#ifndef FASTFLOAT_GENERIC_DECIMAL_TO_BINARY_H #define FASTFLOAT_GENERIC_DECIMAL_TO_BINARY_H /** * This code is meant to handle the case where we have more than 19 digits. * * It is based on work by Nigel Tao (at https://github.com/google/wuffs/) * who credits Ken Thompson for the design (via a reference to the Go source * code). * * Rob Pike suggested that this algorithm be called "Simple Decimal Conversion". * * It is probably not very fast but it is a fallback that should almost never * be used in real life. Though it is not fast, it is "easily" understood and debugged. **/ #include "ascii_number.h" #include "decimal_to_binary.h" #include namespace fast_float { namespace detail { // remove all final zeroes inline void trim(decimal &h) { while ((h.num_digits > 0) && (h.digits[h.num_digits - 1] == 0)) { h.num_digits--; } } inline uint32_t number_of_digits_decimal_left_shift(const decimal &h, uint32_t shift) { shift &= 63; const static uint16_t number_of_digits_decimal_left_shift_table[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, }; uint32_t x_a = number_of_digits_decimal_left_shift_table[shift]; uint32_t x_b = number_of_digits_decimal_left_shift_table[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 static uint8_t number_of_digits_decimal_left_shift_table_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, }; const uint8_t *pow5 = &number_of_digits_decimal_left_shift_table_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; } inline uint64_t round(decimal &h) { if ((h.num_digits == 0) || (h.decimal_point < 0)) { return 0; } else if (h.decimal_point > 18) { return UINT64_MAX; } // at this point, we know that h.decimal_point >= 0 uint32_t dp = uint32_t(h.decimal_point); uint64_t n = 0; for (uint32_t i = 0; 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; // normally, we round up // but we may need to round to even! if ((h.digits[dp] == 5) && (dp + 1 == h.num_digits)) { round_up = h.truncated || ((dp > 0) && (1 & h.digits[dp - 1])); } } if (round_up) { n++; } return n; } // computes h * 2^-shift inline void decimal_left_shift(decimal &h, uint32_t shift) { if (h.num_digits == 0) { return; } uint32_t num_new_digits = number_of_digits_decimal_left_shift(h, shift); int32_t read_index = int32_t(h.num_digits - 1); uint32_t write_index = h.num_digits - 1 + num_new_digits; uint64_t n = 0; while (read_index >= 0) { n += uint64_t(h.digits[read_index]) << shift; uint64_t quotient = n / 10; uint64_t remainder = n - (10 * quotient); if (write_index < max_digits) { h.digits[write_index] = uint8_t(remainder); } else if (remainder > 0) { h.truncated = true; } n = quotient; write_index--; read_index--; } while (n > 0) { uint64_t quotient = n / 10; uint64_t remainder = n - (10 * quotient); if (write_index < max_digits) { h.digits[write_index] = uint8_t(remainder); } else if (remainder > 0) { h.truncated = true; } n = quotient; write_index--; } h.num_digits += num_new_digits; if (h.num_digits > max_digits) { h.num_digits = max_digits; } h.decimal_point += int32_t(num_new_digits); trim(h); } // computes h * 2^shift inline void decimal_right_shift(decimal &h, uint32_t shift) { uint32_t read_index = 0; uint32_t write_index = 0; uint64_t n = 0; while ((n >> shift) == 0) { if (read_index < h.num_digits) { n = (10 * n) + h.digits[read_index++]; } else if (n == 0) { return; } else { while ((n >> shift) == 0) { n = 10 * n; read_index++; } break; } } h.decimal_point -= int32_t(read_index - 1); if (h.decimal_point < -decimal_point_range) { // it is zero h.num_digits = 0; h.decimal_point = 0; h.negative = false; h.truncated = false; return; } uint64_t mask = (uint64_t(1) << shift) - 1; while (read_index < h.num_digits) { uint8_t new_digit = uint8_t(n >> shift); n = (10 * (n & mask)) + h.digits[read_index++]; h.digits[write_index++] = new_digit; } while (n > 0) { uint8_t new_digit = uint8_t(n >> shift); n = 10 * (n & mask); if (write_index < max_digits) { h.digits[write_index++] = new_digit; } else if (new_digit > 0) { h.truncated = true; } } h.num_digits = write_index; trim(h); } } // namespace detail template adjusted_mantissa compute_float(decimal &d) { adjusted_mantissa answer; if (d.num_digits == 0) { // should be zero answer.power2 = 0; answer.mantissa = 0; return answer; } // At this point, going further, we can assume that d.num_digits > 0. // // We want to guard against excessive decimal point values because // they can result in long running times. Indeed, we do // shifts by at most 60 bits. We have that log(10**400)/log(2**60) ~= 22 // which is fine, but log(10**299995)/log(2**60) ~= 16609 which is not // fine (runs for a long time). // if(d.decimal_point < -324) { // We have something smaller than 1e-324 which is always zero // in binary64 and binary32. // It should be zero. answer.power2 = 0; answer.mantissa = 0; return answer; } else if(d.decimal_point >= 310) { // We have something at least as large as 0.1e310 which is // always infinite. answer.power2 = binary::infinite_power(); answer.mantissa = 0; return answer; } static const uint32_t max_shift = 60; static const uint32_t num_powers = 19; static const uint8_t decimal_powers[19] = { 0, 3, 6, 9, 13, 16, 19, 23, 26, 29, // 33, 36, 39, 43, 46, 49, 53, 56, 59, // }; int32_t exp2 = 0; while (d.decimal_point > 0) { uint32_t n = uint32_t(d.decimal_point); uint32_t shift = (n < num_powers) ? decimal_powers[n] : max_shift; detail::decimal_right_shift(d, shift); if (d.decimal_point < -decimal_point_range) { // should be zero answer.power2 = 0; answer.mantissa = 0; return answer; } exp2 += int32_t(shift); } // We shift left toward [1/2 ... 1]. while (d.decimal_point <= 0) { uint32_t shift; if (d.decimal_point == 0) { if (d.digits[0] >= 5) { break; } shift = (d.digits[0] < 2) ? 2 : 1; } else { uint32_t n = uint32_t(-d.decimal_point); shift = (n < num_powers) ? decimal_powers[n] : max_shift; } detail::decimal_left_shift(d, shift); if (d.decimal_point > decimal_point_range) { // we want to get infinity: answer.power2 = binary::infinite_power(); answer.mantissa = 0; return answer; } exp2 -= int32_t(shift); } // We are now in the range [1/2 ... 1] but the binary format uses [1 ... 2]. exp2--; constexpr int32_t minimum_exponent = binary::minimum_exponent(); while ((minimum_exponent + 1) > exp2) { uint32_t n = uint32_t((minimum_exponent + 1) - exp2); if (n > max_shift) { n = max_shift; } detail::decimal_right_shift(d, n); exp2 += int32_t(n); } if ((exp2 - minimum_exponent) >= binary::infinite_power()) { answer.power2 = binary::infinite_power(); answer.mantissa = 0; return answer; } const int mantissa_size_in_bits = binary::mantissa_explicit_bits() + 1; detail::decimal_left_shift(d, mantissa_size_in_bits); uint64_t mantissa = detail::round(d); // It is possible that we have an overflow, in which case we need // to shift back. if(mantissa >= (uint64_t(1) << mantissa_size_in_bits)) { detail::decimal_right_shift(d, 1); exp2 += 1; mantissa = detail::round(d); if ((exp2 - minimum_exponent) >= binary::infinite_power()) { answer.power2 = binary::infinite_power(); answer.mantissa = 0; return answer; } } answer.power2 = exp2 - binary::minimum_exponent(); if(mantissa < (uint64_t(1) << binary::mantissa_explicit_bits())) { answer.power2--; } answer.mantissa = mantissa & ((uint64_t(1) << binary::mantissa_explicit_bits()) - 1); return answer; } template adjusted_mantissa parse_long_mantissa(const char *first, const char* last) { decimal d = parse_decimal(first, last); return compute_float(d); } } // namespace fast_float #endif