611F63A3F0, 0xC1D4CE1F63F57D72}, // 1e269 {0xBC633B39673C8CEC, 0xF24A01A73CF2DCCF}, // 1e270 {0xD5BE0503E085D813, 0x976E41088617CA01}, // 1e271 {0x4B2D8644D8A74E18, 0xBD49D14AA79DBC82}, // 1e272 {0xDDF8E7D60ED1219E, 0xEC9C459D51852BA2}, // 1e273 {0xCABB90E5C942B503, 0x93E1AB8252F33B45}, // 1e274 {0x3D6A751F3B936243, 0xB8DA1662E7B00A17}, // 1e275 {0x0CC512670A783AD4, 0xE7109BFBA19C0C9D}, // 1e276 {0x27FB2B80668B24C5, 0x906A617D450187E2}, // 1e277 {0xB1F9F660802DEDF6, 0xB484F9DC9641E9DA}, // 1e278 {0x5E7873F8A0396973, 0xE1A63853BBD26451}, // 1e279 {0xDB0B487B6423E1E8, 0x8D07E33455637EB2}, // 1e280 {0x91CE1A9A3D2CDA62, 0xB049DC016ABC5E5F}, // 1e281 {0x7641A140CC7810FB, 0xDC5C5301C56B75F7}, // 1e282 {0xA9E904C87FCB0A9D, 0x89B9B3E11B6329BA}, // 1e283 {0x546345FA9FBDCD44, 0xAC2820D9623BF429}, // 1e284 {0xA97C177947AD4095, 0xD732290FBACAF133}, // 1e285 {0x49ED8EABCCCC485D, 0x867F59A9D4BED6C0}, // 1e286 {0x5C68F256BFFF5A74, 0xA81F301449EE8C70}, // 1e287 {0x73832EEC6FFF3111, 0xD226FC195C6A2F8C}, // 1e288 }; #if PK_FLOATCONV_HAS_EXACT_DOUBLE_ARITHMETIC // [pocketpy] Deviation 4. // wuffs_private_impl__f64_powers_of_10 holds powers of 10 that can be exactly // represented by a float64 (what C calls a double). static const double wuffs_private_impl__f64_powers_of_10[23] = { 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22, }; #endif // -------- #define WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE 2047 #define WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION 800 // WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL is the largest N such that // ((10 << N) < (1 << 64)). #define WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL 60 // wuffs_private_impl__high_prec_dec (abbreviated as HPD) is a fixed precision // floating point decimal number, augmented with ±infinity values, but it // cannot represent NaN (Not a Number). // // "High precision" means that the mantissa holds 800 decimal digits. 800 is // WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION. // // An HPD isn't for general purpose arithmetic, only for conversions to and // from IEEE 754 double-precision floating point, where the largest and // smallest positive, finite values are approximately 1.8e+308 and 4.9e-324. // HPD exponents above +2047 mean infinity, below -2047 mean zero. The ±2047 // bounds are further away from zero than ±(324 + 800), where 800 and 2047 is // WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION and // WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE. // // digits[.. num_digits] are the number's digits in big-endian order. The // uint8_t values are in the range [0 ..= 9], not ['0' ..= '9'], where e.g. '7' // is the ASCII value 0x37. // // decimal_point is the index (within digits) of the decimal point. It may be // negative or be larger than num_digits, in which case the explicit digits are // padded with implicit zeroes. // // For example, if num_digits is 3 and digits is "\x07\x08\x09": // - A decimal_point of -2 means ".00789" // - A decimal_point of -1 means ".0789" // - A decimal_point of +0 means ".789" // - A decimal_point of +1 means "7.89" // - A decimal_point of +2 means "78.9" // - A decimal_point of +3 means "789." // - A decimal_point of +4 means "7890." // - A decimal_point of +5 means "78900." // // As above, a decimal_point higher than +2047 means that the overall value is // infinity, lower than -2047 means zero. // // negative is a sign bit. An HPD can distinguish positive and negative zero. // // truncated is whether there are more than // WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION digits, and at least one of those // extra digits are non-zero. The existence of long-tail digits can affect // rounding. // // The "all fields are zero" value is valid, and represents the number +0. typedef struct wuffs_private_impl__high_prec_dec__struct { uint32_t num_digits; int32_t decimal_point; bool negative; bool truncated; uint8_t digits[WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION]; } wuffs_private_impl__high_prec_dec; // wuffs_private_impl__high_prec_dec__trim trims trailing zeroes from the // h->digits[.. h->num_digits] slice. They have no benefit, since we explicitly // track h->decimal_point. // // Preconditions: // - h is non-NULL. static inline void // wuffs_private_impl__high_prec_dec__trim(wuffs_private_impl__high_prec_dec* h) { while ((h->num_digits > 0) && (h->digits[h->num_digits - 1] == 0)) { h->num_digits--; } } // wuffs_private_impl__high_prec_dec__assign sets h to represent the number x. // // Preconditions: // - h is non-NULL. static void // wuffs_private_impl__high_prec_dec__assign(wuffs_private_impl__high_prec_dec* h, uint64_t x, bool negative) { uint32_t n = 0; // Set h->digits. if (x > 0) { // Calculate the digits, working right-to-left. After we determine n (how // many digits there are), copy from buf to h->digits. // // UINT64_MAX, 18446744073709551615, is 20 digits long. It can be faster to // copy a constant number of bytes than a variable number (20 instead of // n). Make buf large enough (and start writing to it from the middle) so // that can we always copy 20 bytes: the slice buf[(20-n) .. (40-n)]. uint8_t buf[40] = {0}; uint8_t* ptr = &buf[20]; do { uint64_t remaining = x / 10; x -= remaining * 10; ptr--; *ptr = (uint8_t)x; n++; x = remaining; } while (x > 0); memcpy(h->digits, ptr, 20); } // Set h's other fields. h->num_digits = n; h->decimal_point = (int32_t)n; h->negative = negative; h->truncated = false; wuffs_private_impl__high_prec_dec__trim(h); } static wuffs_base__status // wuffs_private_impl__high_prec_dec__parse(wuffs_private_impl__high_prec_dec* h, wuffs_base__slice_u8 s, uint32_t options) { if (!h) { return wuffs_base__make_status(wuffs_base__error__bad_receiver); } h->num_digits = 0; h->decimal_point = 0; h->negative = false; h->truncated = false; uint8_t* p = s.ptr; uint8_t* q = s.ptr + s.len; if (options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES) { for (;; p++) { if (p >= q) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } else if (*p != '_') { break; } } } // Parse sign. do { if (*p == '+') { p++; } else if (*p == '-') { h->negative = true; p++; } else { break; } if (options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES) { for (;; p++) { if (p >= q) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } else if (*p != '_') { break; } } } } while (0); // Parse digits, up to (and including) a '.', 'E' or 'e'. Examples for each // limb in this if-else chain: // - "0.789" // - "1002.789" // - ".789" // - Other (invalid input). uint32_t nd = 0; int32_t dp = 0; bool no_digits_before_separator = false; if (('0' == *p) && !(options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_MULTIPLE_LEADING_ZEROES)) { p++; for (;; p++) { if (p >= q) { goto after_all; } else if (*p == ((options & WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.')) { p++; goto after_sep; } else if ((*p == 'E') || (*p == 'e')) { p++; goto after_exp; } else if ((*p != '_') || !(options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES)) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } } } else if (('0' <= *p) && (*p <= '9')) { if (*p == '0') { for (; (p < q) && (*p == '0'); p++) { } } else { h->digits[nd++] = (uint8_t)(*p - '0'); dp = (int32_t)nd; p++; } for (;; p++) { if (p >= q) { goto after_all; } else if (('0' <= *p) && (*p <= '9')) { if (nd < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[nd++] = (uint8_t)(*p - '0'); dp = (int32_t)nd; } else if ('0' != *p) { // Long-tail non-zeroes set the truncated bit. h->truncated = true; } } else if (*p == ((options & WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.')) { p++; goto after_sep; } else if ((*p == 'E') || (*p == 'e')) { p++; goto after_exp; } else if ((*p != '_') || !(options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES)) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } } } else if (*p == ((options & WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.')) { p++; no_digits_before_separator = true; } else { return wuffs_base__make_status(wuffs_base__error__bad_argument); } after_sep: for (;; p++) { if (p >= q) { goto after_all; } else if ('0' == *p) { if (nd == 0) { // Track leading zeroes implicitly. dp--; } else if (nd < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[nd++] = (uint8_t)(*p - '0'); } } else if (('0' < *p) && (*p <= '9')) { if (nd < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[nd++] = (uint8_t)(*p - '0'); } else { // Long-tail non-zeroes set the truncated bit. h->truncated = true; } } else if ((*p == 'E') || (*p == 'e')) { p++; goto after_exp; } else if ((*p != '_') || !(options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES)) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } } after_exp: do { if (options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES) { for (;; p++) { if (p >= q) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } else if (*p != '_') { break; } } } int32_t exp_sign = +1; if (*p == '+') { p++; } else if (*p == '-') { exp_sign = -1; p++; } int32_t exp = 0; const int32_t exp_large = WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE + WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION; bool saw_exp_digits = false; for (; p < q; p++) { if ((*p == '_') && (options & WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES)) { // No-op. } else if (('0' <= *p) && (*p <= '9')) { saw_exp_digits = true; if (exp < exp_large) { exp = (10 * exp) + ((int32_t)(*p - '0')); } } else { break; } } if (!saw_exp_digits) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } dp += exp_sign * exp; } while (0); after_all: if (p != q) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } h->num_digits = nd; if (nd == 0) { if (no_digits_before_separator) { return wuffs_base__make_status(wuffs_base__error__bad_argument); } h->decimal_point = 0; } else if (dp < -WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE) { h->decimal_point = -WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE - 1; } else if (dp > +WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE) { h->decimal_point = +WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE + 1; } else { h->decimal_point = dp; } wuffs_private_impl__high_prec_dec__trim(h); return wuffs_base__make_status(NULL); } // -------- // wuffs_private_impl__high_prec_dec__lshift_num_new_digits returns the number // of additional decimal digits when left-shifting by shift. // // See below for preconditions. static uint32_t // wuffs_private_impl__high_prec_dec__lshift_num_new_digits( wuffs_private_impl__high_prec_dec* h, uint32_t shift) { // Masking with 0x3F should be unnecessary (assuming the preconditions) but // it's cheap and ensures that we don't overflow the // wuffs_private_impl__hpd_left_shift array. shift &= 63; uint32_t x_a = wuffs_private_impl__hpd_left_shift[shift]; uint32_t x_b = wuffs_private_impl__hpd_left_shift[shift + 1]; uint32_t num_new_digits = x_a >> 11; uint32_t pow5_a = 0x7FF & x_a; uint32_t pow5_b = 0x7FF & x_b; const uint8_t* pow5 = &wuffs_private_impl__powers_of_5[pow5_a]; uint32_t i = 0; uint32_t n = pow5_b - pow5_a; for (; i < n; i++) { if (i >= h->num_digits) { return num_new_digits - 1; } else if (h->digits[i] == pow5[i]) { continue; } else if (h->digits[i] < pow5[i]) { return num_new_digits - 1; } else { return num_new_digits; } } return num_new_digits; } // -------- // wuffs_private_impl__high_prec_dec__rounded_integer returns the integral // (non-fractional) part of h, provided that it is 18 or fewer decimal digits. // For 19 or more digits, it returns UINT64_MAX. Note that: // - (1 << 53) is 9007199254740992, which has 16 decimal digits. // - (1 << 56) is 72057594037927936, which has 17 decimal digits. // - (1 << 59) is 576460752303423488, which has 18 decimal digits. // - (1 << 63) is 9223372036854775808, which has 19 decimal digits. // and that IEEE 754 double precision has 52 mantissa bits. // // That integral part is rounded-to-even: rounding 7.5 or 8.5 both give 8. // // h's negative bit is ignored: rounding -8.6 returns 9. // // See below for preconditions. static uint64_t // wuffs_private_impl__high_prec_dec__rounded_integer( wuffs_private_impl__high_prec_dec* h) { if ((h->num_digits == 0) || (h->decimal_point < 0)) { return 0; } else if (h->decimal_point > 18) { return UINT64_MAX; } uint32_t dp = (uint32_t)(h->decimal_point); uint64_t n = 0; uint32_t i = 0; for (; i < dp; i++) { n = (10 * n) + ((i < h->num_digits) ? h->digits[i] : 0); } bool round_up = false; if (dp < h->num_digits) { round_up = h->digits[dp] >= 5; if ((h->digits[dp] == 5) && (dp + 1 == h->num_digits)) { // We are exactly halfway. If we're truncated, round up, otherwise round // to even. round_up = h->truncated || // ((dp > 0) && (1 & h->digits[dp - 1])); } } if (round_up) { n++; } return n; } // wuffs_private_impl__high_prec_dec__small_xshift shifts h's number (where 'x' // is 'l' or 'r' for left or right) by a small shift value. // // Preconditions: // - h is non-NULL. // - h->decimal_point is "not extreme". // - shift is non-zero. // - shift is "a small shift". // // "Not extreme" means within ±WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE. // // "A small shift" means not more than // WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL. // // wuffs_private_impl__high_prec_dec__rounded_integer and // wuffs_private_impl__high_prec_dec__lshift_num_new_digits have the same // preconditions. // // wuffs_private_impl__high_prec_dec__lshift keeps the first two preconditions // but not the last two. Its shift argument is signed and does not need to be // "small": zero is a no-op, positive means left shift and negative means right // shift. static void // wuffs_private_impl__high_prec_dec__small_lshift( wuffs_private_impl__high_prec_dec* h, uint32_t shift) { if (h->num_digits == 0) { return; } uint32_t num_new_digits = wuffs_private_impl__high_prec_dec__lshift_num_new_digits(h, shift); uint32_t rx = h->num_digits - 1; // Read index. uint32_t wx = h->num_digits - 1 + num_new_digits; // Write index. uint64_t n = 0; // Repeat: pick up a digit, put down a digit, right to left. while (((int32_t)rx) >= 0) { n += ((uint64_t)(h->digits[rx])) << shift; uint64_t quo = n / 10; uint64_t rem = n - (10 * quo); if (wx < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[wx] = (uint8_t)rem; } else if (rem > 0) { h->truncated = true; } n = quo; wx--; rx--; } // Put down leading digits, right to left. while (n > 0) { uint64_t quo = n / 10; uint64_t rem = n - (10 * quo); if (wx < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[wx] = (uint8_t)rem; } else if (rem > 0) { h->truncated = true; } n = quo; wx--; } // Finish. h->num_digits += num_new_digits; if (h->num_digits > WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->num_digits = WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION; } h->decimal_point += (int32_t)num_new_digits; wuffs_private_impl__high_prec_dec__trim(h); } static void // wuffs_private_impl__high_prec_dec__small_rshift( wuffs_private_impl__high_prec_dec* h, uint32_t shift) { uint32_t rx = 0; // Read index. uint32_t wx = 0; // Write index. uint64_t n = 0; // Pick up enough leading digits to cover the first shift. while ((n >> shift) == 0) { if (rx < h->num_digits) { // Read a digit. n = (10 * n) + h->digits[rx++]; } else if (n == 0) { // h's number used to be zero and remains zero. return; } else { // Read sufficient implicit trailing zeroes. while ((n >> shift) == 0) { n = 10 * n; rx++; } break; } } h->decimal_point -= ((int32_t)(rx - 1)); if (h->decimal_point < -WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE) { // After the shift, h's number is effectively zero. h->num_digits = 0; h->decimal_point = 0; h->truncated = false; return; } // Repeat: pick up a digit, put down a digit, left to right. uint64_t mask = (((uint64_t)(1)) << shift) - 1; while (rx < h->num_digits) { uint8_t new_digit = ((uint8_t)(n >> shift)); n = (10 * (n & mask)) + h->digits[rx++]; h->digits[wx++] = new_digit; } // Put down trailing digits, left to right. while (n > 0) { uint8_t new_digit = ((uint8_t)(n >> shift)); n = 10 * (n & mask); if (wx < WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION) { h->digits[wx++] = new_digit; } else if (new_digit > 0) { h->truncated = true; } } // Finish. h->num_digits = wx; wuffs_private_impl__high_prec_dec__trim(h); } static void // wuffs_private_impl__high_prec_dec__lshift(wuffs_private_impl__high_prec_dec* h, int32_t shift) { if (shift > 0) { while (shift > +WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL) { wuffs_private_impl__high_prec_dec__small_lshift( h, WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL); shift -= WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL; } wuffs_private_impl__high_prec_dec__small_lshift(h, ((uint32_t)(+shift))); } else if (shift < 0) { while (shift < -WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL) { wuffs_private_impl__high_prec_dec__small_rshift( h, WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL); shift += WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL; } wuffs_private_impl__high_prec_dec__small_rshift(h, ((uint32_t)(-shift))); } } // -------- // wuffs_private_impl__high_prec_dec__round_etc rounds h's number. For those // functions that take an n argument, rounding produces at most n digits (which // is not necessarily at most n decimal places). Negative n values are ignored, // as well as any n greater than or equal to h's number of digits. The // etc__round_just_enough function implicitly chooses an n to implement // WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION. // // Preconditions: // - h is non-NULL. // - h->decimal_point is "not extreme". // // "Not extreme" means within ±WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE. static void // wuffs_private_impl__high_prec_dec__round_down( wuffs_private_impl__high_prec_dec* h, int32_t n) { if ((n < 0) || (h->num_digits <= (uint32_t)n)) { return; } h->num_digits = (uint32_t)(n); wuffs_private_impl__high_prec_dec__trim(h); } static void // wuffs_private_impl__high_prec_dec__round_up( wuffs_private_impl__high_prec_dec* h, int32_t n) { if ((n < 0) || (h->num_digits <= (uint32_t)n)) { return; } for (n--; n >= 0; n--) { if (h->digits[n] < 9) { h->digits[n]++; h->num_digits = (uint32_t)(n + 1); return; } } // The number is all 9s. Change to a single 1 and adjust the decimal point. h->digits[0] = 1; h->num_digits = 1; h->decimal_point++; } static void // wuffs_private_impl__high_prec_dec__round_nearest( wuffs_private_impl__high_prec_dec* h, int32_t n) { if ((n < 0) || (h->num_digits <= (uint32_t)n)) { return; } bool up = h->digits[n] >= 5; if ((h->digits[n] == 5) && ((n + 1) == ((int32_t)(h->num_digits)))) { up = h->truncated || // ((n > 0) && ((h->digits[n - 1] & 1) != 0)); } if (up) { wuffs_private_impl__high_prec_dec__round_up(h, n); } else { wuffs_private_impl__high_prec_dec__round_down(h, n); } } static void // wuffs_private_impl__high_prec_dec__round_just_enough( wuffs_private_impl__high_prec_dec* h, int32_t exp2, uint64_t mantissa) { // The magic numbers 52 and 53 in this function are because IEEE 754 double // precision has 52 mantissa bits. // // Let f be the floating point number represented by exp2 and mantissa (and // also the number in h): the number (mantissa * (2 ** (exp2 - 52))). // // If f is zero or a small integer, we can return early. if ((mantissa == 0) || ((exp2 < 53) && (h->decimal_point >= ((int32_t)(h->num_digits))))) { return; } // The smallest normal f has an exp2 of -1022 and a mantissa of (1 << 52). // Subnormal numbers have the same exp2 but a smaller mantissa. static const int32_t min_incl_normal_exp2 = -1022; static const uint64_t min_incl_normal_mantissa = 0x0010000000000000ul; // Compute lower and upper bounds such that any number between them (possibly // inclusive) will round to f. First, the lower bound. Our number f is: // ((mantissa + 0) * (2 ** ( exp2 - 52))) // // The next lowest floating point number is: // ((mantissa - 1) * (2 ** ( exp2 - 52))) // unless (mantissa - 1) drops the (1 << 52) bit and exp2 is not the // min_incl_normal_exp2. Either way, call it: // ((l_mantissa) * (2 ** (l_exp2 - 52))) // // The lower bound is halfway between them (noting that 52 became 53): // (((2 * l_mantissa) + 1) * (2 ** (l_exp2 - 53))) int32_t l_exp2 = exp2; uint64_t l_mantissa = mantissa - 1; if ((exp2 > min_incl_normal_exp2) && (mantissa <= min_incl_normal_mantissa)) { l_exp2 = exp2 - 1; l_mantissa = (2 * mantissa) - 1; } wuffs_private_impl__high_prec_dec lower; wuffs_private_impl__high_prec_dec__assign(&lower, (2 * l_mantissa) + 1, false); wuffs_private_impl__high_prec_dec__lshift(&lower, l_exp2 - 53); // Next, the upper bound. Our number f is: // ((mantissa + 0) * (2 ** (exp2 - 52))) // // The next highest floating point number is: // ((mantissa + 1) * (2 ** (exp2 - 52))) // // The upper bound is halfway between them (noting that 52 became 53): // (((2 * mantissa) + 1) * (2 ** (exp2 - 53))) wuffs_private_impl__high_prec_dec upper; wuffs_private_impl__high_prec_dec__assign(&upper, (2 * mantissa) + 1, false); wuffs_private_impl__high_prec_dec__lshift(&upper, exp2 - 53); // The lower and upper bounds are possible outputs only if the original // mantissa is even, so that IEEE round-to-even would round to the original // mantissa and not its neighbors. bool inclusive = (mantissa & 1) == 0; // As we walk the digits, we want to know whether rounding up would fall // within the upper bound. This is tracked by upper_delta: // - When -1, the digits of h and upper are the same so far. // - When +0, we saw a difference of 1 between h and upper on a previous // digit and subsequently only 9s for h and 0s for upper. Thus, rounding // up may fall outside of the bound if !inclusive. // - When +1, the difference is greater than 1 and we know that rounding up // falls within the bound. // // This is a state machine with three states. The numerical value for each // state (-1, +0 or +1) isn't important, other than their order. int upper_delta = -1; // We can now figure out the shortest number of digits required. Walk the // digits until h has distinguished itself from lower or upper. // // The zi and zd variables are indexes and digits, for z in l (lower), h (the // number) and u (upper). // // The lower, h and upper numbers may have their decimal points at different // places. In this case, upper is the longest, so we iterate ui starting from // 0 and iterate li and hi starting from either 0 or -1. int32_t ui = 0; for (;; ui++) { // Calculate hd, the middle number's digit. int32_t hi = ui - upper.decimal_point + h->decimal_point; if (hi >= ((int32_t)(h->num_digits))) { break; } uint8_t hd = (((uint32_t)hi) < h->num_digits) ? h->digits[hi] : 0; // Calculate ld, the lower bound's digit. int32_t li = ui - upper.decimal_point + lower.decimal_point; uint8_t ld = (((uint32_t)li) < lower.num_digits) ? lower.digits[li] : 0; // We can round down (truncate) if lower has a different digit than h or if // lower is inclusive and is exactly the result of rounding down (i.e. we // have reached the final digit of lower). bool can_round_down = (ld != hd) || // (inclusive && ((li + 1) == ((int32_t)(lower.num_digits)))); // Calculate ud, the upper bound's digit, and update upper_delta. uint8_t ud = (((uint32_t)ui) < upper.num_digits) ? upper.digits[ui] : 0; if (upper_delta < 0) { if ((hd + 1) < ud) { // For example: // h = 12345??? // upper = 12347??? upper_delta = +1; } else if (hd != ud) { // For example: // h = 12345??? // upper = 12346??? upper_delta = +0; } } else if (upper_delta == 0) { if ((hd != 9) || (ud != 0)) { // For example: // h = 1234598? // upper = 1234600? upper_delta = +1; } } // We can round up if upper has a different digit than h and either upper // is inclusive or upper is bigger than the result of rounding up. bool can_round_up = (upper_delta > 0) || // ((upper_delta == 0) && // (inclusive || ((ui + 1) < ((int32_t)(upper.num_digits))))); // If we can round either way, round to nearest. If we can round only one // way, do it. If we can't round, continue the loop. if (can_round_down) { if (can_round_up) { wuffs_private_impl__high_prec_dec__round_nearest(h, hi + 1); return; } else { wuffs_private_impl__high_prec_dec__round_down(h, hi + 1); return; } } else { if (can_round_up) { wuffs_private_impl__high_prec_dec__round_up(h, hi + 1); return; } } } } // -------- // wuffs_private_impl__parse_number_f64_eisel_lemire produces the IEEE 754 // double-precision value for an exact mantissa and base-10 exponent. For // example: // - when parsing "12345.678e+02", man is 12345678 and exp10 is -1. // - when parsing "-12", man is 12 and exp10 is 0. Processing the leading // minus sign is the responsibility of the caller, not this function. // // On success, it returns a non-negative int64_t such that the low 63 bits hold // the 11-bit exponent and 52-bit mantissa. // // On failure, it returns a negative value. // // The algorithm is based on an original idea by Michael Eisel that was refined // by Daniel Lemire. See // https://lemire.me/blog/2020/03/10/fast-float-parsing-in-practice/ // and // https://nigeltao.github.io/blog/2020/eisel-lemire.html // // Preconditions: // - man is non-zero. // - exp10 is in the range [-307 ..= 288], the same range of the // wuffs_private_impl__powers_of_10 array. // // The exp10 range (and the fact that man is in the range [1 ..= UINT64_MAX], // approximately [1 ..= 1.85e+19]) means that (man * (10 ** exp10)) is in the // range [1e-307 ..= 1.85e+307]. This is entirely within the range of normal // (neither subnormal nor non-finite) f64 values: DBL_MIN and DBL_MAX are // approximately 2.23e–308 and 1.80e+308. static int64_t // wuffs_private_impl__parse_number_f64_eisel_lemire(uint64_t man, int32_t exp10) { // Look up the (possibly truncated) base-2 representation of (10 ** exp10). // The look-up table was constructed so that it is already normalized: the // table entry's mantissa's MSB (most significant bit) is on. const uint64_t* po10 = &wuffs_private_impl__powers_of_10[exp10 + 307][0]; // Normalize the man argument. The (man != 0) precondition means that a // non-zero bit exists. uint32_t clz = wuffs_base__count_leading_zeroes_u64(man); man <<= clz; // Calculate the return value's base-2 exponent. We might tweak it by ±1 // later, but its initial value comes from a linear scaling of exp10, // converting from power-of-10 to power-of-2, and adjusting by clz. // // The magic constants are: // - 1087 = 1023 + 64. The 1023 is the f64 exponent bias. The 64 is because // the look-up table uses 64-bit mantissas. // - 217706 is such that the ratio 217706 / 65536 ≈ 3.321930 is close enough // (over the practical range of exp10) to log(10) / log(2) ≈ 3.321928. // - 65536 = 1<<16 is arbitrary but a power of 2, so division is a shift. // // Equality of the linearly-scaled value and the actual power-of-2, over the // range of exp10 arguments that this function accepts, is confirmed by // script/print-mpb-powers-of-10.go uint64_t ret_exp2 = ((uint64_t)(((217706 * exp10) >> 16) + 1087)) - ((uint64_t)clz); // Multiply the two mantissas. Normalization means that both mantissas are at // least (1<<63), so the 128-bit product must be at least (1<<126). The high // 64 bits of the product, x_hi, must therefore be at least (1<<62). // // As a consequence, x_hi has either 0 or 1 leading zeroes. Shifting x_hi // right by either 9 or 10 bits (depending on x_hi's MSB) will therefore // leave the top 10 MSBs (bits 54 ..= 63) off and the 11th MSB (bit 53) on. wuffs_base__multiply_u64__output x = wuffs_base__multiply_u64(man, po10[1]); uint64_t x_hi = x.hi; uint64_t x_lo = x.lo; // Before we shift right by at least 9 bits, recall that the look-up table // entry was possibly truncated. We have so far only calculated a lower bound // for the product (man * e), where e is (10 ** exp10). The upper bound would // add a further (man * 1) to the 128-bit product, which overflows the lower // 64-bit limb if ((x_lo + man) < man). // // If overflow occurs, that adds 1 to x_hi. Since we're about to shift right // by at least 9 bits, that carried 1 can be ignored unless the higher 64-bit // limb's low 9 bits are all on. // // For example, parsing "9999999999999999999" will take the if-true branch // here, since: // - x_hi = 0x4563918244F3FFFF // - x_lo = 0x8000000000000000 // - man = 0x8AC7230489E7FFFF if (((x_hi & 0x1FF) == 0x1FF) && ((x_lo + man) < man)) { // Refine our calculation of (man * e). Before, our approximation of e used // a "low resolution" 64-bit mantissa. Now use a "high resolution" 128-bit // mantissa. We've already calculated x = (man * bits_0_to_63_incl_of_e). // Now calculate y = (man * bits_64_to_127_incl_of_e). wuffs_base__multiply_u64__output y = wuffs_base__multiply_u64(man, po10[0]); uint64_t y_hi = y.hi; uint64_t y_lo = y.lo; // Merge the 128-bit x and 128-bit y, which overlap by 64 bits, to // calculate the 192-bit product of the 64-bit man by the 128-bit e. // As we exit this if-block, we only care about the high 128 bits // (merged_hi and merged_lo) of that 192-bit product. // // For example, parsing "1.234e-45" will take the if-true branch here, // since: // - x_hi = 0x70B7E3696DB29FFF // - x_lo = 0xE040000000000000 // - y_hi = 0x33718BBEAB0E0D7A // - y_lo = 0xA880000000000000 uint64_t merged_hi = x_hi; uint64_t merged_lo = x_lo + y_hi; if (merged_lo < x_lo) { merged_hi++; // Carry the overflow bit. } // The "high resolution" approximation of e is still a lower bound. Once // again, see if the upper bound is large enough to produce a different // result. This time, if it does, give up instead of reaching for an even // more precise approximation to e. // // This three-part check is similar to the two-part check that guarded the // if block that we're now in, but it has an extra term for the middle 64 // bits (checking that adding 1 to merged_lo would overflow). // // For example, parsing "5.9604644775390625e-8" will take the if-true // branch here, since: // - merged_hi = 0x7FFFFFFFFFFFFFFF // - merged_lo = 0xFFFFFFFFFFFFFFFF // - y_lo = 0x4DB3FFC120988200 // - man = 0xD3C21BCECCEDA100 if (((merged_hi & 0x1FF) == 0x1FF) && ((merged_lo + 1) == 0) && (y_lo + man < man)) { return -1; } // Replace the 128-bit x with merged. x_hi = merged_hi; x_lo = merged_lo; } // As mentioned above, shifting x_hi right by either 9 or 10 bits will leave // the top 10 MSBs (bits 54 ..= 63) off and the 11th MSB (bit 53) on. If the // MSB (before shifting) was on, adjust ret_exp2 for the larger shift. // // Having bit 53 on (and higher bits off) means that ret_mantissa is a 54-bit // number. uint64_t msb = x_hi >> 63; uint64_t ret_mantissa = x_hi >> (msb + 9); ret_exp2 -= 1 ^ msb; // IEEE 754 rounds to-nearest with ties rounded to-even. Rounding to-even can // be tricky. If we're half-way between two exactly representable numbers // (x's low 73 bits are zero and the next 2 bits that matter are "01"), give // up instead of trying to pick the winner. // // Technically, we could tighten the condition by changing "73" to "73 or 74, // depending on msb", but a flat "73" is simpler. // // For example, parsing "1e+23" will take the if-true branch here, since: // - x_hi = 0x54B40B1F852BDA00 // - ret_mantissa = 0x002A5A058FC295ED if ((x_lo == 0) && ((x_hi & 0x1FF) == 0) && ((ret_mantissa & 3) == 1)) { return -1; } // If we're not halfway then it's rounding to-nearest. Starting with a 54-bit // number, carry the lowest bit (bit 0) up if it's on. Regardless of whether // it was on or off, shifting right by one then produces a 53-bit number. If // carrying up overflowed, shift again. ret_mantissa += ret_mantissa & 1; ret_mantissa >>= 1; // This if block is equivalent to (but benchmarks slightly faster than) the // following branchless form: // uint64_t overflow_adjustment = ret_mantissa >> 53; // ret_mantissa >>= overflow_adjustment; // ret_exp2 += overflow_adjustment; // // For example, parsing "7.2057594037927933e+16" will take the if-true // branch here, since: // - x_hi = 0x7FFFFFFFFFFFFE80 // - ret_mantissa = 0x0020000000000000 if ((ret_mantissa >> 53) > 0) { ret_mantissa >>= 1; ret_exp2++; } // Starting with a 53-bit number, IEEE 754 double-precision normal numbers // have an implicit mantissa bit. Mask that away and keep the low 52 bits. ret_mantissa &= 0x000FFFFFFFFFFFFF; // Pack the bits and return. return ((int64_t)(ret_mantissa | (ret_exp2 << 52))); } // -------- static wuffs_base__result_f64 // wuffs_private_impl__parse_number_f64_special(wuffs_base__slice_u8 s, uint32_t options) { do { if (options & WUFFS_BASE__PARSE_NUMBER_FXX__REJECT_INF_AND_NAN) { goto fail; } uint8_t* p = s.ptr; uint8_t* q = s.ptr + s.len; for (; (p < q) && (*p == '_'); p++) { } if (p >= q) { goto fail; } // Parse sign. bool negative = false; do { if (*p == '+') { p++; } else if (*p == '-') { negative = true; p++; } else { break; } for (; (p < q) && (*p == '_'); p++) { } } while (0); if (p >= q) { goto fail; } bool nan = false; switch (p[0]) { case 'I': case 'i': if (((q - p) < 3) || // ((p[1] != 'N') && (p[1] != 'n')) || // ((p[2] != 'F') && (p[2] != 'f'))) { goto fail; } p += 3; if ((p >= q) || (*p == '_')) { break; } else if (((q - p) < 5) || // ((p[0] != 'I') && (p[0] != 'i')) || // ((p[1] != 'N') && (p[1] != 'n')) || // ((p[2] != 'I') && (p[2] != 'i')) || // ((p[3] != 'T') && (p[3] != 't')) || // ((p[4] != 'Y') && (p[4] != 'y'))) { goto fail; } p += 5; if ((p >= q) || (*p == '_')) { break; } goto fail; case 'N': case 'n': if (((q - p) < 3) || // ((p[1] != 'A') && (p[1] != 'a')) || // ((p[2] != 'N') && (p[2] != 'n'))) { goto fail; } p += 3; if ((p >= q) || (*p == '_')) { nan = true; break; } goto fail; default: goto fail; } // Finish. for (; (p < q) && (*p == '_'); p++) { } if (p != q) { goto fail; } wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64( (nan ? 0x7FFFFFFFFFFFFFFF : 0x7FF0000000000000) | (negative ? 0x8000000000000000 : 0)); return ret; } while (0); fail: do { wuffs_base__result_f64 ret; ret.status.repr = wuffs_base__error__bad_argument; ret.value = 0; return ret; } while (0); } WUFFS_BASE__MAYBE_STATIC wuffs_base__result_f64 // wuffs_private_impl__high_prec_dec__to_f64(wuffs_private_impl__high_prec_dec* h, uint32_t options) { do { // powers converts decimal powers of 10 to binary powers of 2. For example, // (10000 >> 13) is 1. It stops before the elements exceed 60, also known // as WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL. // // This rounds down (1<<13 is a lower bound for 1e4). Adding 1 to the array // element value rounds up (1<<14 is an upper bound for 1e4) while staying // at or below WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL. // // When starting in the range [1e+1 .. 1e+2] (i.e. h->decimal_point == +2), // powers[2] == 6 and so: // - Right shifting by 6+0 produces the range [10/64 .. 100/64] = // [0.156250 .. 1.56250]. The resultant h->decimal_point is +0 or +1. // - Right shifting by 6+1 produces the range [10/128 .. 100/128] = // [0.078125 .. 0.78125]. The resultant h->decimal_point is -1 or -0. // // When starting in the range [1e-3 .. 1e-2] (i.e. h->decimal_point == -2), // powers[2] == 6 and so: // - Left shifting by 6+0 produces the range [0.001*64 .. 0.01*64] = // [0.064 .. 0.64]. The resultant h->decimal_point is -1 or -0. // - Left shifting by 6+1 produces the range [0.001*128 .. 0.01*128] = // [0.128 .. 1.28]. The resultant h->decimal_point is +0 or +1. // // Thus, when targeting h->decimal_point being +0 or +1, use (powers[n]+0) // when right shifting but (powers[n]+1) when left shifting. static const uint32_t num_powers = 19; static const uint8_t powers[19] = { 0, 3, 6, 9, 13, 16, 19, 23, 26, 29, // 33, 36, 39, 43, 46, 49, 53, 56, 59, // }; // Handle zero and obvious extremes. The largest and smallest positive // finite f64 values are approximately 1.8e+308 and 4.9e-324. if ((h->num_digits == 0) || (h->decimal_point < -326)) { goto zero; } else if (h->decimal_point > 310) { goto infinity; } // Try the fast Eisel-Lemire algorithm again. Calculating the (man, exp10) // pair from the high_prec_dec h is more correct but slower than the // approach taken in wuffs_base__parse_number_f64. The latter is optimized // for the common cases (e.g. assuming no underscores or a leading '+' // sign) rather than the full set of cases allowed by the Wuffs API. // // When we have 19 or fewer mantissa digits, run Eisel-Lemire once (trying // for an exact result). When we have more than 19 mantissa digits, run it // twice to get a lower and upper bound. We still have an exact result // (within f64's rounding margin) if both bounds are equal (and valid). uint32_t i_max = h->num_digits; if (i_max > 19) { i_max = 19; } int32_t exp10 = h->decimal_point - ((int32_t)i_max); if ((-307 <= exp10) && (exp10 <= 288)) { uint64_t man = 0; uint32_t i; for (i = 0; i < i_max; i++) { man = (10 * man) + h->digits[i]; } while (man != 0) { // The 'while' is just an 'if' that we can 'break'. int64_t r0 = wuffs_private_impl__parse_number_f64_eisel_lemire(man + 0, exp10); if (r0 < 0) { break; } else if (h->num_digits > 19) { int64_t r1 = wuffs_private_impl__parse_number_f64_eisel_lemire(man + 1, exp10); if (r1 != r0) { break; } } wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64( ((uint64_t)r0) | (((uint64_t)(h->negative)) << 63)); return ret; } } // When Eisel-Lemire fails, fall back to Simple Decimal Conversion. See // https://nigeltao.github.io/blog/2020/parse-number-f64-simple.html // // Scale by powers of 2 until we're in the range [0.1 .. 10]. Equivalently, // that h->decimal_point is +0 or +1. // // First we shift right while at or above 10... const int32_t f64_bias = -1023; int32_t exp2 = 0; while (h->decimal_point > 1) { uint32_t n = (uint32_t)(+h->decimal_point); uint32_t shift = (n < num_powers) ? powers[n] : WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL; wuffs_private_impl__high_prec_dec__small_rshift(h, shift); if (h->decimal_point < -WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE) { goto zero; } exp2 += (int32_t)shift; } // ...then we shift left while below 0.1. while (h->decimal_point < 0) { uint32_t shift; uint32_t n = (uint32_t)(-h->decimal_point); shift = (n < num_powers) // The +1 is per "when targeting h->decimal_point being +0 or // +1... when left shifting" in the powers comment above. ? (powers[n] + 1u) : WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL; wuffs_private_impl__high_prec_dec__small_lshift(h, shift); if (h->decimal_point > +WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE) { goto infinity; } exp2 -= (int32_t)shift; } // To get from "in the range [0.1 .. 10]" to "in the range [1 .. 2]" (which // will give us our exponent in base-2), the mantissa's first 3 digits will // determine the final left shift, equal to 52 (the number of explicit f64 // bits) plus an additional adjustment. int man3 = (100 * h->digits[0]) + ((h->num_digits > 1) ? (10 * h->digits[1]) : 0) + ((h->num_digits > 2) ? h->digits[2] : 0); int32_t additional_lshift = 0; if (h->decimal_point == 0) { // The value is in [0.1 .. 1]. if (man3 < 125) { additional_lshift = +4; } else if (man3 < 250) { additional_lshift = +3; } else if (man3 < 500) { additional_lshift = +2; } else { additional_lshift = +1; } } else { // The value is in [1 .. 10]. if (man3 < 200) { additional_lshift = -0; } else if (man3 < 400) { additional_lshift = -1; } else if (man3 < 800) { additional_lshift = -2; } else { additional_lshift = -3; } } exp2 -= additional_lshift; uint32_t final_lshift = (uint32_t)(52 + additional_lshift); // The minimum normal exponent is (f64_bias + 1). while ((f64_bias + 1) > exp2) { uint32_t n = (uint32_t)((f64_bias + 1) - exp2); if (n > WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL) { n = WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL; } wuffs_private_impl__high_prec_dec__small_rshift(h, n); exp2 += (int32_t)n; } // Check for overflow. if ((exp2 - f64_bias) >= 0x07FF) { // (1 << 11) - 1. goto infinity; } // Extract 53 bits for the mantissa (in base-2). wuffs_private_impl__high_prec_dec__small_lshift(h, final_lshift); uint64_t man2 = wuffs_private_impl__high_prec_dec__rounded_integer(h); // Rounding might have added one bit. If so, shift and re-check overflow. if ((man2 >> 53) != 0) { man2 >>= 1; exp2++; if ((exp2 - f64_bias) >= 0x07FF) { // (1 << 11) - 1. goto infinity; } } // Handle subnormal numbers. if ((man2 >> 52) == 0) { exp2 = f64_bias; } // Pack the bits and return. uint64_t exp2_bits = (uint64_t)((exp2 - f64_bias) & 0x07FF); // (1 << 11) - 1. uint64_t bits = (man2 & 0x000FFFFFFFFFFFFF) | // (1 << 52) - 1. (exp2_bits << 52) | // (h->negative ? 0x8000000000000000 : 0); // (1 << 63). wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64(bits); return ret; } while (0); zero: do { uint64_t bits = h->negative ? 0x8000000000000000 : 0; wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64(bits); return ret; } while (0); infinity: do { if (options & WUFFS_BASE__PARSE_NUMBER_FXX__REJECT_INF_AND_NAN) { wuffs_base__result_f64 ret; ret.status.repr = wuffs_base__error__bad_argument; ret.value = 0; return ret; } uint64_t bits = h->negative ? 0xFFF0000000000000 : 0x7FF0000000000000; wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64(bits); return ret; } while (0); } static inline bool // wuffs_private_impl__is_decimal_digit(uint8_t c) { return ('0' <= c) && (c <= '9'); } WUFFS_BASE__MAYBE_STATIC wuffs_base__result_f64 // wuffs_base__parse_number_f64(wuffs_base__slice_u8 s, uint32_t options) { // In practice, almost all "dd.ddddE±xxx" numbers can be represented // losslessly by a uint64_t mantissa "dddddd" and an int32_t base-10 // exponent, adjusting "xxx" for the position (if present) of the decimal // separator '.' or ','. // // This (u64 man, i32 exp10) data structure is superficially similar to the // "Do It Yourself Floating Point" type from Loitsch (†), but the exponent // here is base-10, not base-2. // // If s's number fits in a (man, exp10), parse that pair with the // Eisel-Lemire algorithm. If not, or if Eisel-Lemire fails, parsing s with // the fallback algorithm is slower but comprehensive. // // † "Printing Floating-Point Numbers Quickly and Accurately with Integers" // (https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf). // Florian Loitsch is also the primary contributor to // https://github.com/google/double-conversion do { // Calculating that (man, exp10) pair needs to stay within s's bounds. // Provided that s isn't extremely long, work on a NUL-terminated copy of // s's contents. The NUL byte isn't a valid part of "±dd.ddddE±xxx". // // As the pointer p walks the contents, it's faster to repeatedly check "is // *p a valid digit" than "is p within bounds and *p a valid digit". if (s.len >= 256) { goto fallback; } uint8_t z[256]; memcpy(&z[0], s.ptr, s.len); z[s.len] = 0; const uint8_t* p = &z[0]; // Look for a leading minus sign. Technically, we could also look for an // optional plus sign, but the "script/process-json-numbers.c with -p" // benchmark is noticably slower if we do. It's optional and, in practice, // usually absent. Let the fallback catch it. bool negative = (*p == '-'); if (negative) { p++; } // After walking "dd.dddd", comparing p later with p now will produce the // number of "d"s and "."s. const uint8_t* const start_of_digits_ptr = p; // Walk the "d"s before a '.', 'E', NUL byte, etc. If it starts with '0', // it must be a single '0'. If it starts with a non-zero decimal digit, it // can be a sequence of decimal digits. // // Update the man variable during the walk. It's OK if man overflows now. // We'll detect that later. uint64_t man; if (*p == '0') { man = 0; p++; if (wuffs_private_impl__is_decimal_digit(*p)) { goto fallback; } } else if (wuffs_private_impl__is_decimal_digit(*p)) { man = ((uint8_t)(*p - '0')); p++; for (; wuffs_private_impl__is_decimal_digit(*p); p++) { man = (10 * man) + ((uint8_t)(*p - '0')); } } else { goto fallback; } // Walk the "d"s after the optional decimal separator ('.' or ','), // updating the man and exp10 variables. int32_t exp10 = 0; if (*p == ((options & WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.')) { p++; const uint8_t* first_after_separator_ptr = p; if (!wuffs_private_impl__is_decimal_digit(*p)) { goto fallback; } man = (10 * man) + ((uint8_t)(*p - '0')); p++; for (; wuffs_private_impl__is_decimal_digit(*p); p++) { man = (10 * man) + ((uint8_t)(*p - '0')); } exp10 = ((int32_t)(first_after_separator_ptr - p)); } // Count the number of digits: // - for an input of "314159", digit_count is 6. // - for an input of "3.14159", digit_count is 7. // // This is off-by-one if there is a decimal separator. That's OK for now. // We'll correct for that later. The "script/process-json-numbers.c with // -p" benchmark is noticably slower if we try to correct for that now. uint32_t digit_count = (uint32_t)(p - start_of_digits_ptr); // Update exp10 for the optional exponent, starting with 'E' or 'e'. if ((*p | 0x20) == 'e') { p++; int32_t exp_sign = +1; if (*p == '-') { p++; exp_sign = -1; } else if (*p == '+') { p++; } if (!wuffs_private_impl__is_decimal_digit(*p)) { goto fallback; } int32_t exp_num = ((uint8_t)(*p - '0')); p++; // The rest of the exp_num walking has a peculiar control flow but, once // again, the "script/process-json-numbers.c with -p" benchmark is // sensitive to alternative formulations. if (wuffs_private_impl__is_decimal_digit(*p)) { exp_num = (10 * exp_num) + ((uint8_t)(*p - '0')); p++; } if (wuffs_private_impl__is_decimal_digit(*p)) { exp_num = (10 * exp_num) + ((uint8_t)(*p - '0')); p++; } while (wuffs_private_impl__is_decimal_digit(*p)) { if (exp_num > 0x1000000) { goto fallback; } exp_num = (10 * exp_num) + ((uint8_t)(*p - '0')); p++; } exp10 += exp_sign * exp_num; } // The Wuffs API is that the original slice has no trailing data. It also // allows underscores, which we don't catch here but the fallback should. if (p != &z[s.len]) { goto fallback; } // Check that the uint64_t typed man variable has not overflowed, based on // digit_count. // // For reference: // - (1 << 63) is 9223372036854775808, which has 19 decimal digits. // - (1 << 64) is 18446744073709551616, which has 20 decimal digits. // - 19 nines, 9999999999999999999, is 0x8AC7230489E7FFFF, which has 64 // bits and 16 hexadecimal digits. // - 20 nines, 99999999999999999999, is 0x56BC75E2D630FFFFF, which has 67 // bits and 17 hexadecimal digits. if (digit_count > 19) { // Even if we have more than 19 pseudo-digits, it's not yet definitely an // overflow. Recall that digit_count might be off-by-one (too large) if // there's a decimal separator. It will also over-report the number of // meaningful digits if the input looks something like "0.000dddExxx". // // We adjust by the number of leading '0's and '.'s and re-compare to 19. // Once again, technically, we could skip ','s too, but that perturbs the // "script/process-json-numbers.c with -p" benchmark. const uint8_t* q = start_of_digits_ptr; for (; (*q == '0') || (*q == '.'); q++) { } digit_count -= (uint32_t)(q - start_of_digits_ptr); if (digit_count > 19) { goto fallback; } } // The wuffs_private_impl__parse_number_f64_eisel_lemire preconditions // include that exp10 is in the range [-307 ..= 288]. if ((exp10 < -307) || (288 < exp10)) { goto fallback; } #if PK_FLOATCONV_HAS_EXACT_DOUBLE_ARITHMETIC // [pocketpy] Deviation 4. // If both man and (10 ** exp10) are exactly representable by a double, we // don't need to run the Eisel-Lemire algorithm. if ((-22 <= exp10) && (exp10 <= 22) && ((man >> 53) == 0)) { double d = (double)man; if (exp10 >= 0) { d *= wuffs_private_impl__f64_powers_of_10[+exp10]; } else { d /= wuffs_private_impl__f64_powers_of_10[-exp10]; } wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = negative ? -d : +d; return ret; } #endif // The wuffs_private_impl__parse_number_f64_eisel_lemire preconditions // include that man is non-zero. Parsing "0" should be caught by the "If // both man and (10 ** exp10)" above, but "0e99" might not. if (man == 0) { goto fallback; } // Our man and exp10 are in range. Run the Eisel-Lemire algorithm. int64_t r = wuffs_private_impl__parse_number_f64_eisel_lemire(man, exp10); if (r < 0) { goto fallback; } wuffs_base__result_f64 ret; ret.status.repr = NULL; ret.value = wuffs_base__ieee_754_bit_representation__from_u64_to_f64( ((uint64_t)r) | (((uint64_t)negative) << 63)); return ret; } while (0); fallback: do { wuffs_private_impl__high_prec_dec h; wuffs_base__status status = wuffs_private_impl__high_prec_dec__parse(&h, s, options); if (status.repr) { return wuffs_private_impl__parse_number_f64_special(s, options); } return wuffs_private_impl__high_prec_dec__to_f64(&h, options); } while (0); } // -------- static inline size_t // wuffs_private_impl__render_inf(wuffs_base__slice_u8 dst, bool neg, uint32_t options) { if (neg) { if (dst.len < 4) { return 0; } wuffs_base__poke_u32le__no_bounds_check(dst.ptr, 0x666E492D); // '-Inf'le. return 4; } if (options & WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN) { if (dst.len < 4) { return 0; } wuffs_base__poke_u32le__no_bounds_check(dst.ptr, 0x666E492B); // '+Inf'le. return 4; } if (dst.len < 3) { return 0; } wuffs_base__poke_u24le__no_bounds_check(dst.ptr, 0x666E49); // 'Inf'le. return 3; } static inline size_t // wuffs_private_impl__render_nan(wuffs_base__slice_u8 dst) { if (dst.len < 3) { return 0; } wuffs_base__poke_u24le__no_bounds_check(dst.ptr, 0x4E614E); // 'NaN'le. return 3; } static size_t // wuffs_private_impl__high_prec_dec__render_exponent_absent( wuffs_base__slice_u8 dst, wuffs_private_impl__high_prec_dec* h, uint32_t precision, uint32_t options) { size_t n = (h->negative || (options & WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN)) ? 1 : 0; if (h->decimal_point <= 0) { n += 1; } else { n += (size_t)(h->decimal_point); } if (precision > 0) { n += precision + 1; // +1 for the '.'. } // Don't modify dst if the formatted number won't fit. if (n > dst.len) { return 0; } // Align-left or align-right. uint8_t* ptr = (options & WUFFS_BASE__RENDER_NUMBER_XXX__ALIGN_RIGHT) ? &dst.ptr[dst.len - n] : &dst.ptr[0]; // Leading "±". if (h->negative) { *ptr++ = '-'; } else if (options & WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN) { *ptr++ = '+'; } // Integral digits. if (h->decimal_point <= 0) { *ptr++ = '0'; } else { uint32_t m = wuffs_base__u32__min(h->num_digits, (uint32_t)(h->decimal_point)); uint32_t i = 0; for (; i < m; i++) { *ptr++ = (uint8_t)('0' | h->digits[i]); } for (; i < (uint32_t)(h->decimal_point); i++) { *ptr++ = '0'; } } // Separator and then fractional digits. if (precision > 0) { *ptr++ = (options & WUFFS_BASE__RENDER_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.'; uint32_t i = 0; for (; i < precision; i++) { uint32_t j = ((uint32_t)(h->decimal_point)) + i; *ptr++ = (uint8_t)('0' | ((j < h->num_digits) ? h->digits[j] : 0)); } } return n; } static size_t // wuffs_private_impl__high_prec_dec__render_exponent_present( wuffs_base__slice_u8 dst, wuffs_private_impl__high_prec_dec* h, uint32_t precision, uint32_t options) { int32_t exp = 0; if (h->num_digits > 0) { exp = h->decimal_point - 1; } bool negative_exp = exp < 0; if (negative_exp) { exp = -exp; } size_t n = (h->negative || (options & WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN)) ? 4 : 3; // Mininum 3 bytes: first digit and then "e±". if (precision > 0) { n += precision + 1; // +1 for the '.'. } n += (exp < 100) ? 2 : 3; // Don't modify dst if the formatted number won't fit. if (n > dst.len) { return 0; } // Align-left or align-right. uint8_t* ptr = (options & WUFFS_BASE__RENDER_NUMBER_XXX__ALIGN_RIGHT) ? &dst.ptr[dst.len - n] : &dst.ptr[0]; // Leading "±". if (h->negative) { *ptr++ = '-'; } else if (options & WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN) { *ptr++ = '+'; } // Integral digit. if (h->num_digits > 0) { *ptr++ = (uint8_t)('0' | h->digits[0]); } else { *ptr++ = '0'; } // Separator and then fractional digits. if (precision > 0) { *ptr++ = (options & WUFFS_BASE__RENDER_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA) ? ',' : '.'; uint32_t i = 1; uint32_t j = wuffs_base__u32__min(h->num_digits, precision + 1); for (; i < j; i++) { *ptr++ = (uint8_t)('0' | h->digits[i]); } for (; i <= precision; i++) { *ptr++ = '0'; } } // Exponent: "e±" and then 2 or 3 digits. *ptr++ = 'e'; *ptr++ = negative_exp ? '-' : '+'; if (exp < 10) { *ptr++ = '0'; *ptr++ = (uint8_t)('0' | exp); } else if (exp < 100) { *ptr++ = (uint8_t)('0' | (exp / 10)); *ptr++ = (uint8_t)('0' | (exp % 10)); } else { int32_t e = exp / 100; exp -= e * 100; *ptr++ = (uint8_t)('0' | e); *ptr++ = (uint8_t)('0' | (exp / 10)); *ptr++ = (uint8_t)('0' | (exp % 10)); } return n; } WUFFS_BASE__MAYBE_STATIC size_t // wuffs_base__render_number_f64(wuffs_base__slice_u8 dst, double x, uint32_t precision, uint32_t options) { // Decompose x (64 bits) into negativity (1 bit), base-2 exponent (11 bits // with a -1023 bias) and mantissa (52 bits). uint64_t bits = wuffs_base__ieee_754_bit_representation__from_f64_to_u64(x); bool neg = (bits >> 63) != 0; int32_t exp2 = ((int32_t)(bits >> 52)) & 0x7FF; uint64_t man = bits & 0x000FFFFFFFFFFFFFul; // Apply the exponent bias and set the implicit top bit of the mantissa, // unless x is subnormal. Also take care of Inf and NaN. if (exp2 == 0x7FF) { if (man != 0) { return wuffs_private_impl__render_nan(dst); } return wuffs_private_impl__render_inf(dst, neg, options); } else if (exp2 == 0) { exp2 = -1022; } else { exp2 -= 1023; man |= 0x0010000000000000ul; } // Ensure that precision isn't too large. if (precision > 4095) { precision = 4095; } // Convert from the (neg, exp2, man) tuple to an HPD. wuffs_private_impl__high_prec_dec h; wuffs_private_impl__high_prec_dec__assign(&h, man, neg); if (h.num_digits > 0) { wuffs_private_impl__high_prec_dec__lshift(&h, exp2 - 52); // 52 mantissa bits. } // Handle the "%e" and "%f" formats. switch (options & (WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_ABSENT | WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_PRESENT)) { case WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_ABSENT: // The "%"f" format. if (options & WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION) { wuffs_private_impl__high_prec_dec__round_just_enough(&h, exp2, man); int32_t p = ((int32_t)(h.num_digits)) - h.decimal_point; precision = ((uint32_t)(wuffs_base__i32__max(0, p))); } else { wuffs_private_impl__high_prec_dec__round_nearest( &h, ((int32_t)precision) + h.decimal_point); } return wuffs_private_impl__high_prec_dec__render_exponent_absent( dst, &h, precision, options); case WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_PRESENT: // The "%e" format. if (options & WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION) { wuffs_private_impl__high_prec_dec__round_just_enough(&h, exp2, man); precision = (h.num_digits > 0) ? (h.num_digits - 1) : 0; } else { wuffs_private_impl__high_prec_dec__round_nearest( &h, ((int32_t)precision) + 1); } return wuffs_private_impl__high_prec_dec__render_exponent_present( dst, &h, precision, options); } // We have the "%g" format and so precision means the number of significant // digits, not the number of digits after the decimal separator. Perform // rounding and determine whether to use "%e" or "%f". int32_t e_threshold = 0; if (options & WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION) { wuffs_private_impl__high_prec_dec__round_just_enough(&h, exp2, man); precision = h.num_digits; e_threshold = PK_FLOATCONV_REPR_E_THRESHOLD; // [pocketpy] Deviation 3; was 6. } else { if (precision == 0) { precision = 1; } wuffs_private_impl__high_prec_dec__round_nearest(&h, ((int32_t)precision)); e_threshold = ((int32_t)precision); int32_t nd = ((int32_t)(h.num_digits)); if ((e_threshold > nd) && (nd >= h.decimal_point)) { e_threshold = nd; } } // Use the "%e" format if the exponent is large. int32_t e = h.decimal_point - 1; if ((e < -4) || (e_threshold <= e)) { uint32_t p = wuffs_base__u32__min(precision, h.num_digits); return wuffs_private_impl__high_prec_dec__render_exponent_present( dst, &h, (p > 0) ? (p - 1) : 0, options); } // Use the "%f" format otherwise. int32_t p = ((int32_t)precision); if (p > h.decimal_point) { p = ((int32_t)(h.num_digits)); } precision = ((uint32_t)(wuffs_base__i32__max(0, p - h.decimal_point))); return wuffs_private_impl__high_prec_dec__render_exponent_absent( dst, &h, precision, options); } /* ---------------- [pocketpy] public API ---------------- * * Parse options. Wuffs' defaults are stricter than C's `strtod`, so we opt * back in to redundant leading zeroes: `float("007")` and the literal `00.7` * both have to keep working. * * Underscores stay rejected. The lexer never puts one inside a number token, * `float("1_0")` was already an error under `strtod`, and Wuffs' rule is looser * than PEP 515's anyway (it would accept a leading `_`). * * Infinities and NaNs stay accepted, so `float("nan")` keeps working and * `1e999` keeps overflowing to `inf` the way CPython does, rather than raising. */ #define PK_FLOATCONV_PARSE_OPTIONS (WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_MULTIPLE_LEADING_ZEROES) static bool pk_floatconv__is_digit(char c) { return ('0' <= c) && (c <= '9'); } // Whether the NUL-terminated `p` starts with `word`, which must be lowercase // ASCII. Case insensitive. static bool pk_floatconv__starts_with(const char* p, const char* word) { for(; *word != '\0'; word++, p++) { if((*p | 0x20) != *word) return false; } return true; } bool c11__parse_f64(const char* data, int size, double* out) { if(size <= 0) return false; wuffs_base__slice_u8 s; s.ptr = (uint8_t*)data; // The vendored parser only reads through this. s.len = (size_t)size; wuffs_base__result_f64 res = wuffs_base__parse_number_f64(s, PK_FLOATCONV_PARSE_OPTIONS); if(res.status.repr != NULL) return false; *out = res.value; return true; } double strtod1(const char* s, char** p_end) { const char* p = s; // strtod() skips leading whitespace. Spelled out rather than via isspace() // so that it cannot pick up a locale's extra space characters. while(*p == ' ' || (*p >= '\t' && *p <= '\r')) p++; const char* start = p; if(*p == '+' || *p == '-') p++; // Find the longest prefix that c11__parse_f64 will accept. It only takes // whole slices, so the scanning strtod() does implicitly happens here. const char* end; if((*p | 0x20) == 'i') { if(!pk_floatconv__starts_with(p, "inf")) goto fail; end = p + 3; if(pk_floatconv__starts_with(end, "inity")) end += 5; } else if((*p | 0x20) == 'n') { if(!pk_floatconv__starts_with(p, "nan")) goto fail; end = p + 3; } else { int digits = 0; for(; pk_floatconv__is_digit(*p); p++) digits++; if(*p == '.') { p++; for(; pk_floatconv__is_digit(*p); p++) digits++; } if(digits == 0) goto fail; end = p; // Like strtod(), only consume the exponent if it is well formed. In // "1e+" the 'e' belongs to whatever comes after the number. if((*p | 0x20) == 'e') { const char* q = p + 1; if(*q == '+' || *q == '-') q++; if(pk_floatconv__is_digit(*q)) { for(; pk_floatconv__is_digit(*q); q++) {} end = q; } } } double out; if(!c11__parse_f64(start, (int)(end - start), &out)) goto fail; if(p_end != NULL) *p_end = (char*)end; return out; fail: if(p_end != NULL) *p_end = (char*)s; return 0.0; } int c11__f64_to_shortest(char* dst, int dst_size, double x) { if(dst_size <= 0) return 0; wuffs_base__slice_u8 s; s.ptr = (uint8_t*)dst; s.len = (size_t)dst_size; // Neither EXPONENT_ABSENT nor EXPONENT_PRESENT means "%g", which with // PK_FLOATCONV_REPR_E_THRESHOLD is CPython's repr() notation. return (int)wuffs_base__render_number_f64(s, x, 0, WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION); } int c11__f64_to_fixed(char* dst, int dst_size, double x, int precision) { if(dst_size <= 0) return 0; if(precision < 0) precision = 0; if(precision > C11_F64_MAX_PRECISION) precision = C11_F64_MAX_PRECISION; wuffs_base__slice_u8 s; s.ptr = (uint8_t*)dst; s.len = (size_t)dst_size; return (int)wuffs_base__render_number_f64(s, x, (uint32_t)precision, WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_ABSENT); } #undef WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_MULTIPLE_LEADING_ZEROES #undef WUFFS_BASE__PARSE_NUMBER_XXX__ALLOW_UNDERSCORES #undef WUFFS_PRIVATE_IMPL__HPD__DECIMAL_POINT__RANGE #undef WUFFS_BASE__PARSE_NUMBER_XXX__DEFAULT_OPTIONS #undef WUFFS_BASE__RENDER_NUMBER_XXX__DEFAULT_OPTIONS #undef WUFFS_BASE__PARSE_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA #undef WUFFS_PRIVATE_IMPL__HPD__DIGITS_PRECISION #undef WUFFS_PRIVATE_IMPL__HPD__SHIFT__MAX_INCL #undef WUFFS_BASE__RENDER_NUMBER_FXX__DECIMAL_SEPARATOR_IS_A_COMMA #undef WUFFS_BASE__PARSE_NUMBER_FXX__REJECT_INF_AND_NAN #undef WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_ABSENT #undef PK_FLOATCONV_PARSE_OPTIONS #undef WUFFS_BASE__RENDER_NUMBER_FXX__EXPONENT_PRESENT #undef WUFFS_BASE__RENDER_NUMBER_XXX__ALIGN_RIGHT #undef PK_FLOATCONV_REPR_E_THRESHOLD #undef PK_FLOATCONV_HAS_EXACT_DOUBLE_ARITHMETIC #undef WUFFS_BASE__MAYBE_STATIC #undef WUFFS_BASE__RENDER_NUMBER_FXX__JUST_ENOUGH_PRECISION #undef WUFFS_BASE__RENDER_NUMBER_XXX__LEADING_PLUS_SIGN