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+/*
+ * Copyright (c) 2022, Dan Klishch <danilklishch@gmail.com>
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ */
+
+#include <AK/Array.h>
+#include <AK/BuiltinWrappers.h>
+#include <AK/FloatingPoint.h>
+#include <AK/StringFloatingPointConversions.h>
+#include <AK/UFixedBigInt.h>
+
+namespace AK {
+
+// This entire algorithm is an implementation of the paper: Ryu: Fast Float-to-String Conversion
+// by Ulf Adams, available at https://dl.acm.org/doi/pdf/10.1145/3192366.3192369 and an implemenetation
+// at https://github.com/ulfjack/ryu . A lot of possible mistakes from the article were corrected, see
+// discussion at https://github.com/SerenityOS/serenity/pull/15796 .
+//
+// Not implemented for float80, as it will require an insane lookup table size (193Kb).
+//
+// Run stress tests from https://github.com/DanShaders/serenity-arithmetic-benchmark after non-trivial
+// modifications.
+
+// These approximations should match the ones used in the Python script.
+static constexpr i64 log10_5_num = 10043;
+static constexpr i64 log10_5_denum = 14369;
+
+static constexpr i64 log10_2_num = 1406;
+static constexpr i64 log10_2_denum = 4671;
+
+static constexpr i64 log2_5_num = 8245;
+static constexpr i64 log2_5_denum = 3551;
+
+template<typename Number, size_t Size1, size_t Size2>
+struct LookupInformation {
+ i32 b0, b1; // B0 and B1 from the paper (accidentally swapped)
+ Number lt[Size1];
+ Number ge[Size2];
+};
+
+template<FloatingPoint>
+int lookup_table;
+
+template<typename FloatingPoint, typename MultiplyAndShiftFunction>
+FloatingPointExponentialForm inner_convert_floating_point_to_decimal_exponential_form(FloatingPoint value, MultiplyAndShiftFunction const& multiply_and_shift)
+{
+ using Extractor = FloatExtractor<FloatingPoint>;
+
+ Extractor bit_representation { .d = value };
+
+ bool sign = bit_representation.sign;
+ i32 exponent = bit_representation.exponent;
+ u64 mantissa = bit_representation.mantissa;
+
+ // For +0, it is {.sign = 0, fraction = 0, exponent = 0},
+ // for -0, is {.sign = 1, fraction = 0, exponent = 0},
+ if (exponent == 0 && mantissa == 0)
+ return { sign, 0, 0 };
+
+ // for +inf, -inf, and NaN is undefined.
+ VERIFY(exponent != Extractor::exponent_max);
+
+ // Step 1. Decode the floating point number, and unify normalized and subnormal cases.
+ u64 real_mantissa = (exponent == 0 ? 0 : (1ull << Extractor::mantissa_bits)) + mantissa;
+ i32 real_exponent = (exponent == 0 ? 1 : exponent) - Extractor::exponent_bias - Extractor::mantissa_bits;
+ // abs(value) = real_mantissa * 2 ^ real_exponent
+
+ // Step 2. Determine the interval of information-preserving outputs.
+ // u, v, w are, respectively, lower bound for answer, exact value and upper bound for answer.
+ i32 synthetic_exponent = real_exponent - 2;
+ u64 u = 4 * real_mantissa - (mantissa == 0 && exponent > 1 ? 1 : 2);
+ u64 v = 4 * real_mantissa;
+ u64 w = 4 * real_mantissa + 2;
+ // u * 2 ^ synthetic_exponent < abs(answer) < w * 2 ^ synthetic_exponent (1)
+ // abs(value) = v * 2 ^ synthetic_exponent (yet another representation)
+
+ // Step 3'. Convert to a decimal power base and simultaneously remove most digits.
+ // We want to skip `skipped_iters' iterations of the main conversion loop and find out if
+ // last `skipped_iters' digits of u, v and w would have been zeroes.
+ i32 skipped_iters;
+ bool all_u_zero, all_v_zero, all_w_zero;
+
+ if (synthetic_exponent < 0) {
+ skipped_iters = max(0, -synthetic_exponent * log10_5_num / log10_5_denum - 1);
+
+ all_u_zero = count_trailing_zeroes(u) >= skipped_iters;
+ all_v_zero = count_trailing_zeroes(v) >= skipped_iters;
+ all_w_zero = count_trailing_zeroes(w) >= skipped_iters;
+
+ auto multiplier = lookup_table<FloatingPoint>.lt[-synthetic_exponent - skipped_iters];
+ i32 k_numerator = (log2_5_num + 1) * (-synthetic_exponent - skipped_iters);
+ i32 k = max(0, (k_numerator + log2_5_denum - 1) / log2_5_denum + lookup_table<FloatingPoint>.b0);
+ u = multiply_and_shift(u, multiplier, skipped_iters - k);
+ v = multiply_and_shift(v, multiplier, skipped_iters - k);
+ w = multiply_and_shift(w, multiplier, skipped_iters - k);
+ } else {
+ skipped_iters = max(0, synthetic_exponent * log10_2_num / log10_2_denum - 1);
+
+ // Checks if value is divisible by 5 ^ power.
+ auto is_divisible_by_pow_5 = [](u64 value, i32 power) {
+ constexpr Array<u64, 5> powers_of_five = { { 5, 25, 625, 390625, 152587890625 } };
+
+ if (power <= 0 || value == 0)
+ return true;
+ if (power >= 28) // 2 ^ 64 - 1 < 5 ^ 28
+ return false;
+
+ i32 result = 0;
+ for (i32 i = 5; i--;) {
+ if (value % powers_of_five[i] == 0) {
+ value /= powers_of_five[i];
+ result += 1 << i;
+ }
+ }
+ return result >= power;
+ };
+
+ all_u_zero = is_divisible_by_pow_5(u, skipped_iters);
+ all_v_zero = is_divisible_by_pow_5(v, skipped_iters);
+ all_w_zero = is_divisible_by_pow_5(w, skipped_iters);
+
+ auto multiplier = lookup_table<FloatingPoint>.ge[skipped_iters];
+ i32 k = log2_5_num * skipped_iters / log2_5_denum + lookup_table<FloatingPoint>.b1;
+ u = multiply_and_shift(u, multiplier, skipped_iters + k - synthetic_exponent);
+ v = multiply_and_shift(v, multiplier, skipped_iters + k - synthetic_exponent);
+ w = multiply_and_shift(w, multiplier, skipped_iters + k - synthetic_exponent);
+ }
+
+ // Step 4'. Find the shortest, correctly-rounded decimal representation in the interval.
+ bool is_even = ~mantissa & 1;
+ bool accept_smaller = is_even && all_u_zero;
+ bool accept_larger = is_even || !all_w_zero;
+
+ if (!accept_larger)
+ --w;
+
+ bool all_a_zero = accept_smaller;
+ bool all_b_zero = all_v_zero;
+ int last_digit = 0;
+
+ int exponent10 = skipped_iters - max(-synthetic_exponent, 0);
+
+ while (u / 10 < w / 10) {
+ all_a_zero &= u % 10 == 0;
+ all_b_zero &= last_digit == 0;
+ last_digit = v % 10;
+
+ u /= 10;
+ v /= 10;
+ w /= 10;
+ ++exponent10;
+ }
+ if (all_a_zero) {
+ while (u % 10 == 0) {
+ all_b_zero &= last_digit == 0;
+ last_digit = v % 10;
+
+ u /= 10;
+ v /= 10;
+ w /= 10;
+ ++exponent10;
+ }
+ }
+
+ bool is_tie = all_b_zero && last_digit == 5;
+ bool want_round_down = last_digit < 5 || (is_tie && v % 2 == 0);
+ bool round_down = (want_round_down && (u != v || all_a_zero)) || (v + 1 > w);
+ return { sign, round_down ? v : v + 1, exponent10 };
+}
+
+static u128 multiply(u64 a, u64 b)
+{
+#ifdef __SIZEOF_INT128__
+ unsigned __int128 result = (unsigned __int128)a * b;
+ u64 low = result;
+ u64 high = result >> 64;
+ return u128 { low, high };
+#else
+ return u128 { a }.wide_multiply(u128 { b }).low;
+#endif
+}
+
+template<>
+FloatingPointExponentialForm convert_floating_point_to_decimal_exponential_form<float>(float value)
+{
+ auto multiply_and_shift = [](u64 operand, u64 multiplier, i32 shift) {
+#ifdef __SIZEOF_INT128__
+ auto result = (unsigned __int128)operand * multiplier;
+#else
+ auto result = multiply(operand, multiplier);
+#endif
+ if (shift < 0)
+ return static_cast<u64>(result << static_cast<u32>(-shift));
+ else
+ return static_cast<u64>(result >> static_cast<u32>(shift));
+ };
+
+ return inner_convert_floating_point_to_decimal_exponential_form(value, multiply_and_shift);
+}
+
+template<>
+FloatingPointExponentialForm convert_floating_point_to_decimal_exponential_form<double>(double value)
+{
+ auto multiply_and_shift = [](u64 operand, u64 const multiplier[2], i32 shift) {
+ u128 a = multiply(operand, multiplier[0]);
+ u128 b = multiply(operand, multiplier[1]) + a.high();
+ u64 c = a.low();
+
+ if (0 <= shift && shift < 64) {
+ return (c >> shift) | (b << static_cast<u32>(64 - shift)).low();
+ } else if (shift < 0) {
+ return c << static_cast<u32>(-shift);
+ } else {
+ VERIFY(64 <= shift && shift <= 128);
+ return (b >> static_cast<u32>(shift - 64)).low();
+ }
+ };
+
+ return inner_convert_floating_point_to_decimal_exponential_form(value, multiply_and_shift);
+}
+
+// Step 0. Precompute lookup tables for the given floating point type.
+// Lookup tables was generated using the following Python script.
+/*
+from math import *
+from more_itertools import chunked
+
+
+def ifloor(x, y):
+ assert y > 0
+ if x < 0:
+ return (x - y + 1) // y
+ else:
+ return x // y
+
+
+def iceil(x, y):
+ assert y > 0
+ if x < 0:
+ return x // y
+ else:
+ return (x + y - 1) // y
+
+
+# Finds X = min(a * x % b) and Y = max(a * x % b) where 1 <= x <= N and returns (X, Y)
+# Algorithm is from https://github.com/jk-jeon/Grisu-Exact/blob/master/other_files/Grisu-Exact.pdf , p. 22
+def minmax_euclid(a, b, N):
+ a_i, b_i = a, b
+ s_i, u_i = 1, 0
+
+ while True:
+ q_i = iceil(b_i, a_i) - 1
+ b_i1 = b_i - q_i * a_i
+ u_i1 = u_i + q_i * s_i
+
+ if N < u_i1:
+ k = ifloor(N - u_i, s_i)
+ return (a_i, b - b_i + k * a_i)
+
+ p_i = iceil(a_i, b_i1) - 1
+ a_i1 = a_i - p_i * b_i1
+ s_i1 = s_i + p_i * u_i1
+
+ if N < s_i1:
+ k = ifloor(N - s_i, u_i1)
+ return (a_i - k * b_i1, b - b_i1)
+
+ if b_i1 == b_i and a_i1 == a_i:
+ if N < s_i1 + u_i1:
+ return (a_i1, b - b_i1)
+ else:
+ return (0, b - b_i1)
+
+ b_i, u_i, a_i, s_i = b_i1, u_i1, a_i1, s_i1
+
+
+assert minmax_euclid(3, 8, 5) == (1, 7)
+
+
+def calculate_lookup_tables(mantissa_bits, exponent_bits, nibbles_per_wide_digit, wide_digits_count, digit_suffix):
+ def split_by_wide_digits_and_print(value):
+ length = wide_digits_count * nibbles_per_wide_digit
+ number = reversed(list(chunked(f"{value:0{length}x}", nibbles_per_wide_digit)))
+ number = ", ".join(map(lambda x: "0x" + "".join(x) + digit_suffix, number))
+ print(f"{{ {number} }},")
+
+ mantissa_bias = 1 << mantissa_bits
+ mantissa_max = (1 << mantissa_bits) - 1
+ exponent_bias = (1 << (exponent_bits - 1)) - 1
+ exponent_max = (1 << exponent_bits) - 1
+
+ real_exponent_min = 1 - exponent_bias - mantissa_bits
+ real_exponent_max = exponent_max - exponent_bias - mantissa_bits
+ # real_exponent_min <= ef < real_exponent_max
+
+ synthetic_exponent_min = real_exponent_min - 2
+ synthetic_exponent_max = real_exponent_max - 2
+ # synthetic_exponent_min <= e2 < synthetic_exponent_max
+
+ max_synthetic_mantissa = 4 * (mantissa_bias + mantissa_max) + 2
+
+ # The following are some random approximations. Absolutely nothing special with these exact numbers.
+ LOG10_5_NUM = 10043
+ LOG10_5_DENUM = 14369
+ assert LOG10_5_NUM / LOG10_5_DENUM < log(5, 10)
+
+ LOG10_2_NUM = 1406
+ LOG10_2_DENUM = 4671
+ assert LOG10_2_NUM / LOG10_2_DENUM < log(2, 10)
+
+ LOG2_5_NUM = 8245
+ LOG2_5_DENUM = 3551
+ assert LOG2_5_NUM / LOG2_5_DENUM < log(5, 2)
+ assert (LOG2_5_NUM + 1) / LOG2_5_DENUM > log(5, 2)
+
+ # We want to find maximal b0, such that ceil(log(5, 2) * (-e2 - q)) + b0 <= k. One might plot (-e2 - q, k) from the
+ # iterations of the following loop and k = (-e2 - q) * log(5, 2) to understand the motivation behind this.
+ b0 = 0
+ q0max = 0
+
+ for e2 in range(synthetic_exponent_min, 0):
+ # q = max(0, floor(-e2 * log(5, 10)) - 1)
+ q = max(0, ifloor(-e2 * LOG10_5_NUM, LOG10_5_DENUM) - 1)
+
+ q0max = max(q0max, -e2 - q)
+ a = 5 ** (-e2 - q)
+ b = 2 ** q
+
+ [min_modular_product, _] = minmax_euclid(a, b, max_synthetic_mantissa)
+
+ # Directly via lemma 3.4 we obtain
+ # k = floor(log2(min_modular_product / max_synthetic_mantissa))
+ # But computing this directly might result in OverflowError, so we approximate the value
+ k = (min_modular_product.bit_length() - 1) - max_synthetic_mantissa.bit_length()
+
+ # "It is never wrong just to use 0"
+ # -- Some Guy
+ k = max(k, 0)
+
+ # coefficient = 5 ** (-e2 - q) // 2 ** k
+
+ # ceil(log(5, 2) * (-e2 - q)) + b0 <= k
+ # b0 <= k - ceil(log(5, 2) * (-e2 - q))
+ b0 = min(b0, k - iceil((-e2 - q) * (LOG2_5_NUM + 1), LOG2_5_DENUM))
+
+ print('b0 =', b0)
+ print('q0max =', q0max)
+ for q in range(0, q0max + 1):
+ k = max(0, iceil((LOG2_5_NUM + 1) * q, LOG2_5_DENUM) + b0)
+ coefficient = 5 ** q // 2 ** k
+ split_by_wide_digits_and_print(coefficient)
+
+
+ # Finding minimal b1, such that floor(log(5, 2) * q) + b1 >= k.
+ b1 = 0
+ q1max = 0
+
+ for e2 in range(0, synthetic_exponent_max):
+ # q = max(0, floor(e2 * log(2, 10)) - 1)
+ q = max(0, ifloor(e2 * LOG10_2_NUM, LOG10_2_DENUM) - 1)
+
+ q1max = max(q1max, q)
+ a = 2 ** (e2 - q)
+ b = 5 ** q
+
+ [_, max_modular_product] = minmax_euclid(a, b, max_synthetic_mantissa)
+
+ # Via lemma 3.3:
+ # k = ceil(log2(max_synthetic_mantissa * a * b / (b - max_modular_product)))
+ numerator = max_synthetic_mantissa * a * b
+ denumerator = b - max_modular_product
+ k = numerator.bit_length() - denumerator.bit_length() + 1
+
+ # coefficient = 2 ** k // 5 ** q + 1
+
+ # b1 = max(b1, k - floor(log(5, 2) * q))
+ b1 = max(b1, k - ifloor(q * LOG2_5_NUM, LOG2_5_DENUM))
+
+ print('b1 =', b1)
+ print('q1max =', q1max)
+ for q in range(0, q1max + 1):
+ k = ifloor(LOG2_5_NUM * q, LOG2_5_DENUM) + b1
+ coefficient = 2 ** k // 5 ** q + 1
+ split_by_wide_digits_and_print(coefficient)
+
+
+# float:
+print("float:")
+calculate_lookup_tables(
+ 23, 8,
+ 16, 1, "ULL"
+)
+
+# double:
+print("double:")
+calculate_lookup_tables(
+ 52, 11,
+ 16, 2, "ULL"
+)
+
+# long double:
+# print("long double:")
+# calculate_lookup_tables(
+# 64, 15,
+# 8, 5, "U"
+# )
+*/
+template<>
+constexpr LookupInformation<u64, 48, 30> lookup_table<float> {
+ .b0 = -64,
+ .b1 = 62,
+ .lt = {
+ 0x0000000000000001ULL,
+ 0x0000000000000005ULL,
+ 0x0000000000000019ULL,
+ 0x000000000000007dULL,
+ 0x0000000000000271ULL,
+ 0x0000000000000c35ULL,
+ 0x0000000000003d09ULL,
+ 0x000000000001312dULL,
+ 0x000000000005f5e1ULL,
+ 0x00000000001dcd65ULL,
+ 0x00000000009502f9ULL,
+ 0x0000000002e90eddULL,
+ 0x000000000e8d4a51ULL,
+ 0x0000000048c27395ULL,
+ 0x000000016bcc41e9ULL,
+ 0x000000071afd498dULL,
+ 0x0000002386f26fc1ULL,
+ 0x000000b1a2bc2ec5ULL,
+ 0x000003782dace9d9ULL,
+ 0x00001158e460913dULL,
+ 0x000056bc75e2d631ULL,
+ 0x0001b1ae4d6e2ef5ULL,
+ 0x000878678326eac9ULL,
+ 0x002a5a058fc295edULL,
+ 0x00d3c21bcecceda1ULL,
+ 0x0422ca8b0a00a425ULL,
+ 0x14adf4b7320334b9ULL,
+ 0x6765c793fa10079dULL,
+ 0x813f3978f8940984ULL,
+ 0xa18f07d736b90be5ULL,
+ 0xc9f2c9cd04674edeULL,
+ 0xfc6f7c4045812296ULL,
+ 0x9dc5ada82b70b59dULL,
+ 0xc5371912364ce305ULL,
+ 0xf684df56c3e01bc6ULL,
+ 0x9a130b963a6c115cULL,
+ 0xc097ce7bc90715b3ULL,
+ 0xf0bdc21abb48db20ULL,
+ 0x96769950b50d88f4ULL,
+ 0xbc143fa4e250eb31ULL,
+ 0xeb194f8e1ae525fdULL,
+ 0x92efd1b8d0cf37beULL,
+ 0xb7abc627050305adULL,
+ 0xe596b7b0c643c719ULL,
+ 0x8f7e32ce7bea5c6fULL,
+ 0xb35dbf821ae4f38bULL,
+ 0xe0352f62a19e306eULL,
+ 0x8c213d9da502de45ULL,
+ },
+ .ge = {
+ 0x4000000000000001ULL,
+ 0x3333333333333334ULL,
+ 0x28f5c28f5c28f5c3ULL,
+ 0x20c49ba5e353f7cfULL,
+ 0x346dc5d63886594bULL,
+ 0x29f16b11c6d1e109ULL,
+ 0x218def416bdb1a6eULL,
+ 0x35afe535795e90b0ULL,
+ 0x2af31dc4611873c0ULL,
+ 0x225c17d04dad2966ULL,
+ 0x36f9bfb3af7b7570ULL,
+ 0x2bfaffc2f2c92ac0ULL,
+ 0x232f33025bd42233ULL,
+ 0x384b84d092ed0385ULL,
+ 0x2d09370d42573604ULL,
+ 0x24075f3dceac2b37ULL,
+ 0x39a5652fb1137857ULL,
+ 0x2e1dea8c8da92d13ULL,
+ 0x24e4bba3a4875742ULL,
+ 0x3b07929f6da5586aULL,
+ 0x2f394219248446bbULL,
+ 0x25c768141d369efcULL,
+ 0x3c7240202ebdcb2dULL,
+ 0x305b66802564a28aULL,
+ 0x26af8533511d4ed5ULL,
+ 0x3de5a1ebb4fbb155ULL,
+ 0x318481895d962777ULL,
+ 0x279d346de4781f93ULL,
+ 0x3f61ed7ca0c03284ULL,
+ 0x32b4bdfd4d668ed0ULL,
+ },
+};
+
+template<>
+constexpr LookupInformation<u64[2], 326, 291> lookup_table<double> {
+ .b0 = -125,
+ .b1 = 125,
+ .lt = {
+ { 0x0000000000000001ULL, 0x0000000000000000ULL },
+ { 0x0000000000000005ULL, 0x0000000000000000ULL },
+ { 0x0000000000000019ULL, 0x0000000000000000ULL },
+ { 0x000000000000007dULL, 0x0000000000000000ULL },
+ { 0x0000000000000271ULL, 0x0000000000000000ULL },
+ { 0x0000000000000c35ULL, 0x0000000000000000ULL },
+ { 0x0000000000003d09ULL, 0x0000000000000000ULL },
+ { 0x000000000001312dULL, 0x0000000000000000ULL },
+ { 0x000000000005f5e1ULL, 0x0000000000000000ULL },
+ { 0x00000000001dcd65ULL, 0x0000000000000000ULL },
+ { 0x00000000009502f9ULL, 0x0000000000000000ULL },
+ { 0x0000000002e90eddULL, 0x0000000000000000ULL },
+ { 0x000000000e8d4a51ULL, 0x0000000000000000ULL },
+ { 0x0000000048c27395ULL, 0x0000000000000000ULL },
+ { 0x000000016bcc41e9ULL, 0x0000000000000000ULL },
+ { 0x000000071afd498dULL, 0x0000000000000000ULL },
+ { 0x0000002386f26fc1ULL, 0x0000000000000000ULL },
+ { 0x000000b1a2bc2ec5ULL, 0x0000000000000000ULL },
+ { 0x000003782dace9d9ULL, 0x0000000000000000ULL },
+ { 0x00001158e460913dULL, 0x0000000000000000ULL },
+ { 0x000056bc75e2d631ULL, 0x0000000000000000ULL },
+ { 0x0001b1ae4d6e2ef5ULL, 0x0000000000000000ULL },
+ { 0x000878678326eac9ULL, 0x0000000000000000ULL },
+ { 0x002a5a058fc295edULL, 0x0000000000000000ULL },
+ { 0x00d3c21bcecceda1ULL, 0x0000000000000000ULL },
+ { 0x0422ca8b0a00a425ULL, 0x0000000000000000ULL },
+ { 0x14adf4b7320334b9ULL, 0x0000000000000000ULL },
+ { 0x6765c793fa10079dULL, 0x0000000000000000ULL },
+ { 0x04fce5e3e2502611ULL, 0x0000000000000002ULL },
+ { 0x18f07d736b90be55ULL, 0x000000000000000aULL },
+ { 0x7cb2734119d3b7a9ULL, 0x0000000000000032ULL },
+ { 0x6f7c40458122964dULL, 0x00000000000000fcULL },
+ { 0x2d6d415b85acef81ULL, 0x00000000000004eeULL },
+ { 0xe32246c99c60ad85ULL, 0x00000000000018a6ULL },
+ { 0x6fab61f00de36399ULL, 0x0000000000007b42ULL },
+ { 0x2e58e9b04570f1fdULL, 0x000000000002684cULL },
+ { 0xe7bc90715b34b9f1ULL, 0x00000000000c097cULL },
+ { 0x86aed236c807a1b5ULL, 0x00000000003c2f70ULL },
+ { 0xa16a1b11e8262889ULL, 0x00000000012ced32ULL },
+ { 0x2712875988becaadULL, 0x0000000005e0a1fdULL },
+ { 0xc35ca4bfabb9f561ULL, 0x000000001d6329f1ULL },
+ { 0xd0cf37be5aa1cae5ULL, 0x0000000092efd1b8ULL },
+ { 0x140c16b7c528f679ULL, 0x00000002deaf189cULL },
+ { 0x643c7196d9ccd05dULL, 0x0000000e596b7b0cULL },
+ { 0xf52e37f2410011d1ULL, 0x00000047bf19673dULL },
+ { 0xc9e717bb45005915ULL, 0x00000166bb7f0435ULL },
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+};
+
+}
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-17 06:50:21 +0000