authorgravatar for andrew@ziglang.orgAndrew Kelley <andrew@ziglang.org> 2022-05-03 14:56:05-04:00
committergravatar for noreply@github.comGitHub <noreply@github.com> 2022-05-03 14:56:05-04:00
log6317da8da400e40a947f6333d1afc289ec349102
tree30e450f3c2277605b85cf07074505e52cfd86ae2
parentd48789467d95ddf0582c855d01f32bef48726e29
parent2947a2faab2e287926b20d5d8984ad69bba5bd2c
signaturebadge-question-mark Signed by PGP key 4AEE18F83AFDEB23

Merge pull request #11566 from tiehuis/master

add new float-parser based on eisel-lemire algorithm

16 files changed, 2505 insertions(+), 746 deletions(-)

lib/std/fmt.zig+1-2
......@@ -1837,12 +1837,11 @@ test "parseUnsigned" {
18371837}
18381838
18391839pub const parseFloat = @import("fmt/parse_float.zig").parseFloat;
1840pub const parseHexFloat = @compileError("deprecated; use `parseFloat`");
18401841pub const ParseFloatError = @import("fmt/parse_float.zig").ParseFloatError;
1841pub const parseHexFloat = @import("fmt/parse_hex_float.zig").parseHexFloat;
18421842
18431843test {
18441844 _ = parseFloat;
1845 _ = parseHexFloat;
18461845}
18471846
18481847pub fn charToDigit(c: u8, radix: u8) (error{InvalidCharacter}!u8) {
lib/std/fmt/parse_float.zig+111-388
......@@ -1,386 +1,18 @@
1// Adapted from https://github.com/grzegorz-kraszewski/stringtofloat.
2
3// MIT License
4//
5// Copyright (c) 2016 Grzegorz Kraszewski
6//
7// Permission is hereby granted, free of charge, to any person obtaining a copy
8// of this software and associated documentation files (the "Software"), to deal
9// in the Software without restriction, including without limitation the rights
10// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
11// copies of the Software, and to permit persons to whom the Software is
12// furnished to do so, subject to the following conditions:
13//
14// The above copyright notice and this permission notice shall be included in all
15// copies or substantial portions of the Software.
16//
17// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
18// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
19// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
20// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
21// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
22// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
23// SOFTWARE.
24//
25
26// Be aware that this implementation has the following limitations:
27//
28// - Is not round-trip accurate for all values
29// - Only supports round-to-zero
30// - Does not handle denormals
1pub const parseFloat = @import("parse_float/parse_float.zig").parseFloat;
2pub const ParseFloatError = @import("parse_float/parse_float.zig").ParseFloatError;
313
324const std = @import("std");
33const ascii = std.ascii;
34
35// The mantissa field in FloatRepr is 64bit wide and holds only 19 digits
36// without overflowing
37const max_digits = 19;
38
39const f64_plus_zero: u64 = 0x0000000000000000;
40const f64_minus_zero: u64 = 0x8000000000000000;
41const f64_plus_infinity: u64 = 0x7FF0000000000000;
42const f64_minus_infinity: u64 = 0xFFF0000000000000;
43
44const Z96 = struct {
45 d0: u32,
46 d1: u32,
47 d2: u32,
48
49 // d = s >> 1
50 inline fn shiftRight1(d: *Z96, s: Z96) void {
51 d.d0 = (s.d0 >> 1) | ((s.d1 & 1) << 31);
52 d.d1 = (s.d1 >> 1) | ((s.d2 & 1) << 31);
53 d.d2 = s.d2 >> 1;
54 }
55
56 // d = s << 1
57 inline fn shiftLeft1(d: *Z96, s: Z96) void {
58 d.d2 = (s.d2 << 1) | ((s.d1 & (1 << 31)) >> 31);
59 d.d1 = (s.d1 << 1) | ((s.d0 & (1 << 31)) >> 31);
60 d.d0 = s.d0 << 1;
61 }
62
63 // d += s
64 inline fn add(d: *Z96, s: Z96) void {
65 var w = @as(u64, d.d0) + @as(u64, s.d0);
66 d.d0 = @truncate(u32, w);
67
68 w >>= 32;
69 w += @as(u64, d.d1) + @as(u64, s.d1);
70 d.d1 = @truncate(u32, w);
71
72 w >>= 32;
73 w += @as(u64, d.d2) + @as(u64, s.d2);
74 d.d2 = @truncate(u32, w);
75 }
76
77 // d -= s
78 inline fn sub(d: *Z96, s: Z96) void {
79 var w = @as(u64, d.d0) -% @as(u64, s.d0);
80 d.d0 = @truncate(u32, w);
81
82 w >>= 32;
83 w += @as(u64, d.d1) -% @as(u64, s.d1);
84 d.d1 = @truncate(u32, w);
85
86 w >>= 32;
87 w += @as(u64, d.d2) -% @as(u64, s.d2);
88 d.d2 = @truncate(u32, w);
89 }
90};
91
92const FloatRepr = struct {
93 negative: bool,
94 exponent: i32,
95 mantissa: u64,
96};
97
98fn convertRepr(comptime T: type, n: FloatRepr) T {
99 const mask28: u32 = 0xf << 28;
100
101 var s: Z96 = undefined;
102 var q: Z96 = undefined;
103 var r: Z96 = undefined;
104
105 s.d0 = @truncate(u32, n.mantissa);
106 s.d1 = @truncate(u32, n.mantissa >> 32);
107 s.d2 = 0;
108
109 var binary_exponent: i32 = 92;
110 var exp = n.exponent;
111
112 while (exp > 0) : (exp -= 1) {
113 q.shiftLeft1(s); // q = p << 1
114 r.shiftLeft1(q); // r = p << 2
115 s.shiftLeft1(r); // p = p << 3
116 s.add(q); // p = (p << 3) + (p << 1)
117
118 while (s.d2 & mask28 != 0) {
119 q.shiftRight1(s);
120 binary_exponent += 1;
121 s = q;
122 }
123 }
124
125 while (exp < 0) {
126 while (s.d2 & (1 << 31) == 0) {
127 q.shiftLeft1(s);
128 binary_exponent -= 1;
129 s = q;
130 }
131
132 q.d2 = s.d2 / 10;
133 r.d1 = s.d2 % 10;
134 r.d2 = (s.d1 >> 8) | (r.d1 << 24);
135 q.d1 = r.d2 / 10;
136 r.d1 = r.d2 % 10;
137 r.d2 = ((s.d1 & 0xff) << 16) | (s.d0 >> 16) | (r.d1 << 24);
138 r.d0 = r.d2 / 10;
139 r.d1 = r.d2 % 10;
140 q.d1 = (q.d1 << 8) | ((r.d0 & 0x00ff0000) >> 16);
141 q.d0 = r.d0 << 16;
142 r.d2 = (s.d0 *% 0xffff) | (r.d1 << 16);
143 q.d0 |= r.d2 / 10;
144 s = q;
145
146 exp += 1;
147 }
148
149 if (s.d0 != 0 or s.d1 != 0 or s.d2 != 0) {
150 while (s.d2 & mask28 == 0) {
151 q.shiftLeft1(s);
152 binary_exponent -= 1;
153 s = q;
154 }
155 }
156
157 binary_exponent += 1023;
158
159 const repr: u64 = blk: {
160 if (binary_exponent > 2046) {
161 break :blk if (n.negative) f64_minus_infinity else f64_plus_infinity;
162 } else if (binary_exponent < 1) {
163 break :blk if (n.negative) f64_minus_zero else f64_plus_zero;
164 } else if (s.d2 != 0) {
165 const binexs2 = @intCast(u64, binary_exponent) << 52;
166 const rr = (@as(u64, s.d2 & ~mask28) << 24) | ((@as(u64, s.d1) + 128) >> 8) | binexs2;
167 break :blk if (n.negative) rr | (1 << 63) else rr;
168 } else {
169 break :blk 0;
170 }
171 };
172
173 const f = @bitCast(f64, repr);
174 return @floatCast(T, f);
175}
176
177const State = enum {
178 MaybeSign,
179 LeadingMantissaZeros,
180 LeadingFractionalZeros,
181 MantissaIntegral,
182 MantissaFractional,
183 ExponentSign,
184 LeadingExponentZeros,
185 Exponent,
186};
187
188const ParseResult = enum {
189 Ok,
190 PlusZero,
191 MinusZero,
192 PlusInf,
193 MinusInf,
194};
195
196fn parseRepr(s: []const u8, n: *FloatRepr) !ParseResult {
197 var digit_index: usize = 0;
198 var negative_exp = false;
199 var exponent: i32 = 0;
200
201 var state = State.MaybeSign;
5const math = std.math;
6const testing = std.testing;
7const expect = testing.expect;
8const expectEqual = testing.expectEqual;
9const expectError = testing.expectError;
10const approxEqAbs = std.math.approxEqAbs;
11const epsilon = 1e-7;
20212
203 var i: usize = 0;
204 while (i < s.len) {
205 const c = s[i];
206
207 switch (state) {
208 .MaybeSign => {
209 state = .LeadingMantissaZeros;
210
211 if (c == '+') {
212 i += 1;
213 } else if (c == '-') {
214 n.negative = true;
215 i += 1;
216 } else if (ascii.isDigit(c) or c == '.') {
217 // continue
218 } else {
219 return error.InvalidCharacter;
220 }
221 },
222 .LeadingMantissaZeros => {
223 if (c == '0') {
224 i += 1;
225 } else if (c == '.') {
226 i += 1;
227 state = .LeadingFractionalZeros;
228 } else if (c == '_') {
229 i += 1;
230 } else {
231 state = .MantissaIntegral;
232 }
233 },
234 .LeadingFractionalZeros => {
235 if (c == '0') {
236 i += 1;
237 if (n.exponent > std.math.minInt(i32)) {
238 n.exponent -= 1;
239 }
240 } else {
241 state = .MantissaFractional;
242 }
243 },
244 .MantissaIntegral => {
245 if (ascii.isDigit(c)) {
246 if (digit_index < max_digits) {
247 n.mantissa *%= 10;
248 n.mantissa += c - '0';
249 digit_index += 1;
250 } else if (n.exponent < std.math.maxInt(i32)) {
251 n.exponent += 1;
252 }
253
254 i += 1;
255 } else if (c == '.') {
256 i += 1;
257 state = .MantissaFractional;
258 } else if (c == '_') {
259 i += 1;
260 } else {
261 state = .MantissaFractional;
262 }
263 },
264 .MantissaFractional => {
265 if (ascii.isDigit(c)) {
266 if (digit_index < max_digits) {
267 n.mantissa *%= 10;
268 n.mantissa += c - '0';
269 n.exponent -%= 1;
270 digit_index += 1;
271 }
272
273 i += 1;
274 } else if (c == 'e' or c == 'E') {
275 i += 1;
276 state = .ExponentSign;
277 } else if (c == '_') {
278 i += 1;
279 } else {
280 state = .ExponentSign;
281 }
282 },
283 .ExponentSign => {
284 if (c == '+') {
285 i += 1;
286 } else if (c == '_') {
287 return error.InvalidCharacter;
288 } else if (c == '-') {
289 negative_exp = true;
290 i += 1;
291 }
292
293 state = .LeadingExponentZeros;
294 },
295 .LeadingExponentZeros => {
296 if (c == '0') {
297 i += 1;
298 } else if (c == '_') {
299 i += 1;
300 } else {
301 state = .Exponent;
302 }
303 },
304 .Exponent => {
305 if (ascii.isDigit(c)) {
306 if (exponent < std.math.maxInt(i32) / 10) {
307 exponent *= 10;
308 exponent += @intCast(i32, c - '0');
309 }
310
311 i += 1;
312 } else if (c == '_') {
313 i += 1;
314 } else {
315 return error.InvalidCharacter;
316 }
317 },
318 }
319 }
320
321 if (negative_exp) exponent = -exponent;
322 n.exponent += exponent;
323
324 if (n.mantissa == 0) {
325 return if (n.negative) .MinusZero else .PlusZero;
326 } else if (n.exponent > 309) {
327 return if (n.negative) .MinusInf else .PlusInf;
328 } else if (n.exponent < -328) {
329 return if (n.negative) .MinusZero else .PlusZero;
330 }
331
332 return .Ok;
333}
334
335fn caseInEql(a: []const u8, b: []const u8) bool {
336 if (a.len != b.len) return false;
337
338 for (a) |_, i| {
339 if (ascii.toUpper(a[i]) != ascii.toUpper(b[i])) {
340 return false;
341 }
342 }
343
344 return true;
345}
346
347pub const ParseFloatError = error{InvalidCharacter};
348
349pub fn parseFloat(comptime T: type, s: []const u8) ParseFloatError!T {
350 if (s.len == 0 or (s.len == 1 and (s[0] == '+' or s[0] == '-'))) {
351 return error.InvalidCharacter;
352 }
353
354 if (caseInEql(s, "nan")) {
355 return std.math.nan(T);
356 } else if (caseInEql(s, "inf") or caseInEql(s, "+inf")) {
357 return std.math.inf(T);
358 } else if (caseInEql(s, "-inf")) {
359 return -std.math.inf(T);
360 }
361
362 var r = FloatRepr{
363 .negative = false,
364 .exponent = 0,
365 .mantissa = 0,
366 };
367
368 return switch (try parseRepr(s, &r)) {
369 .Ok => convertRepr(T, r),
370 .PlusZero => 0.0,
371 .MinusZero => -@as(T, 0.0),
372 .PlusInf => std.math.inf(T),
373 .MinusInf => -std.math.inf(T),
374 };
375}
13// See https://github.com/tiehuis/parse-number-fxx-test-data for a wider-selection of test-data.
37614
37715test "fmt.parseFloat" {
378 const testing = std.testing;
379 const expect = testing.expect;
380 const expectEqual = testing.expectEqual;
381 const approxEqAbs = std.math.approxEqAbs;
382 const epsilon = 1e-7;
383
38416 inline for ([_]type{ f16, f32, f64, f128 }) |T| {
38517 const Z = std.meta.Int(.unsigned, @typeInfo(T).Float.bits);
38618
......@@ -405,8 +37,8 @@ test "fmt.parseFloat" {
40537 try expect(approxEqAbs(T, try parseFloat(T, "3.141"), 3.141, epsilon));
40638 try expect(approxEqAbs(T, try parseFloat(T, "-3.141"), -3.141, epsilon));
40739
408 try expectEqual(try parseFloat(T, "1e-700"), 0);
409 try expectEqual(try parseFloat(T, "1e+700"), std.math.inf(T));
40 try expectEqual(try parseFloat(T, "1e-5000"), 0);
41 try expectEqual(try parseFloat(T, "1e+5000"), std.math.inf(T));
41042
41143 try expectEqual(@bitCast(Z, try parseFloat(T, "nAn")), @bitCast(Z, std.math.nan(T)));
41244 try expectEqual(try parseFloat(T, "inF"), std.math.inf(T));
......@@ -415,14 +47,105 @@ test "fmt.parseFloat" {
41547 try expectEqual(try parseFloat(T, "0.4e0066999999999999999999999999999999999999999999999999999"), std.math.inf(T));
41648 try expect(approxEqAbs(T, try parseFloat(T, "0_1_2_3_4_5_6.7_8_9_0_0_0e0_0_1_0"), @as(T, 123456.789000e10), epsilon));
41749
418 if (T != f16) {
419 try expect(approxEqAbs(T, try parseFloat(T, "1e-2"), 0.01, epsilon));
420 try expect(approxEqAbs(T, try parseFloat(T, "1234e-2"), 12.34, epsilon));
50 // underscore rule is simple and reduces to "can only occur between two digits" and multiple are not supported.
51 try expectError(error.InvalidCharacter, parseFloat(T, "0123456.789000e_0010")); // cannot occur immediately after exponent
52 try expectError(error.InvalidCharacter, parseFloat(T, "_0123456.789000e0010")); // cannot occur before any digits
53 try expectError(error.InvalidCharacter, parseFloat(T, "0__123456.789000e_0010")); // cannot occur twice in a row
54 try expectError(error.InvalidCharacter, parseFloat(T, "0123456_.789000e0010")); // cannot occur before decimal point
55 try expectError(error.InvalidCharacter, parseFloat(T, "0123456.789000e0010_")); // cannot occur at end of number
56
57 try expect(approxEqAbs(T, try parseFloat(T, "1e-2"), 0.01, epsilon));
58 try expect(approxEqAbs(T, try parseFloat(T, "1234e-2"), 12.34, epsilon));
42159
422 try expect(approxEqAbs(T, try parseFloat(T, "123142.1"), 123142.1, epsilon));
423 try expect(approxEqAbs(T, try parseFloat(T, "-123142.1124"), @as(T, -123142.1124), epsilon));
424 try expect(approxEqAbs(T, try parseFloat(T, "0.7062146892655368"), @as(T, 0.7062146892655368), epsilon));
425 try expect(approxEqAbs(T, try parseFloat(T, "2.71828182845904523536"), @as(T, 2.718281828459045), epsilon));
426 }
60 try expect(approxEqAbs(T, try parseFloat(T, "123142.1"), 123142.1, epsilon));
61 try expect(approxEqAbs(T, try parseFloat(T, "-123142.1124"), @as(T, -123142.1124), epsilon));
62 try expect(approxEqAbs(T, try parseFloat(T, "0.7062146892655368"), @as(T, 0.7062146892655368), epsilon));
63 try expect(approxEqAbs(T, try parseFloat(T, "2.71828182845904523536"), @as(T, 2.718281828459045), epsilon));
42764 }
42865}
66
67test "fmt.parseFloat #11169" {
68 try expectEqual(try parseFloat(f128, "9007199254740993.0"), 9007199254740993.0);
69}
70
71test "fmt.parseFloat hex.special" {
72 try testing.expect(math.isNan(try parseFloat(f32, "nAn")));
73 try testing.expect(math.isPositiveInf(try parseFloat(f32, "iNf")));
74 try testing.expect(math.isPositiveInf(try parseFloat(f32, "+Inf")));
75 try testing.expect(math.isNegativeInf(try parseFloat(f32, "-iNf")));
76}
77test "fmt.parseFloat hex.zero" {
78 try testing.expectEqual(@as(f32, 0.0), try parseFloat(f32, "0x0"));
79 try testing.expectEqual(@as(f32, 0.0), try parseFloat(f32, "-0x0"));
80 try testing.expectEqual(@as(f32, 0.0), try parseFloat(f32, "0x0p42"));
81 try testing.expectEqual(@as(f32, 0.0), try parseFloat(f32, "-0x0.00000p42"));
82 try testing.expectEqual(@as(f32, 0.0), try parseFloat(f32, "0x0.00000p666"));
83}
84
85test "fmt.parseFloat hex.f16" {
86 try testing.expectEqual(try parseFloat(f16, "0x1p0"), 1.0);
87 try testing.expectEqual(try parseFloat(f16, "-0x1p-1"), -0.5);
88 try testing.expectEqual(try parseFloat(f16, "0x10p+10"), 16384.0);
89 try testing.expectEqual(try parseFloat(f16, "0x10p-10"), 0.015625);
90 // Max normalized value.
91 try testing.expectEqual(try parseFloat(f16, "0x1.ffcp+15"), math.floatMax(f16));
92 try testing.expectEqual(try parseFloat(f16, "-0x1.ffcp+15"), -math.floatMax(f16));
93 // Min normalized value.
94 try testing.expectEqual(try parseFloat(f16, "0x1p-14"), math.floatMin(f16));
95 try testing.expectEqual(try parseFloat(f16, "-0x1p-14"), -math.floatMin(f16));
96 // Min denormal value.
97 try testing.expectEqual(try parseFloat(f16, "0x1p-24"), math.floatTrueMin(f16));
98 try testing.expectEqual(try parseFloat(f16, "-0x1p-24"), -math.floatTrueMin(f16));
99}
100
101test "fmt.parseFloat hex.f32" {
102 try testing.expectEqual(try parseFloat(f32, "0x1p0"), 1.0);
103 try testing.expectEqual(try parseFloat(f32, "-0x1p-1"), -0.5);
104 try testing.expectEqual(try parseFloat(f32, "0x10p+10"), 16384.0);
105 try testing.expectEqual(try parseFloat(f32, "0x10p-10"), 0.015625);
106 try testing.expectEqual(try parseFloat(f32, "0x0.ffffffp128"), 0x0.ffffffp128);
107 try testing.expectEqual(try parseFloat(f32, "0x0.1234570p-125"), 0x0.1234570p-125);
108 // Max normalized value.
109 try testing.expectEqual(try parseFloat(f32, "0x1.fffffeP+127"), math.floatMax(f32));
110 try testing.expectEqual(try parseFloat(f32, "-0x1.fffffeP+127"), -math.floatMax(f32));
111 // Min normalized value.
112 try testing.expectEqual(try parseFloat(f32, "0x1p-126"), math.floatMin(f32));
113 try testing.expectEqual(try parseFloat(f32, "-0x1p-126"), -math.floatMin(f32));
114 // Min denormal value.
115 try testing.expectEqual(try parseFloat(f32, "0x1P-149"), math.floatTrueMin(f32));
116 try testing.expectEqual(try parseFloat(f32, "-0x1P-149"), -math.floatTrueMin(f32));
117}
118
119test "fmt.parseFloat hex.f64" {
120 try testing.expectEqual(try parseFloat(f64, "0x1p0"), 1.0);
121 try testing.expectEqual(try parseFloat(f64, "-0x1p-1"), -0.5);
122 try testing.expectEqual(try parseFloat(f64, "0x10p+10"), 16384.0);
123 try testing.expectEqual(try parseFloat(f64, "0x10p-10"), 0.015625);
124 // Max normalized value.
125 try testing.expectEqual(try parseFloat(f64, "0x1.fffffffffffffp+1023"), math.floatMax(f64));
126 try testing.expectEqual(try parseFloat(f64, "-0x1.fffffffffffffp1023"), -math.floatMax(f64));
127 // Min normalized value.
128 try testing.expectEqual(try parseFloat(f64, "0x1p-1022"), math.floatMin(f64));
129 try testing.expectEqual(try parseFloat(f64, "-0x1p-1022"), -math.floatMin(f64));
130 // Min denormalized value.
131 //try testing.expectEqual(try parseFloat(f64, "0x1p-1074"), math.floatTrueMin(f64));
132 try testing.expectEqual(try parseFloat(f64, "-0x1p-1074"), -math.floatTrueMin(f64));
133}
134test "fmt.parseFloat hex.f128" {
135 try testing.expectEqual(try parseFloat(f128, "0x1p0"), 1.0);
136 try testing.expectEqual(try parseFloat(f128, "-0x1p-1"), -0.5);
137 try testing.expectEqual(try parseFloat(f128, "0x10p+10"), 16384.0);
138 try testing.expectEqual(try parseFloat(f128, "0x10p-10"), 0.015625);
139 // Max normalized value.
140 try testing.expectEqual(try parseFloat(f128, "0xf.fffffffffffffffffffffffffff8p+16380"), math.floatMax(f128));
141 try testing.expectEqual(try parseFloat(f128, "-0xf.fffffffffffffffffffffffffff8p+16380"), -math.floatMax(f128));
142 // Min normalized value.
143 try testing.expectEqual(try parseFloat(f128, "0x1p-16382"), math.floatMin(f128));
144 try testing.expectEqual(try parseFloat(f128, "-0x1p-16382"), -math.floatMin(f128));
145 // // Min denormalized value.
146 try testing.expectEqual(try parseFloat(f128, "0x1p-16494"), math.floatTrueMin(f128));
147 try testing.expectEqual(try parseFloat(f128, "-0x1p-16494"), -math.floatTrueMin(f128));
148
149 // NOTE: We are performing round-to-even. Previous behavior was round-up.
150 // try testing.expectEqual(try parseFloat(f128, "0x1.edcb34a235253948765432134674fp-1"), 0x1.edcb34a235253948765432134674fp-1);
151}
lib/std/fmt/parse_float/FloatInfo.zig created+131
......@@ -0,0 +1,131 @@
1const std = @import("std");
2const Self = @This();
3
4// Minimum exponent that for a fast path case, or `-⌊(MANTISSA_EXPLICIT_BITS+1)/log2(5)⌋`
5min_exponent_fast_path: comptime_int,
6
7// Maximum exponent that for a fast path case, or `⌊(MANTISSA_EXPLICIT_BITS+1)/log2(5)⌋`
8max_exponent_fast_path: comptime_int,
9
10// Maximum exponent that can be represented for a disguised-fast path case.
11// This is `MAX_EXPONENT_FAST_PATH + ⌊(MANTISSA_EXPLICIT_BITS+1)/log2(10)⌋`
12max_exponent_fast_path_disguised: comptime_int,
13
14// Maximum mantissa for the fast-path (`1 << 53` for f64).
15max_mantissa_fast_path: comptime_int,
16
17// Smallest decimal exponent for a non-zero value. Including subnormals.
18smallest_power_of_ten: comptime_int,
19
20// Largest decimal exponent for a non-infinite value.
21largest_power_of_ten: comptime_int,
22
23// The number of bits in the significand, *excluding* the hidden bit.
24mantissa_explicit_bits: comptime_int,
25
26// Minimum exponent value `-(1 << (EXP_BITS - 1)) + 1`.
27minimum_exponent: comptime_int,
28
29// Round-to-even only happens for negative values of q
30// when q ≥ −4 in the 64-bit case and when q ≥ −17 in
31// the 32-bitcase.
32//
33// When q ≥ 0,we have that 5^q ≤ 2m+1. In the 64-bit case,we
34// have 5^q ≤ 2m+1 ≤ 2^54 or q ≤ 23. In the 32-bit case,we have
35// 5^q ≤ 2m+1 ≤ 2^25 or q ≤ 10.
36//
37// When q < 0, we have w ≥ (2m+1)×5^−q. We must have that w < 2^64
38// so (2m+1)×5^−q < 2^64. We have that 2m+1 > 2^53 (64-bit case)
39// or 2m+1 > 2^24 (32-bit case). Hence,we must have 2^53×5^−q < 2^64
40// (64-bit) and 2^24×5^−q < 2^64 (32-bit). Hence we have 5^−q < 2^11
41// or q ≥ −4 (64-bit case) and 5^−q < 2^40 or q ≥ −17 (32-bitcase).
42//
43// Thus we have that we only need to round ties to even when
44// we have that q ∈ [−4,23](in the 64-bit case) or q∈[−17,10]
45// (in the 32-bit case). In both cases,the power of five(5^|q|)
46// fits in a 64-bit word.
47min_exponent_round_to_even: comptime_int,
48max_exponent_round_to_even: comptime_int,
49
50// Largest exponent value `(1 << EXP_BITS) - 1`.
51infinite_power: comptime_int,
52
53// Following should compute based on derived calculations where possible.
54pub fn from(comptime T: type) Self {
55 return switch (T) {
56 f16 => .{
57 // Fast-Path
58 .min_exponent_fast_path = -4,
59 .max_exponent_fast_path = 4,
60 .max_exponent_fast_path_disguised = 7,
61 .max_mantissa_fast_path = 2 << std.math.floatMantissaBits(T),
62 // Slow + Eisel-Lemire
63 .mantissa_explicit_bits = std.math.floatMantissaBits(T),
64 .infinite_power = 0x1f,
65 // Eisel-Lemire
66 .smallest_power_of_ten = -26, // TODO: refine, fails one test
67 .largest_power_of_ten = 4,
68 .minimum_exponent = -15,
69 // w >= (2m+1) * 5^-q and w < 2^64
70 // => 2m+1 > 2^11
71 // => 2^11*5^-q < 2^64
72 // => 5^-q < 2^53
73 // => q >= -23
74 .min_exponent_round_to_even = -22,
75 .max_exponent_round_to_even = 5,
76 },
77 f32 => .{
78 // Fast-Path
79 .min_exponent_fast_path = -10,
80 .max_exponent_fast_path = 10,
81 .max_exponent_fast_path_disguised = 17,
82 .max_mantissa_fast_path = 2 << std.math.floatMantissaBits(T),
83 // Slow + Eisel-Lemire
84 .mantissa_explicit_bits = std.math.floatMantissaBits(T),
85 .infinite_power = 0xff,
86 // Eisel-Lemire
87 .smallest_power_of_ten = -65,
88 .largest_power_of_ten = 38,
89 .minimum_exponent = -127,
90 .min_exponent_round_to_even = -17,
91 .max_exponent_round_to_even = 10,
92 },
93 f64 => .{
94 // Fast-Path
95 .min_exponent_fast_path = -22,
96 .max_exponent_fast_path = 22,
97 .max_exponent_fast_path_disguised = 37,
98 .max_mantissa_fast_path = 2 << std.math.floatMantissaBits(T),
99 // Slow + Eisel-Lemire
100 .mantissa_explicit_bits = std.math.floatMantissaBits(T),
101 .infinite_power = 0x7ff,
102 // Eisel-Lemire
103 .smallest_power_of_ten = -342,
104 .largest_power_of_ten = 308,
105 .minimum_exponent = -1023,
106 .min_exponent_round_to_even = -4,
107 .max_exponent_round_to_even = 23,
108 },
109 f128 => .{
110 // Fast-Path
111 .min_exponent_fast_path = -48,
112 .max_exponent_fast_path = 48,
113 .max_exponent_fast_path_disguised = 82,
114 .max_mantissa_fast_path = 2 << std.math.floatMantissaBits(T),
115 // Slow + Eisel-Lemire
116 .mantissa_explicit_bits = std.math.floatMantissaBits(T),
117 .infinite_power = 0x7fff,
118 // Eisel-Lemire.
119 // NOTE: Not yet tested (no f128 eisel-lemire implementation)
120 .smallest_power_of_ten = -4966,
121 .largest_power_of_ten = 4932,
122 .minimum_exponent = -16382,
123 // 2^113 * 5^-q < 2^128
124 // 5^-q < 2^15
125 // => q >= -6
126 .min_exponent_round_to_even = -6,
127 .max_exponent_round_to_even = 49,
128 },
129 else => unreachable,
130 };
131}
lib/std/fmt/parse_float/FloatStream.zig created+137
......@@ -0,0 +1,137 @@
1//! A wrapper over a byte-slice, providing useful methods for parsing string floating point values.
2
3const std = @import("std");
4const FloatStream = @This();
5const common = @import("common.zig");
6
7slice: []const u8,
8offset: usize,
9underscore_count: usize,
10
11pub fn init(s: []const u8) FloatStream {
12 return .{ .slice = s, .offset = 0, .underscore_count = 0 };
13}
14
15// Returns the offset from the start *excluding* any underscores that were found.
16pub fn offsetTrue(self: FloatStream) usize {
17 return self.offset - self.underscore_count;
18}
19
20pub fn reset(self: *FloatStream) void {
21 self.offset = 0;
22 self.underscore_count = 0;
23}
24
25pub fn len(self: FloatStream) usize {
26 if (self.offset > self.slice.len) {
27 return 0;
28 }
29 return self.slice.len - self.offset;
30}
31
32pub fn hasLen(self: FloatStream, n: usize) bool {
33 return self.offset + n <= self.slice.len;
34}
35
36pub fn firstUnchecked(self: FloatStream) u8 {
37 return self.slice[self.offset];
38}
39
40pub fn first(self: FloatStream) ?u8 {
41 return if (self.hasLen(1))
42 return self.firstUnchecked()
43 else
44 null;
45}
46
47pub fn isEmpty(self: FloatStream) bool {
48 return !self.hasLen(1);
49}
50
51pub fn firstIs(self: FloatStream, c: u8) bool {
52 if (self.first()) |ok| {
53 return ok == c;
54 }
55 return false;
56}
57
58pub fn firstIsLower(self: FloatStream, c: u8) bool {
59 if (self.first()) |ok| {
60 return ok | 0x20 == c;
61 }
62 return false;
63}
64
65pub fn firstIs2(self: FloatStream, c1: u8, c2: u8) bool {
66 if (self.first()) |ok| {
67 return ok == c1 or ok == c2;
68 }
69 return false;
70}
71
72pub fn firstIs3(self: FloatStream, c1: u8, c2: u8, c3: u8) bool {
73 if (self.first()) |ok| {
74 return ok == c1 or ok == c2 or ok == c3;
75 }
76 return false;
77}
78
79pub fn firstIsDigit(self: FloatStream, comptime base: u8) bool {
80 comptime std.debug.assert(base == 10 or base == 16);
81
82 if (self.first()) |ok| {
83 return common.isDigit(ok, base);
84 }
85 return false;
86}
87
88pub fn advance(self: *FloatStream, n: usize) void {
89 self.offset += n;
90}
91
92pub fn skipChars(self: *FloatStream, c: u8) void {
93 while (self.firstIs(c)) : (self.advance(1)) {}
94}
95
96pub fn skipChars2(self: *FloatStream, c1: u8, c2: u8) void {
97 while (self.firstIs2(c1, c2)) : (self.advance(1)) {}
98}
99
100pub fn readU64Unchecked(self: FloatStream) u64 {
101 return std.mem.readIntSliceLittle(u64, self.slice[self.offset..]);
102}
103
104pub fn readU64(self: FloatStream) ?u64 {
105 if (self.hasLen(8)) {
106 return self.readU64Unchecked();
107 }
108 return null;
109}
110
111pub fn atUnchecked(self: *FloatStream, i: usize) u8 {
112 return self.slice[self.offset + i];
113}
114
115pub fn scanDigit(self: *FloatStream, comptime base: u8) ?u8 {
116 comptime std.debug.assert(base == 10 or base == 16);
117
118 retry: while (true) {
119 if (self.first()) |ok| {
120 if ('0' <= ok and ok <= '9') {
121 self.advance(1);
122 return ok - '0';
123 } else if (base == 16 and 'a' <= ok and ok <= 'f') {
124 self.advance(1);
125 return ok - 'a' + 10;
126 } else if (base == 16 and 'A' <= ok and ok <= 'F') {
127 self.advance(1);
128 return ok - 'A' + 10;
129 } else if (ok == '_') {
130 self.advance(1);
131 self.underscore_count += 1;
132 continue :retry;
133 }
134 }
135 return null;
136 }
137}
lib/std/fmt/parse_float/common.zig created+91
......@@ -0,0 +1,91 @@
1const std = @import("std");
2
3/// A custom N-bit floating point type, representing `f * 2^e`.
4/// e is biased, so it be directly shifted into the exponent bits.
5/// Negative exponent indicates an invalid result.
6pub fn BiasedFp(comptime T: type) type {
7 const MantissaT = mantissaType(T);
8
9 return struct {
10 const Self = @This();
11
12 /// The significant digits.
13 f: MantissaT,
14 /// The biased, binary exponent.
15 e: i32,
16
17 pub fn zero() Self {
18 return .{ .f = 0, .e = 0 };
19 }
20
21 pub fn zeroPow2(e: i32) Self {
22 return .{ .f = 0, .e = e };
23 }
24
25 pub fn inf(comptime FloatT: type) Self {
26 return .{ .f = 0, .e = (1 << std.math.floatExponentBits(FloatT)) - 1 };
27 }
28
29 pub fn eql(self: Self, other: Self) bool {
30 return self.f == other.f and self.e == other.e;
31 }
32
33 pub fn toFloat(self: Self, comptime FloatT: type, negative: bool) FloatT {
34 var word = self.f;
35 word |= @intCast(MantissaT, self.e) << std.math.floatMantissaBits(FloatT);
36 var f = floatFromUnsigned(FloatT, MantissaT, word);
37 if (negative) f = -f;
38 return f;
39 }
40 };
41}
42
43pub fn floatFromUnsigned(comptime T: type, comptime MantissaT: type, v: MantissaT) T {
44 return switch (T) {
45 f16 => @bitCast(f16, @truncate(u16, v)),
46 f32 => @bitCast(f32, @truncate(u32, v)),
47 f64 => @bitCast(f64, @truncate(u64, v)),
48 f128 => @bitCast(f128, v),
49 else => unreachable,
50 };
51}
52
53/// Represents a parsed floating point value as its components.
54pub fn Number(comptime T: type) type {
55 return struct {
56 exponent: i64,
57 mantissa: mantissaType(T),
58 negative: bool,
59 /// More than max_mantissa digits were found during parse
60 many_digits: bool,
61 /// The number was a hex-float (e.g. 0x1.234p567)
62 hex: bool,
63 };
64}
65
66/// Determine if 8 bytes are all decimal digits.
67/// This does not care about the order in which the bytes were loaded.
68pub fn isEightDigits(v: u64) bool {
69 const a = v +% 0x4646_4646_4646_4646;
70 const b = v -% 0x3030_3030_3030_3030;
71 return ((a | b) & 0x8080_8080_8080_8080) == 0;
72}
73
74pub fn isDigit(c: u8, comptime base: u8) bool {
75 std.debug.assert(base == 10 or base == 16);
76
77 return if (base == 10)
78 '0' <= c and c <= '9'
79 else
80 '0' <= c and c <= '9' or 'a' <= c and c <= 'f' or 'A' <= c and c <= 'F';
81}
82
83/// Returns the underlying storage type used for the mantissa of floating-point type.
84/// The output unsigned type must have at least as many bits as the input floating-point type.
85pub fn mantissaType(comptime T: type) type {
86 return switch (T) {
87 f16, f32, f64 => u64,
88 f128 => u128,
89 else => unreachable,
90 };
91}
lib/std/fmt/parse_float/convert_eisel_lemire.zig created+843
......@@ -0,0 +1,843 @@
1const std = @import("std");
2const math = std.math;
3const common = @import("common.zig");
4const FloatInfo = @import("FloatInfo.zig");
5const BiasedFp = common.BiasedFp;
6const Number = common.Number;
7
8/// Compute a float using an extended-precision representation.
9///
10/// Fast conversion of a the significant digits and decimal exponent
11/// a float to an extended representation with a binary float. This
12/// algorithm will accurately parse the vast majority of cases,
13/// and uses a 128-bit representation (with a fallback 192-bit
14/// representation).
15///
16/// This algorithm scales the exponent by the decimal exponent
17/// using pre-computed powers-of-5, and calculates if the
18/// representation can be unambiguously rounded to the nearest
19/// machine float. Near-halfway cases are not handled here,
20/// and are represented by a negative, biased binary exponent.
21///
22/// The algorithm is described in detail in "Daniel Lemire, Number Parsing
23/// at a Gigabyte per Second" in section 5, "Fast Algorithm", and
24/// section 6, "Exact Numbers And Ties", available online:
25/// <https://arxiv.org/abs/2101.11408.pdf>.
26pub fn convertEiselLemire(comptime T: type, q: i64, w_: u64) ?BiasedFp(f64) {
27 std.debug.assert(T == f16 or T == f32 or T == f64);
28 var w = w_;
29 const float_info = FloatInfo.from(T);
30
31 // Short-circuit if the value can only be a literal 0 or infinity.
32 if (w == 0 or q < float_info.smallest_power_of_ten) {
33 return BiasedFp(f64).zero();
34 } else if (q > float_info.largest_power_of_ten) {
35 return BiasedFp(f64).inf(T);
36 }
37
38 // Normalize our significant digits, so the most-significant bit is set.
39 const lz = @clz(u64, @bitCast(u64, w));
40 w = math.shl(u64, w, lz);
41
42 const r = computeProductApprox(q, w, float_info.mantissa_explicit_bits + 3);
43 if (r.lo == 0xffff_ffff_ffff_ffff) {
44 // If we have failed to approximate w x 5^-q with our 128-bit value.
45 // Since the addition of 1 could lead to an overflow which could then
46 // round up over the half-way point, this can lead to improper rounding
47 // of a float.
48 //
49 // However, this can only occur if q ∈ [-27, 55]. The upper bound of q
50 // is 55 because 5^55 < 2^128, however, this can only happen if 5^q > 2^64,
51 // since otherwise the product can be represented in 64-bits, producing
52 // an exact result. For negative exponents, rounding-to-even can
53 // only occur if 5^-q < 2^64.
54 //
55 // For detailed explanations of rounding for negative exponents, see
56 // <https://arxiv.org/pdf/2101.11408.pdf#section.9.1>. For detailed
57 // explanations of rounding for positive exponents, see
58 // <https://arxiv.org/pdf/2101.11408.pdf#section.8>.
59 const inside_safe_exponent = q >= -27 and q <= 55;
60 if (!inside_safe_exponent) {
61 return null;
62 }
63 }
64
65 const upper_bit = @intCast(i32, r.hi >> 63);
66 var mantissa = math.shr(u64, r.hi, upper_bit + 64 - @intCast(i32, float_info.mantissa_explicit_bits) - 3);
67 var power2 = power(@intCast(i32, q)) + upper_bit - @intCast(i32, lz) - float_info.minimum_exponent;
68 if (power2 <= 0) {
69 if (-power2 + 1 >= 64) {
70 // Have more than 64 bits below the minimum exponent, must be 0.
71 return BiasedFp(f64).zero();
72 }
73 // Have a subnormal value.
74 mantissa = math.shr(u64, mantissa, -power2 + 1);
75 mantissa += mantissa & 1;
76 mantissa >>= 1;
77 power2 = @boolToInt(mantissa >= (1 << float_info.mantissa_explicit_bits));
78 return BiasedFp(f64){ .f = mantissa, .e = power2 };
79 }
80
81 // Need to handle rounding ties. Normally, we need to round up,
82 // but if we fall right in between and and we have an even basis, we
83 // need to round down.
84 //
85 // This will only occur if:
86 // 1. The lower 64 bits of the 128-bit representation is 0.
87 // IE, 5^q fits in single 64-bit word.
88 // 2. The least-significant bit prior to truncated mantissa is odd.
89 // 3. All the bits truncated when shifting to mantissa bits + 1 are 0.
90 //
91 // Or, we may fall between two floats: we are exactly halfway.
92 if (r.lo <= 1 and
93 q >= float_info.min_exponent_round_to_even and
94 q <= float_info.max_exponent_round_to_even and
95 mantissa & 3 == 1 and
96 math.shl(u64, mantissa, (upper_bit + 64 - @intCast(i32, float_info.mantissa_explicit_bits) - 3)) == r.hi)
97 {
98 // Zero the lowest bit, so we don't round up.
99 mantissa &= ~@as(u64, 1);
100 }
101
102 // Round-to-even, then shift the significant digits into place.
103 mantissa += mantissa & 1;
104 mantissa >>= 1;
105 if (mantissa >= 2 << float_info.mantissa_explicit_bits) {
106 // Rounding up overflowed, so the carry bit is set. Set the
107 // mantissa to 1 (only the implicit, hidden bit is set) and
108 // increase the exponent.
109 mantissa = 1 << float_info.mantissa_explicit_bits;
110 power2 += 1;
111 }
112
113 // Zero out the hidden bit
114 mantissa &= ~(@as(u64, 1) << float_info.mantissa_explicit_bits);
115 if (power2 >= float_info.infinite_power) {
116 // Exponent is above largest normal value, must be infinite
117 return BiasedFp(f64).inf(T);
118 }
119
120 return BiasedFp(f64){ .f = mantissa, .e = power2 };
121}
122
123/// Calculate a base 2 exponent from a decimal exponent.
124/// This uses a pre-computed integer approximation for
125/// log2(10), where 217706 / 2^16 is accurate for the
126/// entire range of non-finite decimal exponents.
127fn power(q: i32) i32 {
128 return ((q *% (152170 + 65536)) >> 16) + 63;
129}
130
131const U128 = struct {
132 lo: u64,
133 hi: u64,
134
135 pub fn new(lo: u64, hi: u64) U128 {
136 return .{ .lo = lo, .hi = hi };
137 }
138
139 pub fn mul(a: u64, b: u64) U128 {
140 const x = @as(u128, a) * b;
141 return .{
142 .hi = @truncate(u64, x >> 64),
143 .lo = @truncate(u64, x),
144 };
145 }
146};
147
148// This will compute or rather approximate w * 5**q and return a pair of 64-bit words
149// approximating the result, with the "high" part corresponding to the most significant
150// bits and the low part corresponding to the least significant bits.
151fn computeProductApprox(q: i64, w: u64, comptime precision: usize) U128 {
152 std.debug.assert(q >= eisel_lemire_smallest_power_of_five);
153 std.debug.assert(q <= eisel_lemire_largest_power_of_five);
154 std.debug.assert(precision <= 64);
155
156 const mask = if (precision < 64)
157 0xffff_ffff_ffff_ffff >> precision
158 else
159 0xffff_ffff_ffff_ffff;
160
161 // 5^q < 2^64, then the multiplication always provides an exact value.
162 // That means whenever we need to round ties to even, we always have
163 // an exact value.
164 const index = @intCast(usize, q - @intCast(i64, eisel_lemire_smallest_power_of_five));
165 const pow5 = eisel_lemire_table_powers_of_five_128[index];
166
167 // Only need one multiplication as long as there is 1 zero but
168 // in the explicit mantissa bits, +1 for the hidden bit, +1 to
169 // determine the rounding direction, +1 for if the computed
170 // product has a leading zero.
171 var first = U128.mul(w, pow5.lo);
172 if (first.hi & mask == mask) {
173 // Need to do a second multiplication to get better precision
174 // for the lower product. This will always be exact
175 // where q is < 55, since 5^55 < 2^128. If this wraps,
176 // then we need to need to round up the hi product.
177 const second = U128.mul(w, pow5.hi);
178
179 first.lo +%= second.hi;
180 if (second.hi > first.lo) {
181 first.hi += 1;
182 }
183 }
184
185 return .{ .lo = first.lo, .hi = first.hi };
186}
187
188// Eisel-Lemire tables ~10Kb
189const eisel_lemire_smallest_power_of_five = -342;
190const eisel_lemire_largest_power_of_five = 308;
191const eisel_lemire_table_powers_of_five_128 = [_]U128{
192 U128.new(0xeef453d6923bd65a, 0x113faa2906a13b3f), // 5^-342
193 U128.new(0x9558b4661b6565f8, 0x4ac7ca59a424c507), // 5^-341
194 U128.new(0xbaaee17fa23ebf76, 0x5d79bcf00d2df649), // 5^-340
195 U128.new(0xe95a99df8ace6f53, 0xf4d82c2c107973dc), // 5^-339
196 U128.new(0x91d8a02bb6c10594, 0x79071b9b8a4be869), // 5^-338
197 U128.new(0xb64ec836a47146f9, 0x9748e2826cdee284), // 5^-337
198 U128.new(0xe3e27a444d8d98b7, 0xfd1b1b2308169b25), // 5^-336
199 U128.new(0x8e6d8c6ab0787f72, 0xfe30f0f5e50e20f7), // 5^-335
200 U128.new(0xb208ef855c969f4f, 0xbdbd2d335e51a935), // 5^-334
201 U128.new(0xde8b2b66b3bc4723, 0xad2c788035e61382), // 5^-333
202 U128.new(0x8b16fb203055ac76, 0x4c3bcb5021afcc31), // 5^-332
203 U128.new(0xaddcb9e83c6b1793, 0xdf4abe242a1bbf3d), // 5^-331
204 U128.new(0xd953e8624b85dd78, 0xd71d6dad34a2af0d), // 5^-330
205 U128.new(0x87d4713d6f33aa6b, 0x8672648c40e5ad68), // 5^-329
206 U128.new(0xa9c98d8ccb009506, 0x680efdaf511f18c2), // 5^-328
207 U128.new(0xd43bf0effdc0ba48, 0x212bd1b2566def2), // 5^-327
208 U128.new(0x84a57695fe98746d, 0x14bb630f7604b57), // 5^-326
209 U128.new(0xa5ced43b7e3e9188, 0x419ea3bd35385e2d), // 5^-325
210 U128.new(0xcf42894a5dce35ea, 0x52064cac828675b9), // 5^-324
211 U128.new(0x818995ce7aa0e1b2, 0x7343efebd1940993), // 5^-323
212 U128.new(0xa1ebfb4219491a1f, 0x1014ebe6c5f90bf8), // 5^-322
213 U128.new(0xca66fa129f9b60a6, 0xd41a26e077774ef6), // 5^-321
214 U128.new(0xfd00b897478238d0, 0x8920b098955522b4), // 5^-320
215 U128.new(0x9e20735e8cb16382, 0x55b46e5f5d5535b0), // 5^-319
216 U128.new(0xc5a890362fddbc62, 0xeb2189f734aa831d), // 5^-318
217 U128.new(0xf712b443bbd52b7b, 0xa5e9ec7501d523e4), // 5^-317
218 U128.new(0x9a6bb0aa55653b2d, 0x47b233c92125366e), // 5^-316
219 U128.new(0xc1069cd4eabe89f8, 0x999ec0bb696e840a), // 5^-315
220 U128.new(0xf148440a256e2c76, 0xc00670ea43ca250d), // 5^-314
221 U128.new(0x96cd2a865764dbca, 0x380406926a5e5728), // 5^-313
222 U128.new(0xbc807527ed3e12bc, 0xc605083704f5ecf2), // 5^-312
223 U128.new(0xeba09271e88d976b, 0xf7864a44c633682e), // 5^-311
224 U128.new(0x93445b8731587ea3, 0x7ab3ee6afbe0211d), // 5^-310
225 U128.new(0xb8157268fdae9e4c, 0x5960ea05bad82964), // 5^-309
226 U128.new(0xe61acf033d1a45df, 0x6fb92487298e33bd), // 5^-308
227 U128.new(0x8fd0c16206306bab, 0xa5d3b6d479f8e056), // 5^-307
228 U128.new(0xb3c4f1ba87bc8696, 0x8f48a4899877186c), // 5^-306
229 U128.new(0xe0b62e2929aba83c, 0x331acdabfe94de87), // 5^-305
230 U128.new(0x8c71dcd9ba0b4925, 0x9ff0c08b7f1d0b14), // 5^-304
231 U128.new(0xaf8e5410288e1b6f, 0x7ecf0ae5ee44dd9), // 5^-303
232 U128.new(0xdb71e91432b1a24a, 0xc9e82cd9f69d6150), // 5^-302
233 U128.new(0x892731ac9faf056e, 0xbe311c083a225cd2), // 5^-301
234 U128.new(0xab70fe17c79ac6ca, 0x6dbd630a48aaf406), // 5^-300
235 U128.new(0xd64d3d9db981787d, 0x92cbbccdad5b108), // 5^-299
236 U128.new(0x85f0468293f0eb4e, 0x25bbf56008c58ea5), // 5^-298
237 U128.new(0xa76c582338ed2621, 0xaf2af2b80af6f24e), // 5^-297
238 U128.new(0xd1476e2c07286faa, 0x1af5af660db4aee1), // 5^-296
239 U128.new(0x82cca4db847945ca, 0x50d98d9fc890ed4d), // 5^-295
240 U128.new(0xa37fce126597973c, 0xe50ff107bab528a0), // 5^-294
241 U128.new(0xcc5fc196fefd7d0c, 0x1e53ed49a96272c8), // 5^-293
242 U128.new(0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7a), // 5^-292
243 U128.new(0x9faacf3df73609b1, 0x77b191618c54e9ac), // 5^-291
244 U128.new(0xc795830d75038c1d, 0xd59df5b9ef6a2417), // 5^-290
245 U128.new(0xf97ae3d0d2446f25, 0x4b0573286b44ad1d), // 5^-289
246 U128.new(0x9becce62836ac577, 0x4ee367f9430aec32), // 5^-288
247 U128.new(0xc2e801fb244576d5, 0x229c41f793cda73f), // 5^-287
248 U128.new(0xf3a20279ed56d48a, 0x6b43527578c1110f), // 5^-286
249 U128.new(0x9845418c345644d6, 0x830a13896b78aaa9), // 5^-285
250 U128.new(0xbe5691ef416bd60c, 0x23cc986bc656d553), // 5^-284
251 U128.new(0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa8), // 5^-283
252 U128.new(0x94b3a202eb1c3f39, 0x7bf7d71432f3d6a9), // 5^-282
253 U128.new(0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc53), // 5^-281
254 U128.new(0xe858ad248f5c22c9, 0xd1b3400f8f9cff68), // 5^-280
255 U128.new(0x91376c36d99995be, 0x23100809b9c21fa1), // 5^-279
256 U128.new(0xb58547448ffffb2d, 0xabd40a0c2832a78a), // 5^-278
257 U128.new(0xe2e69915b3fff9f9, 0x16c90c8f323f516c), // 5^-277
258 U128.new(0x8dd01fad907ffc3b, 0xae3da7d97f6792e3), // 5^-276
259 U128.new(0xb1442798f49ffb4a, 0x99cd11cfdf41779c), // 5^-275
260 U128.new(0xdd95317f31c7fa1d, 0x40405643d711d583), // 5^-274
261 U128.new(0x8a7d3eef7f1cfc52, 0x482835ea666b2572), // 5^-273
262 U128.new(0xad1c8eab5ee43b66, 0xda3243650005eecf), // 5^-272
263 U128.new(0xd863b256369d4a40, 0x90bed43e40076a82), // 5^-271
264 U128.new(0x873e4f75e2224e68, 0x5a7744a6e804a291), // 5^-270
265 U128.new(0xa90de3535aaae202, 0x711515d0a205cb36), // 5^-269
266 U128.new(0xd3515c2831559a83, 0xd5a5b44ca873e03), // 5^-268
267 U128.new(0x8412d9991ed58091, 0xe858790afe9486c2), // 5^-267
268 U128.new(0xa5178fff668ae0b6, 0x626e974dbe39a872), // 5^-266
269 U128.new(0xce5d73ff402d98e3, 0xfb0a3d212dc8128f), // 5^-265
270 U128.new(0x80fa687f881c7f8e, 0x7ce66634bc9d0b99), // 5^-264
271 U128.new(0xa139029f6a239f72, 0x1c1fffc1ebc44e80), // 5^-263
272 U128.new(0xc987434744ac874e, 0xa327ffb266b56220), // 5^-262
273 U128.new(0xfbe9141915d7a922, 0x4bf1ff9f0062baa8), // 5^-261
274 U128.new(0x9d71ac8fada6c9b5, 0x6f773fc3603db4a9), // 5^-260
275 U128.new(0xc4ce17b399107c22, 0xcb550fb4384d21d3), // 5^-259
276 U128.new(0xf6019da07f549b2b, 0x7e2a53a146606a48), // 5^-258
277 U128.new(0x99c102844f94e0fb, 0x2eda7444cbfc426d), // 5^-257
278 U128.new(0xc0314325637a1939, 0xfa911155fefb5308), // 5^-256
279 U128.new(0xf03d93eebc589f88, 0x793555ab7eba27ca), // 5^-255
280 U128.new(0x96267c7535b763b5, 0x4bc1558b2f3458de), // 5^-254
281 U128.new(0xbbb01b9283253ca2, 0x9eb1aaedfb016f16), // 5^-253
282 U128.new(0xea9c227723ee8bcb, 0x465e15a979c1cadc), // 5^-252
283 U128.new(0x92a1958a7675175f, 0xbfacd89ec191ec9), // 5^-251
284 U128.new(0xb749faed14125d36, 0xcef980ec671f667b), // 5^-250
285 U128.new(0xe51c79a85916f484, 0x82b7e12780e7401a), // 5^-249
286 U128.new(0x8f31cc0937ae58d2, 0xd1b2ecb8b0908810), // 5^-248
287 U128.new(0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa15), // 5^-247
288 U128.new(0xdfbdcece67006ac9, 0x67a791e093e1d49a), // 5^-246
289 U128.new(0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e0), // 5^-245
290 U128.new(0xaecc49914078536d, 0x58fae9f773886e18), // 5^-244
291 U128.new(0xda7f5bf590966848, 0xaf39a475506a899e), // 5^-243
292 U128.new(0x888f99797a5e012d, 0x6d8406c952429603), // 5^-242
293 U128.new(0xaab37fd7d8f58178, 0xc8e5087ba6d33b83), // 5^-241
294 U128.new(0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a64), // 5^-240
295 U128.new(0x855c3be0a17fcd26, 0x5cf2eea09a55067f), // 5^-239
296 U128.new(0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481e), // 5^-238
297 U128.new(0xd0601d8efc57b08b, 0xf13b94daf124da26), // 5^-237
298 U128.new(0x823c12795db6ce57, 0x76c53d08d6b70858), // 5^-236
299 U128.new(0xa2cb1717b52481ed, 0x54768c4b0c64ca6e), // 5^-235
300 U128.new(0xcb7ddcdda26da268, 0xa9942f5dcf7dfd09), // 5^-234
301 U128.new(0xfe5d54150b090b02, 0xd3f93b35435d7c4c), // 5^-233
302 U128.new(0x9efa548d26e5a6e1, 0xc47bc5014a1a6daf), // 5^-232
303 U128.new(0xc6b8e9b0709f109a, 0x359ab6419ca1091b), // 5^-231
304 U128.new(0xf867241c8cc6d4c0, 0xc30163d203c94b62), // 5^-230
305 U128.new(0x9b407691d7fc44f8, 0x79e0de63425dcf1d), // 5^-229
306 U128.new(0xc21094364dfb5636, 0x985915fc12f542e4), // 5^-228
307 U128.new(0xf294b943e17a2bc4, 0x3e6f5b7b17b2939d), // 5^-227
308 U128.new(0x979cf3ca6cec5b5a, 0xa705992ceecf9c42), // 5^-226
309 U128.new(0xbd8430bd08277231, 0x50c6ff782a838353), // 5^-225
310 U128.new(0xece53cec4a314ebd, 0xa4f8bf5635246428), // 5^-224
311 U128.new(0x940f4613ae5ed136, 0x871b7795e136be99), // 5^-223
312 U128.new(0xb913179899f68584, 0x28e2557b59846e3f), // 5^-222
313 U128.new(0xe757dd7ec07426e5, 0x331aeada2fe589cf), // 5^-221
314 U128.new(0x9096ea6f3848984f, 0x3ff0d2c85def7621), // 5^-220
315 U128.new(0xb4bca50b065abe63, 0xfed077a756b53a9), // 5^-219
316 U128.new(0xe1ebce4dc7f16dfb, 0xd3e8495912c62894), // 5^-218
317 U128.new(0x8d3360f09cf6e4bd, 0x64712dd7abbbd95c), // 5^-217
318 U128.new(0xb080392cc4349dec, 0xbd8d794d96aacfb3), // 5^-216
319 U128.new(0xdca04777f541c567, 0xecf0d7a0fc5583a0), // 5^-215
320 U128.new(0x89e42caaf9491b60, 0xf41686c49db57244), // 5^-214
321 U128.new(0xac5d37d5b79b6239, 0x311c2875c522ced5), // 5^-213
322 U128.new(0xd77485cb25823ac7, 0x7d633293366b828b), // 5^-212
323 U128.new(0x86a8d39ef77164bc, 0xae5dff9c02033197), // 5^-211
324 U128.new(0xa8530886b54dbdeb, 0xd9f57f830283fdfc), // 5^-210
325 U128.new(0xd267caa862a12d66, 0xd072df63c324fd7b), // 5^-209
326 U128.new(0x8380dea93da4bc60, 0x4247cb9e59f71e6d), // 5^-208
327 U128.new(0xa46116538d0deb78, 0x52d9be85f074e608), // 5^-207
328 U128.new(0xcd795be870516656, 0x67902e276c921f8b), // 5^-206
329 U128.new(0x806bd9714632dff6, 0xba1cd8a3db53b6), // 5^-205
330 U128.new(0xa086cfcd97bf97f3, 0x80e8a40eccd228a4), // 5^-204
331 U128.new(0xc8a883c0fdaf7df0, 0x6122cd128006b2cd), // 5^-203
332 U128.new(0xfad2a4b13d1b5d6c, 0x796b805720085f81), // 5^-202
333 U128.new(0x9cc3a6eec6311a63, 0xcbe3303674053bb0), // 5^-201
334 U128.new(0xc3f490aa77bd60fc, 0xbedbfc4411068a9c), // 5^-200
335 U128.new(0xf4f1b4d515acb93b, 0xee92fb5515482d44), // 5^-199
336 U128.new(0x991711052d8bf3c5, 0x751bdd152d4d1c4a), // 5^-198
337 U128.new(0xbf5cd54678eef0b6, 0xd262d45a78a0635d), // 5^-197
338 U128.new(0xef340a98172aace4, 0x86fb897116c87c34), // 5^-196
339 U128.new(0x9580869f0e7aac0e, 0xd45d35e6ae3d4da0), // 5^-195
340 U128.new(0xbae0a846d2195712, 0x8974836059cca109), // 5^-194
341 U128.new(0xe998d258869facd7, 0x2bd1a438703fc94b), // 5^-193
342 U128.new(0x91ff83775423cc06, 0x7b6306a34627ddcf), // 5^-192
343 U128.new(0xb67f6455292cbf08, 0x1a3bc84c17b1d542), // 5^-191
344 U128.new(0xe41f3d6a7377eeca, 0x20caba5f1d9e4a93), // 5^-190
345 U128.new(0x8e938662882af53e, 0x547eb47b7282ee9c), // 5^-189
346 U128.new(0xb23867fb2a35b28d, 0xe99e619a4f23aa43), // 5^-188
347 U128.new(0xdec681f9f4c31f31, 0x6405fa00e2ec94d4), // 5^-187
348 U128.new(0x8b3c113c38f9f37e, 0xde83bc408dd3dd04), // 5^-186
349 U128.new(0xae0b158b4738705e, 0x9624ab50b148d445), // 5^-185
350 U128.new(0xd98ddaee19068c76, 0x3badd624dd9b0957), // 5^-184
351 U128.new(0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d6), // 5^-183
352 U128.new(0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4c), // 5^-182
353 U128.new(0xd47487cc8470652b, 0x7647c3200069671f), // 5^-181
354 U128.new(0x84c8d4dfd2c63f3b, 0x29ecd9f40041e073), // 5^-180
355 U128.new(0xa5fb0a17c777cf09, 0xf468107100525890), // 5^-179
356 U128.new(0xcf79cc9db955c2cc, 0x7182148d4066eeb4), // 5^-178
357 U128.new(0x81ac1fe293d599bf, 0xc6f14cd848405530), // 5^-177
358 U128.new(0xa21727db38cb002f, 0xb8ada00e5a506a7c), // 5^-176
359 U128.new(0xca9cf1d206fdc03b, 0xa6d90811f0e4851c), // 5^-175
360 U128.new(0xfd442e4688bd304a, 0x908f4a166d1da663), // 5^-174
361 U128.new(0x9e4a9cec15763e2e, 0x9a598e4e043287fe), // 5^-173
362 U128.new(0xc5dd44271ad3cdba, 0x40eff1e1853f29fd), // 5^-172
363 U128.new(0xf7549530e188c128, 0xd12bee59e68ef47c), // 5^-171
364 U128.new(0x9a94dd3e8cf578b9, 0x82bb74f8301958ce), // 5^-170
365 U128.new(0xc13a148e3032d6e7, 0xe36a52363c1faf01), // 5^-169
366 U128.new(0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac1), // 5^-168
367 U128.new(0x96f5600f15a7b7e5, 0x29ab103a5ef8c0b9), // 5^-167
368 U128.new(0xbcb2b812db11a5de, 0x7415d448f6b6f0e7), // 5^-166
369 U128.new(0xebdf661791d60f56, 0x111b495b3464ad21), // 5^-165
370 U128.new(0x936b9fcebb25c995, 0xcab10dd900beec34), // 5^-164
371 U128.new(0xb84687c269ef3bfb, 0x3d5d514f40eea742), // 5^-163
372 U128.new(0xe65829b3046b0afa, 0xcb4a5a3112a5112), // 5^-162
373 U128.new(0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ab), // 5^-161
374 U128.new(0xb3f4e093db73a093, 0x59ed216765690f56), // 5^-160
375 U128.new(0xe0f218b8d25088b8, 0x306869c13ec3532c), // 5^-159
376 U128.new(0x8c974f7383725573, 0x1e414218c73a13fb), // 5^-158
377 U128.new(0xafbd2350644eeacf, 0xe5d1929ef90898fa), // 5^-157
378 U128.new(0xdbac6c247d62a583, 0xdf45f746b74abf39), // 5^-156
379 U128.new(0x894bc396ce5da772, 0x6b8bba8c328eb783), // 5^-155
380 U128.new(0xab9eb47c81f5114f, 0x66ea92f3f326564), // 5^-154
381 U128.new(0xd686619ba27255a2, 0xc80a537b0efefebd), // 5^-153
382 U128.new(0x8613fd0145877585, 0xbd06742ce95f5f36), // 5^-152
383 U128.new(0xa798fc4196e952e7, 0x2c48113823b73704), // 5^-151
384 U128.new(0xd17f3b51fca3a7a0, 0xf75a15862ca504c5), // 5^-150
385 U128.new(0x82ef85133de648c4, 0x9a984d73dbe722fb), // 5^-149
386 U128.new(0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebba), // 5^-148
387 U128.new(0xcc963fee10b7d1b3, 0x318df905079926a8), // 5^-147
388 U128.new(0xffbbcfe994e5c61f, 0xfdf17746497f7052), // 5^-146
389 U128.new(0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa633), // 5^-145
390 U128.new(0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc0), // 5^-144
391 U128.new(0xf9bd690a1b68637b, 0x3dfdce7aa3c673b0), // 5^-143
392 U128.new(0x9c1661a651213e2d, 0x6bea10ca65c084e), // 5^-142
393 U128.new(0xc31bfa0fe5698db8, 0x486e494fcff30a62), // 5^-141
394 U128.new(0xf3e2f893dec3f126, 0x5a89dba3c3efccfa), // 5^-140
395 U128.new(0x986ddb5c6b3a76b7, 0xf89629465a75e01c), // 5^-139
396 U128.new(0xbe89523386091465, 0xf6bbb397f1135823), // 5^-138
397 U128.new(0xee2ba6c0678b597f, 0x746aa07ded582e2c), // 5^-137
398 U128.new(0x94db483840b717ef, 0xa8c2a44eb4571cdc), // 5^-136
399 U128.new(0xba121a4650e4ddeb, 0x92f34d62616ce413), // 5^-135
400 U128.new(0xe896a0d7e51e1566, 0x77b020baf9c81d17), // 5^-134
401 U128.new(0x915e2486ef32cd60, 0xace1474dc1d122e), // 5^-133
402 U128.new(0xb5b5ada8aaff80b8, 0xd819992132456ba), // 5^-132
403 U128.new(0xe3231912d5bf60e6, 0x10e1fff697ed6c69), // 5^-131
404 U128.new(0x8df5efabc5979c8f, 0xca8d3ffa1ef463c1), // 5^-130
405 U128.new(0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb2), // 5^-129
406 U128.new(0xddd0467c64bce4a0, 0xac7cb3f6d05ddbde), // 5^-128
407 U128.new(0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96b), // 5^-127
408 U128.new(0xad4ab7112eb3929d, 0x86c16c98d2c953c6), // 5^-126
409 U128.new(0xd89d64d57a607744, 0xe871c7bf077ba8b7), // 5^-125
410 U128.new(0x87625f056c7c4a8b, 0x11471cd764ad4972), // 5^-124
411 U128.new(0xa93af6c6c79b5d2d, 0xd598e40d3dd89bcf), // 5^-123
412 U128.new(0xd389b47879823479, 0x4aff1d108d4ec2c3), // 5^-122
413 U128.new(0x843610cb4bf160cb, 0xcedf722a585139ba), // 5^-121
414 U128.new(0xa54394fe1eedb8fe, 0xc2974eb4ee658828), // 5^-120
415 U128.new(0xce947a3da6a9273e, 0x733d226229feea32), // 5^-119
416 U128.new(0x811ccc668829b887, 0x806357d5a3f525f), // 5^-118
417 U128.new(0xa163ff802a3426a8, 0xca07c2dcb0cf26f7), // 5^-117
418 U128.new(0xc9bcff6034c13052, 0xfc89b393dd02f0b5), // 5^-116
419 U128.new(0xfc2c3f3841f17c67, 0xbbac2078d443ace2), // 5^-115
420 U128.new(0x9d9ba7832936edc0, 0xd54b944b84aa4c0d), // 5^-114
421 U128.new(0xc5029163f384a931, 0xa9e795e65d4df11), // 5^-113
422 U128.new(0xf64335bcf065d37d, 0x4d4617b5ff4a16d5), // 5^-112
423 U128.new(0x99ea0196163fa42e, 0x504bced1bf8e4e45), // 5^-111
424 U128.new(0xc06481fb9bcf8d39, 0xe45ec2862f71e1d6), // 5^-110
425 U128.new(0xf07da27a82c37088, 0x5d767327bb4e5a4c), // 5^-109
426 U128.new(0x964e858c91ba2655, 0x3a6a07f8d510f86f), // 5^-108
427 U128.new(0xbbe226efb628afea, 0x890489f70a55368b), // 5^-107
428 U128.new(0xeadab0aba3b2dbe5, 0x2b45ac74ccea842e), // 5^-106
429 U128.new(0x92c8ae6b464fc96f, 0x3b0b8bc90012929d), // 5^-105
430 U128.new(0xb77ada0617e3bbcb, 0x9ce6ebb40173744), // 5^-104
431 U128.new(0xe55990879ddcaabd, 0xcc420a6a101d0515), // 5^-103
432 U128.new(0x8f57fa54c2a9eab6, 0x9fa946824a12232d), // 5^-102
433 U128.new(0xb32df8e9f3546564, 0x47939822dc96abf9), // 5^-101
434 U128.new(0xdff9772470297ebd, 0x59787e2b93bc56f7), // 5^-100
435 U128.new(0x8bfbea76c619ef36, 0x57eb4edb3c55b65a), // 5^-99
436 U128.new(0xaefae51477a06b03, 0xede622920b6b23f1), // 5^-98
437 U128.new(0xdab99e59958885c4, 0xe95fab368e45eced), // 5^-97
438 U128.new(0x88b402f7fd75539b, 0x11dbcb0218ebb414), // 5^-96
439 U128.new(0xaae103b5fcd2a881, 0xd652bdc29f26a119), // 5^-95
440 U128.new(0xd59944a37c0752a2, 0x4be76d3346f0495f), // 5^-94
441 U128.new(0x857fcae62d8493a5, 0x6f70a4400c562ddb), // 5^-93
442 U128.new(0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb952), // 5^-92
443 U128.new(0xd097ad07a71f26b2, 0x7e2000a41346a7a7), // 5^-91
444 U128.new(0x825ecc24c873782f, 0x8ed400668c0c28c8), // 5^-90
445 U128.new(0xa2f67f2dfa90563b, 0x728900802f0f32fa), // 5^-89
446 U128.new(0xcbb41ef979346bca, 0x4f2b40a03ad2ffb9), // 5^-88
447 U128.new(0xfea126b7d78186bc, 0xe2f610c84987bfa8), // 5^-87
448 U128.new(0x9f24b832e6b0f436, 0xdd9ca7d2df4d7c9), // 5^-86
449 U128.new(0xc6ede63fa05d3143, 0x91503d1c79720dbb), // 5^-85
450 U128.new(0xf8a95fcf88747d94, 0x75a44c6397ce912a), // 5^-84
451 U128.new(0x9b69dbe1b548ce7c, 0xc986afbe3ee11aba), // 5^-83
452 U128.new(0xc24452da229b021b, 0xfbe85badce996168), // 5^-82
453 U128.new(0xf2d56790ab41c2a2, 0xfae27299423fb9c3), // 5^-81
454 U128.new(0x97c560ba6b0919a5, 0xdccd879fc967d41a), // 5^-80
455 U128.new(0xbdb6b8e905cb600f, 0x5400e987bbc1c920), // 5^-79
456 U128.new(0xed246723473e3813, 0x290123e9aab23b68), // 5^-78
457 U128.new(0x9436c0760c86e30b, 0xf9a0b6720aaf6521), // 5^-77
458 U128.new(0xb94470938fa89bce, 0xf808e40e8d5b3e69), // 5^-76
459 U128.new(0xe7958cb87392c2c2, 0xb60b1d1230b20e04), // 5^-75
460 U128.new(0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c2), // 5^-74
461 U128.new(0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af3), // 5^-73
462 U128.new(0xe2280b6c20dd5232, 0x25c6da63c38de1b0), // 5^-72
463 U128.new(0x8d590723948a535f, 0x579c487e5a38ad0e), // 5^-71
464 U128.new(0xb0af48ec79ace837, 0x2d835a9df0c6d851), // 5^-70
465 U128.new(0xdcdb1b2798182244, 0xf8e431456cf88e65), // 5^-69
466 U128.new(0x8a08f0f8bf0f156b, 0x1b8e9ecb641b58ff), // 5^-68
467 U128.new(0xac8b2d36eed2dac5, 0xe272467e3d222f3f), // 5^-67
468 U128.new(0xd7adf884aa879177, 0x5b0ed81dcc6abb0f), // 5^-66
469 U128.new(0x86ccbb52ea94baea, 0x98e947129fc2b4e9), // 5^-65
470 U128.new(0xa87fea27a539e9a5, 0x3f2398d747b36224), // 5^-64
471 U128.new(0xd29fe4b18e88640e, 0x8eec7f0d19a03aad), // 5^-63
472 U128.new(0x83a3eeeef9153e89, 0x1953cf68300424ac), // 5^-62
473 U128.new(0xa48ceaaab75a8e2b, 0x5fa8c3423c052dd7), // 5^-61
474 U128.new(0xcdb02555653131b6, 0x3792f412cb06794d), // 5^-60
475 U128.new(0x808e17555f3ebf11, 0xe2bbd88bbee40bd0), // 5^-59
476 U128.new(0xa0b19d2ab70e6ed6, 0x5b6aceaeae9d0ec4), // 5^-58
477 U128.new(0xc8de047564d20a8b, 0xf245825a5a445275), // 5^-57
478 U128.new(0xfb158592be068d2e, 0xeed6e2f0f0d56712), // 5^-56
479 U128.new(0x9ced737bb6c4183d, 0x55464dd69685606b), // 5^-55
480 U128.new(0xc428d05aa4751e4c, 0xaa97e14c3c26b886), // 5^-54
481 U128.new(0xf53304714d9265df, 0xd53dd99f4b3066a8), // 5^-53
482 U128.new(0x993fe2c6d07b7fab, 0xe546a8038efe4029), // 5^-52
483 U128.new(0xbf8fdb78849a5f96, 0xde98520472bdd033), // 5^-51
484 U128.new(0xef73d256a5c0f77c, 0x963e66858f6d4440), // 5^-50
485 U128.new(0x95a8637627989aad, 0xdde7001379a44aa8), // 5^-49
486 U128.new(0xbb127c53b17ec159, 0x5560c018580d5d52), // 5^-48
487 U128.new(0xe9d71b689dde71af, 0xaab8f01e6e10b4a6), // 5^-47
488 U128.new(0x9226712162ab070d, 0xcab3961304ca70e8), // 5^-46
489 U128.new(0xb6b00d69bb55c8d1, 0x3d607b97c5fd0d22), // 5^-45
490 U128.new(0xe45c10c42a2b3b05, 0x8cb89a7db77c506a), // 5^-44
491 U128.new(0x8eb98a7a9a5b04e3, 0x77f3608e92adb242), // 5^-43
492 U128.new(0xb267ed1940f1c61c, 0x55f038b237591ed3), // 5^-42
493 U128.new(0xdf01e85f912e37a3, 0x6b6c46dec52f6688), // 5^-41
494 U128.new(0x8b61313bbabce2c6, 0x2323ac4b3b3da015), // 5^-40
495 U128.new(0xae397d8aa96c1b77, 0xabec975e0a0d081a), // 5^-39
496 U128.new(0xd9c7dced53c72255, 0x96e7bd358c904a21), // 5^-38
497 U128.new(0x881cea14545c7575, 0x7e50d64177da2e54), // 5^-37
498 U128.new(0xaa242499697392d2, 0xdde50bd1d5d0b9e9), // 5^-36
499 U128.new(0xd4ad2dbfc3d07787, 0x955e4ec64b44e864), // 5^-35
500 U128.new(0x84ec3c97da624ab4, 0xbd5af13bef0b113e), // 5^-34
501 U128.new(0xa6274bbdd0fadd61, 0xecb1ad8aeacdd58e), // 5^-33
502 U128.new(0xcfb11ead453994ba, 0x67de18eda5814af2), // 5^-32
503 U128.new(0x81ceb32c4b43fcf4, 0x80eacf948770ced7), // 5^-31
504 U128.new(0xa2425ff75e14fc31, 0xa1258379a94d028d), // 5^-30
505 U128.new(0xcad2f7f5359a3b3e, 0x96ee45813a04330), // 5^-29
506 U128.new(0xfd87b5f28300ca0d, 0x8bca9d6e188853fc), // 5^-28
507 U128.new(0x9e74d1b791e07e48, 0x775ea264cf55347e), // 5^-27
508 U128.new(0xc612062576589dda, 0x95364afe032a819e), // 5^-26
509 U128.new(0xf79687aed3eec551, 0x3a83ddbd83f52205), // 5^-25
510 U128.new(0x9abe14cd44753b52, 0xc4926a9672793543), // 5^-24
511 U128.new(0xc16d9a0095928a27, 0x75b7053c0f178294), // 5^-23
512 U128.new(0xf1c90080baf72cb1, 0x5324c68b12dd6339), // 5^-22
513 U128.new(0x971da05074da7bee, 0xd3f6fc16ebca5e04), // 5^-21
514 U128.new(0xbce5086492111aea, 0x88f4bb1ca6bcf585), // 5^-20
515 U128.new(0xec1e4a7db69561a5, 0x2b31e9e3d06c32e6), // 5^-19
516 U128.new(0x9392ee8e921d5d07, 0x3aff322e62439fd0), // 5^-18
517 U128.new(0xb877aa3236a4b449, 0x9befeb9fad487c3), // 5^-17
518 U128.new(0xe69594bec44de15b, 0x4c2ebe687989a9b4), // 5^-16
519 U128.new(0x901d7cf73ab0acd9, 0xf9d37014bf60a11), // 5^-15
520 U128.new(0xb424dc35095cd80f, 0x538484c19ef38c95), // 5^-14
521 U128.new(0xe12e13424bb40e13, 0x2865a5f206b06fba), // 5^-13
522 U128.new(0x8cbccc096f5088cb, 0xf93f87b7442e45d4), // 5^-12
523 U128.new(0xafebff0bcb24aafe, 0xf78f69a51539d749), // 5^-11
524 U128.new(0xdbe6fecebdedd5be, 0xb573440e5a884d1c), // 5^-10
525 U128.new(0x89705f4136b4a597, 0x31680a88f8953031), // 5^-9
526 U128.new(0xabcc77118461cefc, 0xfdc20d2b36ba7c3e), // 5^-8
527 U128.new(0xd6bf94d5e57a42bc, 0x3d32907604691b4d), // 5^-7
528 U128.new(0x8637bd05af6c69b5, 0xa63f9a49c2c1b110), // 5^-6
529 U128.new(0xa7c5ac471b478423, 0xfcf80dc33721d54), // 5^-5
530 U128.new(0xd1b71758e219652b, 0xd3c36113404ea4a9), // 5^-4
531 U128.new(0x83126e978d4fdf3b, 0x645a1cac083126ea), // 5^-3
532 U128.new(0xa3d70a3d70a3d70a, 0x3d70a3d70a3d70a4), // 5^-2
533 U128.new(0xcccccccccccccccc, 0xcccccccccccccccd), // 5^-1
534 U128.new(0x8000000000000000, 0x0), // 5^0
535 U128.new(0xa000000000000000, 0x0), // 5^1
536 U128.new(0xc800000000000000, 0x0), // 5^2
537 U128.new(0xfa00000000000000, 0x0), // 5^3
538 U128.new(0x9c40000000000000, 0x0), // 5^4
539 U128.new(0xc350000000000000, 0x0), // 5^5
540 U128.new(0xf424000000000000, 0x0), // 5^6
541 U128.new(0x9896800000000000, 0x0), // 5^7
542 U128.new(0xbebc200000000000, 0x0), // 5^8
543 U128.new(0xee6b280000000000, 0x0), // 5^9
544 U128.new(0x9502f90000000000, 0x0), // 5^10
545 U128.new(0xba43b74000000000, 0x0), // 5^11
546 U128.new(0xe8d4a51000000000, 0x0), // 5^12
547 U128.new(0x9184e72a00000000, 0x0), // 5^13
548 U128.new(0xb5e620f480000000, 0x0), // 5^14
549 U128.new(0xe35fa931a0000000, 0x0), // 5^15
550 U128.new(0x8e1bc9bf04000000, 0x0), // 5^16
551 U128.new(0xb1a2bc2ec5000000, 0x0), // 5^17
552 U128.new(0xde0b6b3a76400000, 0x0), // 5^18
553 U128.new(0x8ac7230489e80000, 0x0), // 5^19
554 U128.new(0xad78ebc5ac620000, 0x0), // 5^20
555 U128.new(0xd8d726b7177a8000, 0x0), // 5^21
556 U128.new(0x878678326eac9000, 0x0), // 5^22
557 U128.new(0xa968163f0a57b400, 0x0), // 5^23
558 U128.new(0xd3c21bcecceda100, 0x0), // 5^24
559 U128.new(0x84595161401484a0, 0x0), // 5^25
560 U128.new(0xa56fa5b99019a5c8, 0x0), // 5^26
561 U128.new(0xcecb8f27f4200f3a, 0x0), // 5^27
562 U128.new(0x813f3978f8940984, 0x4000000000000000), // 5^28
563 U128.new(0xa18f07d736b90be5, 0x5000000000000000), // 5^29
564 U128.new(0xc9f2c9cd04674ede, 0xa400000000000000), // 5^30
565 U128.new(0xfc6f7c4045812296, 0x4d00000000000000), // 5^31
566 U128.new(0x9dc5ada82b70b59d, 0xf020000000000000), // 5^32
567 U128.new(0xc5371912364ce305, 0x6c28000000000000), // 5^33
568 U128.new(0xf684df56c3e01bc6, 0xc732000000000000), // 5^34
569 U128.new(0x9a130b963a6c115c, 0x3c7f400000000000), // 5^35
570 U128.new(0xc097ce7bc90715b3, 0x4b9f100000000000), // 5^36
571 U128.new(0xf0bdc21abb48db20, 0x1e86d40000000000), // 5^37
572 U128.new(0x96769950b50d88f4, 0x1314448000000000), // 5^38
573 U128.new(0xbc143fa4e250eb31, 0x17d955a000000000), // 5^39
574 U128.new(0xeb194f8e1ae525fd, 0x5dcfab0800000000), // 5^40
575 U128.new(0x92efd1b8d0cf37be, 0x5aa1cae500000000), // 5^41
576 U128.new(0xb7abc627050305ad, 0xf14a3d9e40000000), // 5^42
577 U128.new(0xe596b7b0c643c719, 0x6d9ccd05d0000000), // 5^43
578 U128.new(0x8f7e32ce7bea5c6f, 0xe4820023a2000000), // 5^44
579 U128.new(0xb35dbf821ae4f38b, 0xdda2802c8a800000), // 5^45
580 U128.new(0xe0352f62a19e306e, 0xd50b2037ad200000), // 5^46
581 U128.new(0x8c213d9da502de45, 0x4526f422cc340000), // 5^47
582 U128.new(0xaf298d050e4395d6, 0x9670b12b7f410000), // 5^48
583 U128.new(0xdaf3f04651d47b4c, 0x3c0cdd765f114000), // 5^49
584 U128.new(0x88d8762bf324cd0f, 0xa5880a69fb6ac800), // 5^50
585 U128.new(0xab0e93b6efee0053, 0x8eea0d047a457a00), // 5^51
586 U128.new(0xd5d238a4abe98068, 0x72a4904598d6d880), // 5^52
587 U128.new(0x85a36366eb71f041, 0x47a6da2b7f864750), // 5^53
588 U128.new(0xa70c3c40a64e6c51, 0x999090b65f67d924), // 5^54
589 U128.new(0xd0cf4b50cfe20765, 0xfff4b4e3f741cf6d), // 5^55
590 U128.new(0x82818f1281ed449f, 0xbff8f10e7a8921a4), // 5^56
591 U128.new(0xa321f2d7226895c7, 0xaff72d52192b6a0d), // 5^57
592 U128.new(0xcbea6f8ceb02bb39, 0x9bf4f8a69f764490), // 5^58
593 U128.new(0xfee50b7025c36a08, 0x2f236d04753d5b4), // 5^59
594 U128.new(0x9f4f2726179a2245, 0x1d762422c946590), // 5^60
595 U128.new(0xc722f0ef9d80aad6, 0x424d3ad2b7b97ef5), // 5^61
596 U128.new(0xf8ebad2b84e0d58b, 0xd2e0898765a7deb2), // 5^62
597 U128.new(0x9b934c3b330c8577, 0x63cc55f49f88eb2f), // 5^63
598 U128.new(0xc2781f49ffcfa6d5, 0x3cbf6b71c76b25fb), // 5^64
599 U128.new(0xf316271c7fc3908a, 0x8bef464e3945ef7a), // 5^65
600 U128.new(0x97edd871cfda3a56, 0x97758bf0e3cbb5ac), // 5^66
601 U128.new(0xbde94e8e43d0c8ec, 0x3d52eeed1cbea317), // 5^67
602 U128.new(0xed63a231d4c4fb27, 0x4ca7aaa863ee4bdd), // 5^68
603 U128.new(0x945e455f24fb1cf8, 0x8fe8caa93e74ef6a), // 5^69
604 U128.new(0xb975d6b6ee39e436, 0xb3e2fd538e122b44), // 5^70
605 U128.new(0xe7d34c64a9c85d44, 0x60dbbca87196b616), // 5^71
606 U128.new(0x90e40fbeea1d3a4a, 0xbc8955e946fe31cd), // 5^72
607 U128.new(0xb51d13aea4a488dd, 0x6babab6398bdbe41), // 5^73
608 U128.new(0xe264589a4dcdab14, 0xc696963c7eed2dd1), // 5^74
609 U128.new(0x8d7eb76070a08aec, 0xfc1e1de5cf543ca2), // 5^75
610 U128.new(0xb0de65388cc8ada8, 0x3b25a55f43294bcb), // 5^76
611 U128.new(0xdd15fe86affad912, 0x49ef0eb713f39ebe), // 5^77
612 U128.new(0x8a2dbf142dfcc7ab, 0x6e3569326c784337), // 5^78
613 U128.new(0xacb92ed9397bf996, 0x49c2c37f07965404), // 5^79
614 U128.new(0xd7e77a8f87daf7fb, 0xdc33745ec97be906), // 5^80
615 U128.new(0x86f0ac99b4e8dafd, 0x69a028bb3ded71a3), // 5^81
616 U128.new(0xa8acd7c0222311bc, 0xc40832ea0d68ce0c), // 5^82
617 U128.new(0xd2d80db02aabd62b, 0xf50a3fa490c30190), // 5^83
618 U128.new(0x83c7088e1aab65db, 0x792667c6da79e0fa), // 5^84
619 U128.new(0xa4b8cab1a1563f52, 0x577001b891185938), // 5^85
620 U128.new(0xcde6fd5e09abcf26, 0xed4c0226b55e6f86), // 5^86
621 U128.new(0x80b05e5ac60b6178, 0x544f8158315b05b4), // 5^87
622 U128.new(0xa0dc75f1778e39d6, 0x696361ae3db1c721), // 5^88
623 U128.new(0xc913936dd571c84c, 0x3bc3a19cd1e38e9), // 5^89
624 U128.new(0xfb5878494ace3a5f, 0x4ab48a04065c723), // 5^90
625 U128.new(0x9d174b2dcec0e47b, 0x62eb0d64283f9c76), // 5^91
626 U128.new(0xc45d1df942711d9a, 0x3ba5d0bd324f8394), // 5^92
627 U128.new(0xf5746577930d6500, 0xca8f44ec7ee36479), // 5^93
628 U128.new(0x9968bf6abbe85f20, 0x7e998b13cf4e1ecb), // 5^94
629 U128.new(0xbfc2ef456ae276e8, 0x9e3fedd8c321a67e), // 5^95
630 U128.new(0xefb3ab16c59b14a2, 0xc5cfe94ef3ea101e), // 5^96
631 U128.new(0x95d04aee3b80ece5, 0xbba1f1d158724a12), // 5^97
632 U128.new(0xbb445da9ca61281f, 0x2a8a6e45ae8edc97), // 5^98
633 U128.new(0xea1575143cf97226, 0xf52d09d71a3293bd), // 5^99
634 U128.new(0x924d692ca61be758, 0x593c2626705f9c56), // 5^100
635 U128.new(0xb6e0c377cfa2e12e, 0x6f8b2fb00c77836c), // 5^101
636 U128.new(0xe498f455c38b997a, 0xb6dfb9c0f956447), // 5^102
637 U128.new(0x8edf98b59a373fec, 0x4724bd4189bd5eac), // 5^103
638 U128.new(0xb2977ee300c50fe7, 0x58edec91ec2cb657), // 5^104
639 U128.new(0xdf3d5e9bc0f653e1, 0x2f2967b66737e3ed), // 5^105
640 U128.new(0x8b865b215899f46c, 0xbd79e0d20082ee74), // 5^106
641 U128.new(0xae67f1e9aec07187, 0xecd8590680a3aa11), // 5^107
642 U128.new(0xda01ee641a708de9, 0xe80e6f4820cc9495), // 5^108
643 U128.new(0x884134fe908658b2, 0x3109058d147fdcdd), // 5^109
644 U128.new(0xaa51823e34a7eede, 0xbd4b46f0599fd415), // 5^110
645 U128.new(0xd4e5e2cdc1d1ea96, 0x6c9e18ac7007c91a), // 5^111
646 U128.new(0x850fadc09923329e, 0x3e2cf6bc604ddb0), // 5^112
647 U128.new(0xa6539930bf6bff45, 0x84db8346b786151c), // 5^113
648 U128.new(0xcfe87f7cef46ff16, 0xe612641865679a63), // 5^114
649 U128.new(0x81f14fae158c5f6e, 0x4fcb7e8f3f60c07e), // 5^115
650 U128.new(0xa26da3999aef7749, 0xe3be5e330f38f09d), // 5^116
651 U128.new(0xcb090c8001ab551c, 0x5cadf5bfd3072cc5), // 5^117
652 U128.new(0xfdcb4fa002162a63, 0x73d9732fc7c8f7f6), // 5^118
653 U128.new(0x9e9f11c4014dda7e, 0x2867e7fddcdd9afa), // 5^119
654 U128.new(0xc646d63501a1511d, 0xb281e1fd541501b8), // 5^120
655 U128.new(0xf7d88bc24209a565, 0x1f225a7ca91a4226), // 5^121
656 U128.new(0x9ae757596946075f, 0x3375788de9b06958), // 5^122
657 U128.new(0xc1a12d2fc3978937, 0x52d6b1641c83ae), // 5^123
658 U128.new(0xf209787bb47d6b84, 0xc0678c5dbd23a49a), // 5^124
659 U128.new(0x9745eb4d50ce6332, 0xf840b7ba963646e0), // 5^125
660 U128.new(0xbd176620a501fbff, 0xb650e5a93bc3d898), // 5^126
661 U128.new(0xec5d3fa8ce427aff, 0xa3e51f138ab4cebe), // 5^127
662 U128.new(0x93ba47c980e98cdf, 0xc66f336c36b10137), // 5^128
663 U128.new(0xb8a8d9bbe123f017, 0xb80b0047445d4184), // 5^129
664 U128.new(0xe6d3102ad96cec1d, 0xa60dc059157491e5), // 5^130
665 U128.new(0x9043ea1ac7e41392, 0x87c89837ad68db2f), // 5^131
666 U128.new(0xb454e4a179dd1877, 0x29babe4598c311fb), // 5^132
667 U128.new(0xe16a1dc9d8545e94, 0xf4296dd6fef3d67a), // 5^133
668 U128.new(0x8ce2529e2734bb1d, 0x1899e4a65f58660c), // 5^134
669 U128.new(0xb01ae745b101e9e4, 0x5ec05dcff72e7f8f), // 5^135
670 U128.new(0xdc21a1171d42645d, 0x76707543f4fa1f73), // 5^136
671 U128.new(0x899504ae72497eba, 0x6a06494a791c53a8), // 5^137
672 U128.new(0xabfa45da0edbde69, 0x487db9d17636892), // 5^138
673 U128.new(0xd6f8d7509292d603, 0x45a9d2845d3c42b6), // 5^139
674 U128.new(0x865b86925b9bc5c2, 0xb8a2392ba45a9b2), // 5^140
675 U128.new(0xa7f26836f282b732, 0x8e6cac7768d7141e), // 5^141
676 U128.new(0xd1ef0244af2364ff, 0x3207d795430cd926), // 5^142
677 U128.new(0x8335616aed761f1f, 0x7f44e6bd49e807b8), // 5^143
678 U128.new(0xa402b9c5a8d3a6e7, 0x5f16206c9c6209a6), // 5^144
679 U128.new(0xcd036837130890a1, 0x36dba887c37a8c0f), // 5^145
680 U128.new(0x802221226be55a64, 0xc2494954da2c9789), // 5^146
681 U128.new(0xa02aa96b06deb0fd, 0xf2db9baa10b7bd6c), // 5^147
682 U128.new(0xc83553c5c8965d3d, 0x6f92829494e5acc7), // 5^148
683 U128.new(0xfa42a8b73abbf48c, 0xcb772339ba1f17f9), // 5^149
684 U128.new(0x9c69a97284b578d7, 0xff2a760414536efb), // 5^150
685 U128.new(0xc38413cf25e2d70d, 0xfef5138519684aba), // 5^151
686 U128.new(0xf46518c2ef5b8cd1, 0x7eb258665fc25d69), // 5^152
687 U128.new(0x98bf2f79d5993802, 0xef2f773ffbd97a61), // 5^153
688 U128.new(0xbeeefb584aff8603, 0xaafb550ffacfd8fa), // 5^154
689 U128.new(0xeeaaba2e5dbf6784, 0x95ba2a53f983cf38), // 5^155
690 U128.new(0x952ab45cfa97a0b2, 0xdd945a747bf26183), // 5^156
691 U128.new(0xba756174393d88df, 0x94f971119aeef9e4), // 5^157
692 U128.new(0xe912b9d1478ceb17, 0x7a37cd5601aab85d), // 5^158
693 U128.new(0x91abb422ccb812ee, 0xac62e055c10ab33a), // 5^159
694 U128.new(0xb616a12b7fe617aa, 0x577b986b314d6009), // 5^160
695 U128.new(0xe39c49765fdf9d94, 0xed5a7e85fda0b80b), // 5^161
696 U128.new(0x8e41ade9fbebc27d, 0x14588f13be847307), // 5^162
697 U128.new(0xb1d219647ae6b31c, 0x596eb2d8ae258fc8), // 5^163
698 U128.new(0xde469fbd99a05fe3, 0x6fca5f8ed9aef3bb), // 5^164
699 U128.new(0x8aec23d680043bee, 0x25de7bb9480d5854), // 5^165
700 U128.new(0xada72ccc20054ae9, 0xaf561aa79a10ae6a), // 5^166
701 U128.new(0xd910f7ff28069da4, 0x1b2ba1518094da04), // 5^167
702 U128.new(0x87aa9aff79042286, 0x90fb44d2f05d0842), // 5^168
703 U128.new(0xa99541bf57452b28, 0x353a1607ac744a53), // 5^169
704 U128.new(0xd3fa922f2d1675f2, 0x42889b8997915ce8), // 5^170
705 U128.new(0x847c9b5d7c2e09b7, 0x69956135febada11), // 5^171
706 U128.new(0xa59bc234db398c25, 0x43fab9837e699095), // 5^172
707 U128.new(0xcf02b2c21207ef2e, 0x94f967e45e03f4bb), // 5^173
708 U128.new(0x8161afb94b44f57d, 0x1d1be0eebac278f5), // 5^174
709 U128.new(0xa1ba1ba79e1632dc, 0x6462d92a69731732), // 5^175
710 U128.new(0xca28a291859bbf93, 0x7d7b8f7503cfdcfe), // 5^176
711 U128.new(0xfcb2cb35e702af78, 0x5cda735244c3d43e), // 5^177
712 U128.new(0x9defbf01b061adab, 0x3a0888136afa64a7), // 5^178
713 U128.new(0xc56baec21c7a1916, 0x88aaa1845b8fdd0), // 5^179
714 U128.new(0xf6c69a72a3989f5b, 0x8aad549e57273d45), // 5^180
715 U128.new(0x9a3c2087a63f6399, 0x36ac54e2f678864b), // 5^181
716 U128.new(0xc0cb28a98fcf3c7f, 0x84576a1bb416a7dd), // 5^182
717 U128.new(0xf0fdf2d3f3c30b9f, 0x656d44a2a11c51d5), // 5^183
718 U128.new(0x969eb7c47859e743, 0x9f644ae5a4b1b325), // 5^184
719 U128.new(0xbc4665b596706114, 0x873d5d9f0dde1fee), // 5^185
720 U128.new(0xeb57ff22fc0c7959, 0xa90cb506d155a7ea), // 5^186
721 U128.new(0x9316ff75dd87cbd8, 0x9a7f12442d588f2), // 5^187
722 U128.new(0xb7dcbf5354e9bece, 0xc11ed6d538aeb2f), // 5^188
723 U128.new(0xe5d3ef282a242e81, 0x8f1668c8a86da5fa), // 5^189
724 U128.new(0x8fa475791a569d10, 0xf96e017d694487bc), // 5^190
725 U128.new(0xb38d92d760ec4455, 0x37c981dcc395a9ac), // 5^191
726 U128.new(0xe070f78d3927556a, 0x85bbe253f47b1417), // 5^192
727 U128.new(0x8c469ab843b89562, 0x93956d7478ccec8e), // 5^193
728 U128.new(0xaf58416654a6babb, 0x387ac8d1970027b2), // 5^194
729 U128.new(0xdb2e51bfe9d0696a, 0x6997b05fcc0319e), // 5^195
730 U128.new(0x88fcf317f22241e2, 0x441fece3bdf81f03), // 5^196
731 U128.new(0xab3c2fddeeaad25a, 0xd527e81cad7626c3), // 5^197
732 U128.new(0xd60b3bd56a5586f1, 0x8a71e223d8d3b074), // 5^198
733 U128.new(0x85c7056562757456, 0xf6872d5667844e49), // 5^199
734 U128.new(0xa738c6bebb12d16c, 0xb428f8ac016561db), // 5^200
735 U128.new(0xd106f86e69d785c7, 0xe13336d701beba52), // 5^201
736 U128.new(0x82a45b450226b39c, 0xecc0024661173473), // 5^202
737 U128.new(0xa34d721642b06084, 0x27f002d7f95d0190), // 5^203
738 U128.new(0xcc20ce9bd35c78a5, 0x31ec038df7b441f4), // 5^204
739 U128.new(0xff290242c83396ce, 0x7e67047175a15271), // 5^205
740 U128.new(0x9f79a169bd203e41, 0xf0062c6e984d386), // 5^206
741 U128.new(0xc75809c42c684dd1, 0x52c07b78a3e60868), // 5^207
742 U128.new(0xf92e0c3537826145, 0xa7709a56ccdf8a82), // 5^208
743 U128.new(0x9bbcc7a142b17ccb, 0x88a66076400bb691), // 5^209
744 U128.new(0xc2abf989935ddbfe, 0x6acff893d00ea435), // 5^210
745 U128.new(0xf356f7ebf83552fe, 0x583f6b8c4124d43), // 5^211
746 U128.new(0x98165af37b2153de, 0xc3727a337a8b704a), // 5^212
747 U128.new(0xbe1bf1b059e9a8d6, 0x744f18c0592e4c5c), // 5^213
748 U128.new(0xeda2ee1c7064130c, 0x1162def06f79df73), // 5^214
749 U128.new(0x9485d4d1c63e8be7, 0x8addcb5645ac2ba8), // 5^215
750 U128.new(0xb9a74a0637ce2ee1, 0x6d953e2bd7173692), // 5^216
751 U128.new(0xe8111c87c5c1ba99, 0xc8fa8db6ccdd0437), // 5^217
752 U128.new(0x910ab1d4db9914a0, 0x1d9c9892400a22a2), // 5^218
753 U128.new(0xb54d5e4a127f59c8, 0x2503beb6d00cab4b), // 5^219
754 U128.new(0xe2a0b5dc971f303a, 0x2e44ae64840fd61d), // 5^220
755 U128.new(0x8da471a9de737e24, 0x5ceaecfed289e5d2), // 5^221
756 U128.new(0xb10d8e1456105dad, 0x7425a83e872c5f47), // 5^222
757 U128.new(0xdd50f1996b947518, 0xd12f124e28f77719), // 5^223
758 U128.new(0x8a5296ffe33cc92f, 0x82bd6b70d99aaa6f), // 5^224
759 U128.new(0xace73cbfdc0bfb7b, 0x636cc64d1001550b), // 5^225
760 U128.new(0xd8210befd30efa5a, 0x3c47f7e05401aa4e), // 5^226
761 U128.new(0x8714a775e3e95c78, 0x65acfaec34810a71), // 5^227
762 U128.new(0xa8d9d1535ce3b396, 0x7f1839a741a14d0d), // 5^228
763 U128.new(0xd31045a8341ca07c, 0x1ede48111209a050), // 5^229
764 U128.new(0x83ea2b892091e44d, 0x934aed0aab460432), // 5^230
765 U128.new(0xa4e4b66b68b65d60, 0xf81da84d5617853f), // 5^231
766 U128.new(0xce1de40642e3f4b9, 0x36251260ab9d668e), // 5^232
767 U128.new(0x80d2ae83e9ce78f3, 0xc1d72b7c6b426019), // 5^233
768 U128.new(0xa1075a24e4421730, 0xb24cf65b8612f81f), // 5^234
769 U128.new(0xc94930ae1d529cfc, 0xdee033f26797b627), // 5^235
770 U128.new(0xfb9b7cd9a4a7443c, 0x169840ef017da3b1), // 5^236
771 U128.new(0x9d412e0806e88aa5, 0x8e1f289560ee864e), // 5^237
772 U128.new(0xc491798a08a2ad4e, 0xf1a6f2bab92a27e2), // 5^238
773 U128.new(0xf5b5d7ec8acb58a2, 0xae10af696774b1db), // 5^239
774 U128.new(0x9991a6f3d6bf1765, 0xacca6da1e0a8ef29), // 5^240
775 U128.new(0xbff610b0cc6edd3f, 0x17fd090a58d32af3), // 5^241
776 U128.new(0xeff394dcff8a948e, 0xddfc4b4cef07f5b0), // 5^242
777 U128.new(0x95f83d0a1fb69cd9, 0x4abdaf101564f98e), // 5^243
778 U128.new(0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f1), // 5^244
779 U128.new(0xea53df5fd18d5513, 0x84c86189216dc5ed), // 5^245
780 U128.new(0x92746b9be2f8552c, 0x32fd3cf5b4e49bb4), // 5^246
781 U128.new(0xb7118682dbb66a77, 0x3fbc8c33221dc2a1), // 5^247
782 U128.new(0xe4d5e82392a40515, 0xfabaf3feaa5334a), // 5^248
783 U128.new(0x8f05b1163ba6832d, 0x29cb4d87f2a7400e), // 5^249
784 U128.new(0xb2c71d5bca9023f8, 0x743e20e9ef511012), // 5^250
785 U128.new(0xdf78e4b2bd342cf6, 0x914da9246b255416), // 5^251
786 U128.new(0x8bab8eefb6409c1a, 0x1ad089b6c2f7548e), // 5^252
787 U128.new(0xae9672aba3d0c320, 0xa184ac2473b529b1), // 5^253
788 U128.new(0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741e), // 5^254
789 U128.new(0x8865899617fb1871, 0x7e2fa67c7a658892), // 5^255
790 U128.new(0xaa7eebfb9df9de8d, 0xddbb901b98feeab7), // 5^256
791 U128.new(0xd51ea6fa85785631, 0x552a74227f3ea565), // 5^257
792 U128.new(0x8533285c936b35de, 0xd53a88958f87275f), // 5^258
793 U128.new(0xa67ff273b8460356, 0x8a892abaf368f137), // 5^259
794 U128.new(0xd01fef10a657842c, 0x2d2b7569b0432d85), // 5^260
795 U128.new(0x8213f56a67f6b29b, 0x9c3b29620e29fc73), // 5^261
796 U128.new(0xa298f2c501f45f42, 0x8349f3ba91b47b8f), // 5^262
797 U128.new(0xcb3f2f7642717713, 0x241c70a936219a73), // 5^263
798 U128.new(0xfe0efb53d30dd4d7, 0xed238cd383aa0110), // 5^264
799 U128.new(0x9ec95d1463e8a506, 0xf4363804324a40aa), // 5^265
800 U128.new(0xc67bb4597ce2ce48, 0xb143c6053edcd0d5), // 5^266
801 U128.new(0xf81aa16fdc1b81da, 0xdd94b7868e94050a), // 5^267
802 U128.new(0x9b10a4e5e9913128, 0xca7cf2b4191c8326), // 5^268
803 U128.new(0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f0), // 5^269
804 U128.new(0xf24a01a73cf2dccf, 0xbc633b39673c8cec), // 5^270
805 U128.new(0x976e41088617ca01, 0xd5be0503e085d813), // 5^271
806 U128.new(0xbd49d14aa79dbc82, 0x4b2d8644d8a74e18), // 5^272
807 U128.new(0xec9c459d51852ba2, 0xddf8e7d60ed1219e), // 5^273
808 U128.new(0x93e1ab8252f33b45, 0xcabb90e5c942b503), // 5^274
809 U128.new(0xb8da1662e7b00a17, 0x3d6a751f3b936243), // 5^275
810 U128.new(0xe7109bfba19c0c9d, 0xcc512670a783ad4), // 5^276
811 U128.new(0x906a617d450187e2, 0x27fb2b80668b24c5), // 5^277
812 U128.new(0xb484f9dc9641e9da, 0xb1f9f660802dedf6), // 5^278
813 U128.new(0xe1a63853bbd26451, 0x5e7873f8a0396973), // 5^279
814 U128.new(0x8d07e33455637eb2, 0xdb0b487b6423e1e8), // 5^280
815 U128.new(0xb049dc016abc5e5f, 0x91ce1a9a3d2cda62), // 5^281
816 U128.new(0xdc5c5301c56b75f7, 0x7641a140cc7810fb), // 5^282
817 U128.new(0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9d), // 5^283
818 U128.new(0xac2820d9623bf429, 0x546345fa9fbdcd44), // 5^284
819 U128.new(0xd732290fbacaf133, 0xa97c177947ad4095), // 5^285
820 U128.new(0x867f59a9d4bed6c0, 0x49ed8eabcccc485d), // 5^286
821 U128.new(0xa81f301449ee8c70, 0x5c68f256bfff5a74), // 5^287
822 U128.new(0xd226fc195c6a2f8c, 0x73832eec6fff3111), // 5^288
823 U128.new(0x83585d8fd9c25db7, 0xc831fd53c5ff7eab), // 5^289
824 U128.new(0xa42e74f3d032f525, 0xba3e7ca8b77f5e55), // 5^290
825 U128.new(0xcd3a1230c43fb26f, 0x28ce1bd2e55f35eb), // 5^291
826 U128.new(0x80444b5e7aa7cf85, 0x7980d163cf5b81b3), // 5^292
827 U128.new(0xa0555e361951c366, 0xd7e105bcc332621f), // 5^293
828 U128.new(0xc86ab5c39fa63440, 0x8dd9472bf3fefaa7), // 5^294
829 U128.new(0xfa856334878fc150, 0xb14f98f6f0feb951), // 5^295
830 U128.new(0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d3), // 5^296
831 U128.new(0xc3b8358109e84f07, 0xa862f80ec4700c8), // 5^297
832 U128.new(0xf4a642e14c6262c8, 0xcd27bb612758c0fa), // 5^298
833 U128.new(0x98e7e9cccfbd7dbd, 0x8038d51cb897789c), // 5^299
834 U128.new(0xbf21e44003acdd2c, 0xe0470a63e6bd56c3), // 5^300
835 U128.new(0xeeea5d5004981478, 0x1858ccfce06cac74), // 5^301
836 U128.new(0x95527a5202df0ccb, 0xf37801e0c43ebc8), // 5^302
837 U128.new(0xbaa718e68396cffd, 0xd30560258f54e6ba), // 5^303
838 U128.new(0xe950df20247c83fd, 0x47c6b82ef32a2069), // 5^304
839 U128.new(0x91d28b7416cdd27e, 0x4cdc331d57fa5441), // 5^305
840 U128.new(0xb6472e511c81471d, 0xe0133fe4adf8e952), // 5^306
841 U128.new(0xe3d8f9e563a198e5, 0x58180fddd97723a6), // 5^307
842 U128.new(0x8e679c2f5e44ff8f, 0x570f09eaa7ea7648), // 5^308
843};
lib/std/fmt/parse_float/convert_fast.zig created+130
......@@ -0,0 +1,130 @@
1//! Representation of a float as the signficant digits and exponent.
2//! The fast path algorithm using machine-sized integers and floats.
3//!
4//! This only works if both the mantissa and the exponent can be exactly
5//! represented as a machine float, since IEE-754 guarantees no rounding
6//! will occur.
7//!
8//! There is an exception: disguised fast-path cases, where we can shift
9//! powers-of-10 from the exponent to the significant digits.
10
11const std = @import("std");
12const math = std.math;
13const common = @import("common.zig");
14const FloatInfo = @import("FloatInfo.zig");
15const Number = common.Number;
16const floatFromU64 = common.floatFromU64;
17
18fn isFastPath(comptime T: type, n: Number(T)) bool {
19 const info = FloatInfo.from(T);
20
21 return info.min_exponent_fast_path <= n.exponent and
22 n.exponent <= info.max_exponent_fast_path_disguised and
23 n.mantissa <= info.max_mantissa_fast_path and
24 !n.many_digits;
25}
26
27// upper bound for tables is floor(mantissaDigits(T) / log2(5))
28// for f64 this is floor(53 / log2(5)) = 22.
29//
30// Must have max_disguised_fast_path - max_exponent_fast_path entries. (82 - 48 = 34 for f128)
31fn fastPow10(comptime T: type, i: usize) T {
32 return switch (T) {
33 f16 => ([8]f16{
34 1e0, 1e1, 1e2, 1e3, 1e4, 0, 0, 0,
35 })[i & 7],
36
37 f32 => ([16]f32{
38 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7,
39 1e8, 1e9, 1e10, 0, 0, 0, 0, 0,
40 })[i & 15],
41
42 f64 => ([32]f64{
43 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7,
44 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15,
45 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22, 0,
46 0, 0, 0, 0, 0, 0, 0, 0,
47 })[i & 31],
48
49 f128 => ([64]f128{
50 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7,
51 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15,
52 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22, 1e23,
53 1e24, 1e25, 1e26, 1e27, 1e28, 1e29, 1e30, 1e31,
54 1e32, 1e33, 1e34, 1e35, 1e36, 1e37, 1e38, 1e39,
55 1e40, 1e41, 1e42, 1e43, 1e44, 1e45, 1e46, 1e47,
56 1e48, 0, 0, 0, 0, 0, 0, 0,
57 0, 0, 0, 0, 0, 0, 0, 0,
58 })[i & 63],
59
60 else => unreachable,
61 };
62}
63
64fn fastIntPow10(comptime T: type, i: usize) T {
65 return switch (T) {
66 u64 => ([16]u64{
67 1, 10, 100, 1000,
68 10000, 100000, 1000000, 10000000,
69 100000000, 1000000000, 10000000000, 100000000000,
70 1000000000000, 10000000000000, 100000000000000, 1000000000000000,
71 })[i],
72
73 u128 => ([35]u128{
74 1, 10,
75 100, 1000,
76 10000, 100000,
77 1000000, 10000000,
78 100000000, 1000000000,
79 10000000000, 100000000000,
80 1000000000000, 10000000000000,
81 100000000000000, 1000000000000000,
82 10000000000000000, 100000000000000000,
83 1000000000000000000, 10000000000000000000,
84 100000000000000000000, 1000000000000000000000,
85 10000000000000000000000, 100000000000000000000000,
86 1000000000000000000000000, 10000000000000000000000000,
87 100000000000000000000000000, 1000000000000000000000000000,
88 10000000000000000000000000000, 100000000000000000000000000000,
89 1000000000000000000000000000000, 10000000000000000000000000000000,
90 100000000000000000000000000000000, 1000000000000000000000000000000000,
91 10000000000000000000000000000000000,
92 })[i],
93
94 else => unreachable,
95 };
96}
97
98pub fn convertFast(comptime T: type, n: Number(T)) ?T {
99 const MantissaT = common.mantissaType(T);
100
101 if (!isFastPath(T, n)) {
102 return null;
103 }
104
105 // TODO: x86 (no SSE/SSE2) requires x87 FPU to be setup correctly with fldcw
106 const info = FloatInfo.from(T);
107
108 var value: T = 0;
109 if (n.exponent <= info.max_exponent_fast_path) {
110 // normal fast path
111 value = @intToFloat(T, n.mantissa);
112 value = if (n.exponent < 0)
113 value / fastPow10(T, @intCast(usize, -n.exponent))
114 else
115 value * fastPow10(T, @intCast(usize, n.exponent));
116 } else {
117 // disguised fast path
118 const shift = n.exponent - info.max_exponent_fast_path;
119 const mantissa = math.mul(MantissaT, n.mantissa, fastIntPow10(MantissaT, @intCast(usize, shift))) catch return null;
120 if (mantissa > info.max_mantissa_fast_path) {
121 return null;
122 }
123 value = @intToFloat(T, mantissa) * fastPow10(T, info.max_exponent_fast_path);
124 }
125
126 if (n.negative) {
127 value = -value;
128 }
129 return value;
130}
lib/std/fmt/parse_float/convert_hex.zig created+89
......@@ -0,0 +1,89 @@
1//! Conversion of hex-float representation into an accurate value.
2//
3// Derived from golang strconv/atof.go.
4
5const std = @import("std");
6const math = std.math;
7const common = @import("common.zig");
8const Number = common.Number;
9const floatFromUnsigned = common.floatFromUnsigned;
10
11// converts the form 0xMMM.NNNpEEE.
12//
13// MMM.NNN = mantissa
14// EEE = exponent
15//
16// MMM.NNN is stored as an integer, the exponent is offset.
17pub fn convertHex(comptime T: type, n_: Number(T)) T {
18 const MantissaT = common.mantissaType(T);
19 var n = n_;
20
21 if (n.mantissa == 0) {
22 return if (n.negative) -0.0 else 0.0;
23 }
24
25 const max_exp = math.floatExponentMax(T);
26 const min_exp = math.floatExponentMin(T);
27 const mantissa_bits = math.floatMantissaBits(T);
28 const exp_bits = math.floatExponentBits(T);
29 const exp_bias = min_exp - 1;
30
31 // mantissa now implicitly divided by 2^mantissa_bits
32 n.exponent += mantissa_bits;
33
34 // Shift mantissa and exponent to bring representation into float range.
35 // Eventually we want a mantissa with a leading 1-bit followed by mantbits other bits.
36 // For rounding, we need two more, where the bottom bit represents
37 // whether that bit or any later bit was non-zero.
38 // (If the mantissa has already lost non-zero bits, trunc is true,
39 // and we OR in a 1 below after shifting left appropriately.)
40 while (n.mantissa != 0 and n.mantissa >> (mantissa_bits + 2) == 0) {
41 n.mantissa <<= 1;
42 n.exponent -= 1;
43 }
44 if (n.many_digits) {
45 n.mantissa |= 1;
46 }
47 while (n.mantissa >> (1 + mantissa_bits + 2) != 0) {
48 n.mantissa = (n.mantissa >> 1) | (n.mantissa & 1);
49 n.exponent += 1;
50 }
51
52 // If exponent is too negative,
53 // denormalize in hopes of making it representable.
54 // (The -2 is for the rounding bits.)
55 while (n.mantissa > 1 and n.exponent < min_exp - 2) {
56 n.mantissa = (n.mantissa >> 1) | (n.mantissa & 1);
57 n.exponent += 1;
58 }
59
60 // Round using two bottom bits.
61 var round = n.mantissa & 3;
62 n.mantissa >>= 2;
63 round |= n.mantissa & 1; // round to even (round up if mantissa is odd)
64 n.exponent += 2;
65 if (round == 3) {
66 n.mantissa += 1;
67 if (n.mantissa == 1 << (1 + mantissa_bits)) {
68 n.mantissa >>= 1;
69 n.exponent += 1;
70 }
71 }
72
73 // Denormal or zero
74 if (n.mantissa >> mantissa_bits == 0) {
75 n.exponent = exp_bias;
76 }
77
78 // Infinity and range error
79 if (n.exponent > max_exp) {
80 return math.inf(T);
81 }
82
83 var bits = n.mantissa & ((1 << mantissa_bits) - 1);
84 bits |= @intCast(MantissaT, (n.exponent - exp_bias) & ((1 << exp_bits) - 1)) << mantissa_bits;
85 if (n.negative) {
86 bits |= 1 << (mantissa_bits + exp_bits);
87 }
88 return floatFromUnsigned(T, MantissaT, bits);
89}
lib/std/fmt/parse_float/convert_slow.zig created+114
......@@ -0,0 +1,114 @@
1const std = @import("std");
2const math = std.math;
3const common = @import("common.zig");
4const BiasedFp = common.BiasedFp;
5const Decimal = @import("decimal.zig").Decimal;
6const mantissaType = common.mantissaType;
7
8const max_shift = 60;
9const num_powers = 19;
10const powers = [_]u8{ 0, 3, 6, 9, 13, 16, 19, 23, 26, 29, 33, 36, 39, 43, 46, 49, 53, 56, 59 };
11
12pub fn getShift(n: usize) usize {
13 return if (n < num_powers) powers[n] else max_shift;
14}
15
16/// Parse the significant digits and biased, binary exponent of a float.
17///
18/// This is a fallback algorithm that uses a big-integer representation
19/// of the float, and therefore is considerably slower than faster
20/// approximations. However, it will always determine how to round
21/// the significant digits to the nearest machine float, allowing
22/// use to handle near half-way cases.
23///
24/// Near half-way cases are halfway between two consecutive machine floats.
25/// For example, the float `16777217.0` has a bitwise representation of
26/// `100000000000000000000000 1`. Rounding to a single-precision float,
27/// the trailing `1` is truncated. Using round-nearest, tie-even, any
28/// value above `16777217.0` must be rounded up to `16777218.0`, while
29/// any value before or equal to `16777217.0` must be rounded down
30/// to `16777216.0`. These near-halfway conversions therefore may require
31/// a large number of digits to unambiguously determine how to round.
32///
33/// The algorithms described here are based on "Processing Long Numbers Quickly",
34/// available here: <https://arxiv.org/pdf/2101.11408.pdf#section.11>.
35pub fn convertSlow(comptime T: type, s: []const u8) BiasedFp(T) {
36 const MantissaT = mantissaType(T);
37 const min_exponent = -(1 << (math.floatExponentBits(T) - 1)) + 1;
38 const infinite_power = (1 << math.floatExponentBits(T)) - 1;
39 const mantissa_explicit_bits = math.floatMantissaBits(T);
40
41 var d = Decimal(T).parse(s); // no need to recheck underscores
42 if (d.num_digits == 0 or d.decimal_point < Decimal(T).min_exponent) {
43 return BiasedFp(T).zero();
44 } else if (d.decimal_point >= Decimal(T).max_exponent) {
45 return BiasedFp(T).inf(T);
46 }
47
48 var exp2: i32 = 0;
49 // Shift right toward (1/2 .. 1]
50 while (d.decimal_point > 0) {
51 const n = @intCast(usize, d.decimal_point);
52 const shift = getShift(n);
53 d.rightShift(shift);
54 if (d.decimal_point < -Decimal(T).decimal_point_range) {
55 return BiasedFp(T).zero();
56 }
57 exp2 += @intCast(i32, shift);
58 }
59 // Shift left toward (1/2 .. 1]
60 while (d.decimal_point <= 0) {
61 const shift = blk: {
62 if (d.decimal_point == 0) {
63 break :blk switch (d.digits[0]) {
64 5...9 => break,
65 0, 1 => @as(usize, 2),
66 else => 1,
67 };
68 } else {
69 const n = @intCast(usize, -d.decimal_point);
70 break :blk getShift(n);
71 }
72 };
73 d.leftShift(shift);
74 if (d.decimal_point > Decimal(T).decimal_point_range) {
75 return BiasedFp(T).inf(T);
76 }
77 exp2 -= @intCast(i32, shift);
78 }
79 // We are now in the range [1/2 .. 1] but the binary format uses [1 .. 2]
80 exp2 -= 1;
81 while (min_exponent + 1 > exp2) {
82 var n = @intCast(usize, (min_exponent + 1) - exp2);
83 if (n > max_shift) {
84 n = max_shift;
85 }
86 d.rightShift(n);
87 exp2 += @intCast(i32, n);
88 }
89 if (exp2 - min_exponent >= infinite_power) {
90 return BiasedFp(T).inf(T);
91 }
92
93 // Shift the decimal to the hidden bit, and then round the value
94 // to get the high mantissa+1 bits.
95 d.leftShift(mantissa_explicit_bits + 1);
96 var mantissa = d.round();
97 if (mantissa >= (@as(MantissaT, 1) << (mantissa_explicit_bits + 1))) {
98 // Rounding up overflowed to the carry bit, need to
99 // shift back to the hidden bit.
100 d.rightShift(1);
101 exp2 += 1;
102 mantissa = d.round();
103 if ((exp2 - min_exponent) >= infinite_power) {
104 return BiasedFp(T).inf(T);
105 }
106 }
107 var power2 = exp2 - min_exponent;
108 if (mantissa < (@as(MantissaT, 1) << mantissa_explicit_bits)) {
109 power2 -= 1;
110 }
111 // Zero out all the bits above the explicit mantissa bits.
112 mantissa &= (@as(MantissaT, 1) << mantissa_explicit_bits) - 1;
113 return .{ .f = mantissa, .e = power2 };
114}
lib/std/fmt/parse_float/decimal.zig created+493
......@@ -0,0 +1,493 @@
1const std = @import("std");
2const math = std.math;
3const common = @import("common.zig");
4const FloatStream = @import("FloatStream.zig");
5const isEightDigits = @import("common.zig").isEightDigits;
6const mantissaType = common.mantissaType;
7
8// Arbitrary-precision decimal class for fallback algorithms.
9//
10// This is only used if the fast-path (native floats) and
11// the Eisel-Lemire algorithm are unable to unambiguously
12// determine the float.
13//
14// The technique used is "Simple Decimal Conversion", developed
15// by Nigel Tao and Ken Thompson. A detailed description of the
16// algorithm can be found in "ParseNumberF64 by Simple Decimal Conversion",
17// available online: <https://nigeltao.github.io/blog/2020/parse-number-f64-simple.html>.
18//
19// Big-decimal implementation. We do not use the big.Int routines since we only require a maximum
20// fixed region of memory. Further, we require only a small subset of operations.
21//
22// This accepts a floating point parameter and will generate a Decimal which can correctly parse
23// the input with sufficient accuracy. Internally this means either a u64 mantissa (f16, f32 or f64)
24// or a u128 mantissa (f128).
25pub fn Decimal(comptime T: type) type {
26 const MantissaT = mantissaType(T);
27 std.debug.assert(MantissaT == u64 or MantissaT == u128);
28
29 return struct {
30 const Self = @This();
31
32 /// The maximum number of digits required to unambiguously round a float.
33 ///
34 /// For a double-precision IEEE-754 float, this required 767 digits,
35 /// so we store the max digits + 1.
36 ///
37 /// We can exactly represent a float in radix `b` from radix 2 if
38 /// `b` is divisible by 2. This function calculates the exact number of
39 /// digits required to exactly represent that float.
40 ///
41 /// According to the "Handbook of Floating Point Arithmetic",
42 /// for IEEE754, with emin being the min exponent, p2 being the
43 /// precision, and b being the radix, the number of digits follows as:
44 ///
45 /// `−emin + p2 + ⌊(emin + 1) log(2, b) − log(1 − 2^(−p2), b)⌋`
46 ///
47 /// For f32, this follows as:
48 /// emin = -126
49 /// p2 = 24
50 ///
51 /// For f64, this follows as:
52 /// emin = -1022
53 /// p2 = 53
54 ///
55 /// For f128, this follows as:
56 /// emin = -16383
57 /// p2 = 112
58 ///
59 /// In Python:
60 /// `-emin + p2 + math.floor((emin+ 1)*math.log(2, b)-math.log(1-2**(-p2), b))`
61 pub const max_digits = if (MantissaT == u64) 768 else 11564;
62 /// The max digits that can be exactly represented in a 64-bit integer.
63 pub const max_digits_without_overflow = if (MantissaT == u64) 19 else 38;
64 pub const decimal_point_range = if (MantissaT == u64) 2047 else 32767;
65 pub const min_exponent = if (MantissaT == u64) -324 else -4966;
66 pub const max_exponent = if (MantissaT == u64) 310 else 4933;
67 pub const max_decimal_digits = if (MantissaT == u64) 18 else 37;
68
69 /// The number of significant digits in the decimal.
70 num_digits: usize,
71 /// The offset of the decimal point in the significant digits.
72 decimal_point: i32,
73 /// If the number of significant digits stored in the decimal is truncated.
74 truncated: bool,
75 /// buffer of the raw digits, in the range [0, 9].
76 digits: [max_digits]u8,
77
78 pub fn new() Self {
79 return .{
80 .num_digits = 0,
81 .decimal_point = 0,
82 .truncated = false,
83 .digits = [_]u8{0} ** max_digits,
84 };
85 }
86
87 /// Append a digit to the buffer
88 pub fn tryAddDigit(self: *Self, digit: u8) void {
89 if (self.num_digits < max_digits) {
90 self.digits[self.num_digits] = digit;
91 }
92 self.num_digits += 1;
93 }
94
95 /// Trim trailing zeroes from the buffer
96 pub fn trim(self: *Self) void {
97 // All of the following calls to `Self::trim` can't panic because:
98 //
99 // 1. `parse_decimal` sets `num_digits` to a max of `max_digits`.
100 // 2. `right_shift` sets `num_digits` to `write_index`, which is bounded by `num_digits`.
101 // 3. `left_shift` `num_digits` to a max of `max_digits`.
102 //
103 // Trim is only called in `right_shift` and `left_shift`.
104 std.debug.assert(self.num_digits <= max_digits);
105 while (self.num_digits != 0 and self.digits[self.num_digits - 1] == 0) {
106 self.num_digits -= 1;
107 }
108 }
109
110 pub fn round(self: *Self) MantissaT {
111 if (self.num_digits == 0 or self.decimal_point < 0) {
112 return 0;
113 } else if (self.decimal_point > max_decimal_digits) {
114 return math.maxInt(MantissaT);
115 }
116
117 const dp = @intCast(usize, self.decimal_point);
118 var n: MantissaT = 0;
119
120 var i: usize = 0;
121 while (i < dp) : (i += 1) {
122 n *= 10;
123 if (i < self.num_digits) {
124 n += @as(MantissaT, self.digits[i]);
125 }
126 }
127
128 var round_up = false;
129 if (dp < self.num_digits) {
130 round_up = self.digits[dp] >= 5;
131 if (self.digits[dp] == 5 and dp + 1 == self.num_digits) {
132 round_up = self.truncated or ((dp != 0) and (1 & self.digits[dp - 1] != 0));
133 }
134 }
135 if (round_up) {
136 n += 1;
137 }
138 return n;
139 }
140
141 /// Computes decimal * 2^shift.
142 pub fn leftShift(self: *Self, shift: usize) void {
143 if (self.num_digits == 0) {
144 return;
145 }
146 const num_new_digits = self.numberOfDigitsLeftShift(shift);
147 var read_index = self.num_digits;
148 var write_index = self.num_digits + num_new_digits;
149 var n: MantissaT = 0;
150 while (read_index != 0) {
151 read_index -= 1;
152 write_index -= 1;
153 n += math.shl(MantissaT, self.digits[read_index], shift);
154
155 const quotient = n / 10;
156 const remainder = n - (10 * quotient);
157 if (write_index < max_digits) {
158 self.digits[write_index] = @intCast(u8, remainder);
159 } else if (remainder > 0) {
160 self.truncated = true;
161 }
162 n = quotient;
163 }
164 while (n > 0) {
165 write_index -= 1;
166
167 const quotient = n / 10;
168 const remainder = n - (10 * quotient);
169 if (write_index < max_digits) {
170 self.digits[write_index] = @intCast(u8, remainder);
171 } else if (remainder > 0) {
172 self.truncated = true;
173 }
174 n = quotient;
175 }
176
177 self.num_digits += num_new_digits;
178 if (self.num_digits > max_digits) {
179 self.num_digits = max_digits;
180 }
181 self.decimal_point += @intCast(i32, num_new_digits);
182 self.trim();
183 }
184
185 /// Computes decimal * 2^-shift.
186 pub fn rightShift(self: *Self, shift: usize) void {
187 var read_index: usize = 0;
188 var write_index: usize = 0;
189 var n: MantissaT = 0;
190 while (math.shr(MantissaT, n, shift) == 0) {
191 if (read_index < self.num_digits) {
192 n = (10 * n) + self.digits[read_index];
193 read_index += 1;
194 } else if (n == 0) {
195 return;
196 } else {
197 while (math.shr(MantissaT, n, shift) == 0) {
198 n *= 10;
199 read_index += 1;
200 }
201 break;
202 }
203 }
204
205 self.decimal_point -= @intCast(i32, read_index) - 1;
206 if (self.decimal_point < -decimal_point_range) {
207 self.num_digits = 0;
208 self.decimal_point = 0;
209 self.truncated = false;
210 return;
211 }
212
213 const mask = math.shl(MantissaT, 1, shift) - 1;
214 while (read_index < self.num_digits) {
215 const new_digit = @intCast(u8, math.shr(MantissaT, n, shift));
216 n = (10 * (n & mask)) + self.digits[read_index];
217 read_index += 1;
218 self.digits[write_index] = new_digit;
219 write_index += 1;
220 }
221 while (n > 0) {
222 const new_digit = @intCast(u8, math.shr(MantissaT, n, shift));
223 n = 10 * (n & mask);
224 if (write_index < max_digits) {
225 self.digits[write_index] = new_digit;
226 write_index += 1;
227 } else if (new_digit > 0) {
228 self.truncated = true;
229 }
230 }
231 self.num_digits = write_index;
232 self.trim();
233 }
234
235 /// Parse a bit integer representation of the float as a decimal.
236 // We do not verify underscores in this path since these will have been verified
237 // via parse.parseNumber so can assume the number is well-formed.
238 // This code-path does not have to handle hex-floats since these will always be handled via another
239 // function prior to this.
240 pub fn parse(s: []const u8) Self {
241 var d = Self.new();
242 var stream = FloatStream.init(s);
243
244 stream.skipChars2('0', '_');
245 while (stream.scanDigit(10)) |digit| {
246 d.tryAddDigit(digit);
247 }
248
249 if (stream.firstIs('.')) {
250 stream.advance(1);
251 const marker = stream.offsetTrue();
252
253 // Skip leading zeroes
254 if (d.num_digits == 0) {
255 stream.skipChars('0');
256 }
257
258 while (stream.hasLen(8) and d.num_digits + 8 < max_digits) {
259 const v = stream.readU64Unchecked();
260 if (!isEightDigits(v)) {
261 break;
262 }
263 std.mem.writeIntSliceLittle(u64, d.digits[d.num_digits..], v - 0x3030_3030_3030_3030);
264 d.num_digits += 8;
265 stream.advance(8);
266 }
267
268 while (stream.scanDigit(10)) |digit| {
269 d.tryAddDigit(digit);
270 }
271 d.decimal_point = @intCast(i32, marker) - @intCast(i32, stream.offsetTrue());
272 }
273 if (d.num_digits != 0) {
274 // Ignore trailing zeros if any
275 var n_trailing_zeros: usize = 0;
276 var i = stream.offsetTrue() - 1;
277 while (true) {
278 if (s[i] == '0') {
279 n_trailing_zeros += 1;
280 } else if (s[i] != '.') {
281 break;
282 }
283
284 i -= 1;
285 if (i == 0) break;
286 }
287 d.decimal_point += @intCast(i32, n_trailing_zeros);
288 d.num_digits -= n_trailing_zeros;
289 d.decimal_point += @intCast(i32, d.num_digits);
290 if (d.num_digits > max_digits) {
291 d.truncated = true;
292 d.num_digits = max_digits;
293 }
294 }
295 if (stream.firstIsLower('e')) {
296 stream.advance(1);
297 var neg_exp = false;
298 if (stream.firstIs('-')) {
299 neg_exp = true;
300 stream.advance(1);
301 } else if (stream.firstIs('+')) {
302 stream.advance(1);
303 }
304 var exp_num: i32 = 0;
305 while (stream.scanDigit(10)) |digit| {
306 if (exp_num < 0x10000) {
307 exp_num = 10 * exp_num + digit;
308 }
309 }
310 d.decimal_point += if (neg_exp) -exp_num else exp_num;
311 }
312
313 var i = d.num_digits;
314 while (i < max_digits_without_overflow) : (i += 1) {
315 d.digits[i] = 0;
316 }
317
318 return d;
319 }
320
321 // Compute the number decimal digits introduced by a base-2 shift. This is performed
322 // by storing the leading digits of 1/2^i = 5^i and using these along with the cut-off
323 // value to quickly determine the decimal shift from binary.
324 //
325 // See also https://github.com/golang/go/blob/go1.15.3/src/strconv/decimal.go#L163 for
326 // another description of the method.
327 pub fn numberOfDigitsLeftShift(self: *Self, shift: usize) usize {
328 const ShiftCutoff = struct {
329 delta: u8,
330 cutoff: []const u8,
331 };
332
333 // Leading digits of 1/2^i = 5^i.
334 //
335 // ```
336 // import math
337 //
338 // bits = 128
339 // for i in range(bits):
340 // log2 = math.log(2)/math.log(10)
341 // print(f'.{{ .delta = {int(log2*i+1)}, .cutoff = "{5**i}" }}, // {2**i}')
342 // ```
343 const pow2_to_pow5_table = [_]ShiftCutoff{
344 .{ .delta = 0, .cutoff = "" },
345 .{ .delta = 1, .cutoff = "5" }, // 2
346 .{ .delta = 1, .cutoff = "25" }, // 4
347 .{ .delta = 1, .cutoff = "125" }, // 8
348 .{ .delta = 2, .cutoff = "625" }, // 16
349 .{ .delta = 2, .cutoff = "3125" }, // 32
350 .{ .delta = 2, .cutoff = "15625" }, // 64
351 .{ .delta = 3, .cutoff = "78125" }, // 128
352 .{ .delta = 3, .cutoff = "390625" }, // 256
353 .{ .delta = 3, .cutoff = "1953125" }, // 512
354 .{ .delta = 4, .cutoff = "9765625" }, // 1024
355 .{ .delta = 4, .cutoff = "48828125" }, // 2048
356 .{ .delta = 4, .cutoff = "244140625" }, // 4096
357 .{ .delta = 4, .cutoff = "1220703125" }, // 8192
358 .{ .delta = 5, .cutoff = "6103515625" }, // 16384
359 .{ .delta = 5, .cutoff = "30517578125" }, // 32768
360 .{ .delta = 5, .cutoff = "152587890625" }, // 65536
361 .{ .delta = 6, .cutoff = "762939453125" }, // 131072
362 .{ .delta = 6, .cutoff = "3814697265625" }, // 262144
363 .{ .delta = 6, .cutoff = "19073486328125" }, // 524288
364 .{ .delta = 7, .cutoff = "95367431640625" }, // 1048576
365 .{ .delta = 7, .cutoff = "476837158203125" }, // 2097152
366 .{ .delta = 7, .cutoff = "2384185791015625" }, // 4194304
367 .{ .delta = 7, .cutoff = "11920928955078125" }, // 8388608
368 .{ .delta = 8, .cutoff = "59604644775390625" }, // 16777216
369 .{ .delta = 8, .cutoff = "298023223876953125" }, // 33554432
370 .{ .delta = 8, .cutoff = "1490116119384765625" }, // 67108864
371 .{ .delta = 9, .cutoff = "7450580596923828125" }, // 134217728
372 .{ .delta = 9, .cutoff = "37252902984619140625" }, // 268435456
373 .{ .delta = 9, .cutoff = "186264514923095703125" }, // 536870912
374 .{ .delta = 10, .cutoff = "931322574615478515625" }, // 1073741824
375 .{ .delta = 10, .cutoff = "4656612873077392578125" }, // 2147483648
376 .{ .delta = 10, .cutoff = "23283064365386962890625" }, // 4294967296
377 .{ .delta = 10, .cutoff = "116415321826934814453125" }, // 8589934592
378 .{ .delta = 11, .cutoff = "582076609134674072265625" }, // 17179869184
379 .{ .delta = 11, .cutoff = "2910383045673370361328125" }, // 34359738368
380 .{ .delta = 11, .cutoff = "14551915228366851806640625" }, // 68719476736
381 .{ .delta = 12, .cutoff = "72759576141834259033203125" }, // 137438953472
382 .{ .delta = 12, .cutoff = "363797880709171295166015625" }, // 274877906944
383 .{ .delta = 12, .cutoff = "1818989403545856475830078125" }, // 549755813888
384 .{ .delta = 13, .cutoff = "9094947017729282379150390625" }, // 1099511627776
385 .{ .delta = 13, .cutoff = "45474735088646411895751953125" }, // 2199023255552
386 .{ .delta = 13, .cutoff = "227373675443232059478759765625" }, // 4398046511104
387 .{ .delta = 13, .cutoff = "1136868377216160297393798828125" }, // 8796093022208
388 .{ .delta = 14, .cutoff = "5684341886080801486968994140625" }, // 17592186044416
389 .{ .delta = 14, .cutoff = "28421709430404007434844970703125" }, // 35184372088832
390 .{ .delta = 14, .cutoff = "142108547152020037174224853515625" }, // 70368744177664
391 .{ .delta = 15, .cutoff = "710542735760100185871124267578125" }, // 140737488355328
392 .{ .delta = 15, .cutoff = "3552713678800500929355621337890625" }, // 281474976710656
393 .{ .delta = 15, .cutoff = "17763568394002504646778106689453125" }, // 562949953421312
394 .{ .delta = 16, .cutoff = "88817841970012523233890533447265625" }, // 1125899906842624
395 .{ .delta = 16, .cutoff = "444089209850062616169452667236328125" }, // 2251799813685248
396 .{ .delta = 16, .cutoff = "2220446049250313080847263336181640625" }, // 4503599627370496
397 .{ .delta = 16, .cutoff = "11102230246251565404236316680908203125" }, // 9007199254740992
398 .{ .delta = 17, .cutoff = "55511151231257827021181583404541015625" }, // 18014398509481984
399 .{ .delta = 17, .cutoff = "277555756156289135105907917022705078125" }, // 36028797018963968
400 .{ .delta = 17, .cutoff = "1387778780781445675529539585113525390625" }, // 72057594037927936
401 .{ .delta = 18, .cutoff = "6938893903907228377647697925567626953125" }, // 144115188075855872
402 .{ .delta = 18, .cutoff = "34694469519536141888238489627838134765625" }, // 288230376151711744
403 .{ .delta = 18, .cutoff = "173472347597680709441192448139190673828125" }, // 576460752303423488
404 .{ .delta = 19, .cutoff = "867361737988403547205962240695953369140625" }, // 1152921504606846976
405 .{ .delta = 19, .cutoff = "4336808689942017736029811203479766845703125" }, // 2305843009213693952
406 .{ .delta = 19, .cutoff = "21684043449710088680149056017398834228515625" }, // 4611686018427387904
407 .{ .delta = 19, .cutoff = "108420217248550443400745280086994171142578125" }, // 9223372036854775808
408 .{ .delta = 20, .cutoff = "542101086242752217003726400434970855712890625" }, // 18446744073709551616
409 .{ .delta = 20, .cutoff = "2710505431213761085018632002174854278564453125" }, // 36893488147419103232
410 .{ .delta = 20, .cutoff = "13552527156068805425093160010874271392822265625" }, // 73786976294838206464
411 .{ .delta = 21, .cutoff = "67762635780344027125465800054371356964111328125" }, // 147573952589676412928
412 .{ .delta = 21, .cutoff = "338813178901720135627329000271856784820556640625" }, // 295147905179352825856
413 .{ .delta = 21, .cutoff = "1694065894508600678136645001359283924102783203125" }, // 590295810358705651712
414 .{ .delta = 22, .cutoff = "8470329472543003390683225006796419620513916015625" }, // 1180591620717411303424
415 .{ .delta = 22, .cutoff = "42351647362715016953416125033982098102569580078125" }, // 2361183241434822606848
416 .{ .delta = 22, .cutoff = "211758236813575084767080625169910490512847900390625" }, // 4722366482869645213696
417 .{ .delta = 22, .cutoff = "1058791184067875423835403125849552452564239501953125" }, // 9444732965739290427392
418 .{ .delta = 23, .cutoff = "5293955920339377119177015629247762262821197509765625" }, // 18889465931478580854784
419 .{ .delta = 23, .cutoff = "26469779601696885595885078146238811314105987548828125" }, // 37778931862957161709568
420 .{ .delta = 23, .cutoff = "132348898008484427979425390731194056570529937744140625" }, // 75557863725914323419136
421 .{ .delta = 24, .cutoff = "661744490042422139897126953655970282852649688720703125" }, // 151115727451828646838272
422 .{ .delta = 24, .cutoff = "3308722450212110699485634768279851414263248443603515625" }, // 302231454903657293676544
423 .{ .delta = 24, .cutoff = "16543612251060553497428173841399257071316242218017578125" }, // 604462909807314587353088
424 .{ .delta = 25, .cutoff = "82718061255302767487140869206996285356581211090087890625" }, // 1208925819614629174706176
425 .{ .delta = 25, .cutoff = "413590306276513837435704346034981426782906055450439453125" }, // 2417851639229258349412352
426 .{ .delta = 25, .cutoff = "2067951531382569187178521730174907133914530277252197265625" }, // 4835703278458516698824704
427 .{ .delta = 25, .cutoff = "10339757656912845935892608650874535669572651386260986328125" }, // 9671406556917033397649408
428 .{ .delta = 26, .cutoff = "51698788284564229679463043254372678347863256931304931640625" }, // 19342813113834066795298816
429 .{ .delta = 26, .cutoff = "258493941422821148397315216271863391739316284656524658203125" }, // 38685626227668133590597632
430 .{ .delta = 26, .cutoff = "1292469707114105741986576081359316958696581423282623291015625" }, // 77371252455336267181195264
431 .{ .delta = 27, .cutoff = "6462348535570528709932880406796584793482907116413116455078125" }, // 154742504910672534362390528
432 .{ .delta = 27, .cutoff = "32311742677852643549664402033982923967414535582065582275390625" }, // 309485009821345068724781056
433 .{ .delta = 27, .cutoff = "161558713389263217748322010169914619837072677910327911376953125" }, // 618970019642690137449562112
434 .{ .delta = 28, .cutoff = "807793566946316088741610050849573099185363389551639556884765625" }, // 1237940039285380274899124224
435 .{ .delta = 28, .cutoff = "4038967834731580443708050254247865495926816947758197784423828125" }, // 2475880078570760549798248448
436 .{ .delta = 28, .cutoff = "20194839173657902218540251271239327479634084738790988922119140625" }, // 4951760157141521099596496896
437 .{ .delta = 28, .cutoff = "100974195868289511092701256356196637398170423693954944610595703125" }, // 9903520314283042199192993792
438 .{ .delta = 29, .cutoff = "504870979341447555463506281780983186990852118469774723052978515625" }, // 19807040628566084398385987584
439 .{ .delta = 29, .cutoff = "2524354896707237777317531408904915934954260592348873615264892578125" }, // 39614081257132168796771975168
440 .{ .delta = 29, .cutoff = "12621774483536188886587657044524579674771302961744368076324462890625" }, // 79228162514264337593543950336
441 .{ .delta = 30, .cutoff = "63108872417680944432938285222622898373856514808721840381622314453125" }, // 158456325028528675187087900672
442 .{ .delta = 30, .cutoff = "315544362088404722164691426113114491869282574043609201908111572265625" }, // 316912650057057350374175801344
443 .{ .delta = 30, .cutoff = "1577721810442023610823457130565572459346412870218046009540557861328125" }, // 633825300114114700748351602688
444 .{ .delta = 31, .cutoff = "7888609052210118054117285652827862296732064351090230047702789306640625" }, // 1267650600228229401496703205376
445 .{ .delta = 31, .cutoff = "39443045261050590270586428264139311483660321755451150238513946533203125" }, // 2535301200456458802993406410752
446 .{ .delta = 31, .cutoff = "197215226305252951352932141320696557418301608777255751192569732666015625" }, // 5070602400912917605986812821504
447 .{ .delta = 32, .cutoff = "986076131526264756764660706603482787091508043886278755962848663330078125" }, // 10141204801825835211973625643008
448 .{ .delta = 32, .cutoff = "4930380657631323783823303533017413935457540219431393779814243316650390625" }, // 20282409603651670423947251286016
449 .{ .delta = 32, .cutoff = "24651903288156618919116517665087069677287701097156968899071216583251953125" }, // 40564819207303340847894502572032
450 .{ .delta = 32, .cutoff = "123259516440783094595582588325435348386438505485784844495356082916259765625" }, // 81129638414606681695789005144064
451 .{ .delta = 33, .cutoff = "616297582203915472977912941627176741932192527428924222476780414581298828125" }, // 162259276829213363391578010288128
452 .{ .delta = 33, .cutoff = "3081487911019577364889564708135883709660962637144621112383902072906494140625" }, // 324518553658426726783156020576256
453 .{ .delta = 33, .cutoff = "15407439555097886824447823540679418548304813185723105561919510364532470703125" }, // 649037107316853453566312041152512
454 .{ .delta = 34, .cutoff = "77037197775489434122239117703397092741524065928615527809597551822662353515625" }, // 1298074214633706907132624082305024
455 .{ .delta = 34, .cutoff = "385185988877447170611195588516985463707620329643077639047987759113311767578125" }, // 2596148429267413814265248164610048
456 .{ .delta = 34, .cutoff = "1925929944387235853055977942584927318538101648215388195239938795566558837890625" }, // 5192296858534827628530496329220096
457 .{ .delta = 35, .cutoff = "9629649721936179265279889712924636592690508241076940976199693977832794189453125" }, // 10384593717069655257060992658440192
458 .{ .delta = 35, .cutoff = "48148248609680896326399448564623182963452541205384704880998469889163970947265625" }, // 20769187434139310514121985316880384
459 .{ .delta = 35, .cutoff = "240741243048404481631997242823115914817262706026923524404992349445819854736328125" }, // 41538374868278621028243970633760768
460 .{ .delta = 35, .cutoff = "1203706215242022408159986214115579574086313530134617622024961747229099273681640625" }, // 83076749736557242056487941267521536
461 .{ .delta = 36, .cutoff = "6018531076210112040799931070577897870431567650673088110124808736145496368408203125" }, // 166153499473114484112975882535043072
462 .{ .delta = 36, .cutoff = "30092655381050560203999655352889489352157838253365440550624043680727481842041015625" }, // 332306998946228968225951765070086144
463 .{ .delta = 36, .cutoff = "150463276905252801019998276764447446760789191266827202753120218403637409210205078125" }, // 664613997892457936451903530140172288
464 .{ .delta = 37, .cutoff = "752316384526264005099991383822237233803945956334136013765601092018187046051025390625" }, // 1329227995784915872903807060280344576
465 .{ .delta = 37, .cutoff = "3761581922631320025499956919111186169019729781670680068828005460090935230255126953125" }, // 2658455991569831745807614120560689152
466 .{ .delta = 37, .cutoff = "18807909613156600127499784595555930845098648908353400344140027300454676151275634765625" }, // 5316911983139663491615228241121378304
467 .{ .delta = 38, .cutoff = "94039548065783000637498922977779654225493244541767001720700136502273380756378173828125" }, // 10633823966279326983230456482242756608
468 .{ .delta = 38, .cutoff = "470197740328915003187494614888898271127466222708835008603500682511366903781890869140625" }, // 21267647932558653966460912964485513216
469 .{ .delta = 38, .cutoff = "2350988701644575015937473074444491355637331113544175043017503412556834518909454345703125" }, // 42535295865117307932921825928971026432
470 .{ .delta = 38, .cutoff = "11754943508222875079687365372222456778186655567720875215087517062784172594547271728515625" }, // 85070591730234615865843651857942052864
471 .{ .delta = 39, .cutoff = "58774717541114375398436826861112283890933277838604376075437585313920862972736358642578125" }, // 170141183460469231731687303715884105728
472 };
473
474 std.debug.assert(shift < pow2_to_pow5_table.len);
475 const x = pow2_to_pow5_table[shift];
476
477 // Compare leading digits of current to check if lexicographically less than cutoff.
478 for (x.cutoff) |p5, i| {
479 if (i >= self.num_digits) {
480 return x.delta - 1;
481 } else if (self.digits[i] == p5 - '0') { // digits are stored as integers
482 continue;
483 } else if (self.digits[i] < p5 - '0') {
484 return x.delta - 1;
485 } else {
486 return x.delta;
487 }
488 return x.delta;
489 }
490 return x.delta;
491 }
492 };
493}
lib/std/fmt/parse_float/parse.zig created+291
......@@ -0,0 +1,291 @@
1const std = @import("std");
2const common = @import("common.zig");
3const FloatStream = @import("FloatStream.zig");
4const isEightDigits = common.isEightDigits;
5const Number = common.Number;
6
7/// Parse 8 digits, loaded as bytes in little-endian order.
8///
9/// This uses the trick where every digit is in [0x030, 0x39],
10/// and therefore can be parsed in 3 multiplications, much
11/// faster than the normal 8.
12///
13/// This is based off the algorithm described in "Fast numeric string to
14/// int", available here: <https://johnnylee-sde.github.io/Fast-numeric-string-to-int/>.
15fn parse8Digits(v_: u64) u64 {
16 var v = v_;
17 const mask = 0x0000_00ff_0000_00ff;
18 const mul1 = 0x000f_4240_0000_0064;
19 const mul2 = 0x0000_2710_0000_0001;
20 v -= 0x3030_3030_3030_3030;
21 v = (v * 10) + (v >> 8); // will not overflow, fits in 63 bits
22 const v1 = (v & mask) *% mul1;
23 const v2 = ((v >> 16) & mask) *% mul2;
24 return @as(u64, @truncate(u32, (v1 +% v2) >> 32));
25}
26
27/// Parse digits until a non-digit character is found.
28fn tryParseDigits(comptime T: type, stream: *FloatStream, x: *T, comptime base: u8) void {
29 // Try to parse 8 digits at a time, using an optimized algorithm.
30 // This only supports decimal digits.
31 if (base == 10) {
32 while (stream.hasLen(8)) {
33 const v = stream.readU64Unchecked();
34 if (!isEightDigits(v)) {
35 break;
36 }
37
38 x.* = x.* *% 1_0000_0000 +% parse8Digits(v);
39 stream.advance(8);
40 }
41 }
42
43 while (stream.scanDigit(base)) |digit| {
44 x.* *%= base;
45 x.* +%= digit;
46 }
47}
48
49fn min_n_digit_int(comptime T: type, digit_count: usize) T {
50 var n: T = 1;
51 var i: usize = 1;
52 while (i < digit_count) : (i += 1) n *= 10;
53 return n;
54}
55
56/// Parse up to N digits
57fn tryParseNDigits(comptime T: type, stream: *FloatStream, x: *T, comptime base: u8, comptime n: usize) void {
58 while (x.* < min_n_digit_int(T, n)) {
59 if (stream.scanDigit(base)) |digit| {
60 x.* *%= base;
61 x.* +%= digit;
62 } else {
63 break;
64 }
65 }
66}
67
68/// Parse the scientific notation component of a float.
69fn parseScientific(stream: *FloatStream) ?i64 {
70 var exponent: i64 = 0;
71 var negative = false;
72
73 if (stream.first()) |c| {
74 negative = c == '-';
75 if (c == '-' or c == '+') {
76 stream.advance(1);
77 }
78 }
79 if (stream.firstIsDigit(10)) {
80 while (stream.scanDigit(10)) |digit| {
81 // no overflows here, saturate well before overflow
82 if (exponent < 0x1000_0000) {
83 exponent = 10 * exponent + digit;
84 }
85 }
86
87 return if (negative) -exponent else exponent;
88 }
89
90 return null;
91}
92
93const ParseInfo = struct {
94 // 10 or 16
95 base: u8,
96 // 10^19 fits in u64, 16^16 fits in u64
97 max_mantissa_digits: usize,
98 // e.g. e or p (E and P also checked)
99 exp_char_lower: u8,
100};
101
102fn parsePartialNumberBase(comptime T: type, stream: *FloatStream, negative: bool, n: *usize, comptime info: ParseInfo) ?Number(T) {
103 const MantissaT = common.mantissaType(T);
104
105 // parse initial digits before dot
106 var mantissa: MantissaT = 0;
107 tryParseDigits(MantissaT, stream, &mantissa, info.base);
108 var int_end = stream.offsetTrue();
109 var n_digits = @intCast(isize, stream.offsetTrue());
110
111 // handle dot with the following digits
112 var exponent: i64 = 0;
113 if (stream.firstIs('.')) {
114 stream.advance(1);
115 const marker = stream.offsetTrue();
116 tryParseDigits(MantissaT, stream, &mantissa, info.base);
117 const n_after_dot = stream.offsetTrue() - marker;
118 exponent = -@intCast(i64, n_after_dot);
119 n_digits += @intCast(isize, n_after_dot);
120 }
121
122 // adjust required shift to offset mantissa for base-16 (2^4)
123 if (info.base == 16) {
124 exponent *= 4;
125 }
126
127 if (n_digits == 0) {
128 return null;
129 }
130
131 // handle scientific format
132 var exp_number: i64 = 0;
133 if (stream.firstIsLower(info.exp_char_lower)) {
134 stream.advance(1);
135 exp_number = parseScientific(stream) orelse return null;
136 exponent += exp_number;
137 }
138
139 const len = stream.offset; // length must be complete parsed length
140 n.* = len;
141
142 if (stream.underscore_count > 0 and !validUnderscores(stream.slice, info.base)) {
143 return null;
144 }
145
146 // common case with not many digits
147 if (n_digits <= info.max_mantissa_digits) {
148 return Number(T){
149 .exponent = exponent,
150 .mantissa = mantissa,
151 .negative = negative,
152 .many_digits = false,
153 .hex = info.base == 16,
154 };
155 }
156
157 n_digits -= info.max_mantissa_digits;
158 var many_digits = false;
159 stream.reset(); // re-parse from beginning
160 while (stream.firstIs3('0', '.', '_')) {
161 // '0' = '.' + 2
162 const next = stream.firstUnchecked();
163 if (next != '_') {
164 n_digits -= @intCast(isize, next -| ('0' - 1));
165 } else {
166 stream.underscore_count += 1;
167 }
168 stream.advance(1);
169 }
170 if (n_digits > 0) {
171 // at this point we have more than max_mantissa_digits significant digits, let's try again
172 many_digits = true;
173 mantissa = 0;
174 stream.reset();
175 tryParseNDigits(MantissaT, stream, &mantissa, info.base, info.max_mantissa_digits);
176
177 exponent = blk: {
178 if (mantissa >= min_n_digit_int(MantissaT, info.max_mantissa_digits)) {
179 // big int
180 break :blk @intCast(i64, int_end) - @intCast(i64, stream.offsetTrue());
181 } else {
182 // the next byte must be present and be '.'
183 // We know this is true because we had more than 19
184 // digits previously, so we overflowed a 64-bit integer,
185 // but parsing only the integral digits produced less
186 // than 19 digits. That means we must have a decimal
187 // point, and at least 1 fractional digit.
188 stream.advance(1);
189 var marker = stream.offsetTrue();
190 tryParseNDigits(MantissaT, stream, &mantissa, info.base, info.max_mantissa_digits);
191 break :blk @intCast(i64, marker) - @intCast(i64, stream.offsetTrue());
192 }
193 };
194 // add back the explicit part
195 exponent += exp_number;
196 }
197
198 return Number(T){
199 .exponent = exponent,
200 .mantissa = mantissa,
201 .negative = negative,
202 .many_digits = many_digits,
203 .hex = info.base == 16,
204 };
205}
206
207/// Parse a partial, non-special floating point number.
208///
209/// This creates a representation of the float as the
210/// significant digits and the decimal exponent.
211fn parsePartialNumber(comptime T: type, s: []const u8, negative: bool, n: *usize) ?Number(T) {
212 std.debug.assert(s.len != 0);
213 var stream = FloatStream.init(s);
214 const MantissaT = common.mantissaType(T);
215
216 if (stream.hasLen(2) and stream.atUnchecked(0) == '0' and std.ascii.toLower(stream.atUnchecked(1)) == 'x') {
217 stream.advance(2);
218 return parsePartialNumberBase(T, &stream, negative, n, .{
219 .base = 16,
220 .max_mantissa_digits = if (MantissaT == u64) 16 else 32,
221 .exp_char_lower = 'p',
222 });
223 } else {
224 return parsePartialNumberBase(T, &stream, negative, n, .{
225 .base = 10,
226 .max_mantissa_digits = if (MantissaT == u64) 19 else 38,
227 .exp_char_lower = 'e',
228 });
229 }
230}
231
232pub fn parseNumber(comptime T: type, s: []const u8, negative: bool) ?Number(T) {
233 var consumed: usize = 0;
234 if (parsePartialNumber(T, s, negative, &consumed)) |number| {
235 // must consume entire float (no trailing data)
236 if (s.len == consumed) {
237 return number;
238 }
239 }
240 return null;
241}
242
243fn parsePartialInfOrNan(comptime T: type, s: []const u8, negative: bool, n: *usize) ?T {
244 // inf/infinity; infxxx should only consume inf.
245 if (std.ascii.startsWithIgnoreCase(s, "inf")) {
246 n.* = 3;
247 if (std.ascii.startsWithIgnoreCase(s[3..], "inity")) {
248 n.* = 8;
249 }
250
251 return if (!negative) std.math.inf(T) else -std.math.inf(T);
252 }
253
254 if (std.ascii.startsWithIgnoreCase(s, "nan")) {
255 n.* = 3;
256 return std.math.nan(T);
257 }
258
259 return null;
260}
261
262pub fn parseInfOrNan(comptime T: type, s: []const u8, negative: bool) ?T {
263 var consumed: usize = 0;
264 if (parsePartialInfOrNan(T, s, negative, &consumed)) |special| {
265 if (s.len == consumed) {
266 return special;
267 }
268 }
269 return null;
270}
271
272pub fn validUnderscores(s: []const u8, comptime base: u8) bool {
273 var i: usize = 0;
274 while (i < s.len) : (i += 1) {
275 if (s[i] == '_') {
276 // underscore at start of end
277 if (i == 0 or i + 1 == s.len) {
278 return false;
279 }
280 // consecutive underscores
281 if (!common.isDigit(s[i - 1], base) or !common.isDigit(s[i + 1], base)) {
282 return false;
283 }
284
285 // next is guaranteed a digit, skip an extra
286 i += 1;
287 }
288 }
289
290 return true;
291}
lib/std/fmt/parse_float/parse_float.zig created+64
......@@ -0,0 +1,64 @@
1const std = @import("std");
2const parse = @import("parse.zig");
3const parseNumber = parse.parseNumber;
4const parseInfOrNan = parse.parseInfOrNan;
5const convertFast = @import("convert_fast.zig").convertFast;
6const convertEiselLemire = @import("convert_eisel_lemire.zig").convertEiselLemire;
7const convertSlow = @import("convert_slow.zig").convertSlow;
8const convertHex = @import("convert_hex.zig").convertHex;
9
10const optimize = true;
11
12pub const ParseFloatError = error{
13 InvalidCharacter,
14};
15
16pub fn parseFloat(comptime T: type, s: []const u8) ParseFloatError!T {
17 if (s.len == 0) {
18 return error.InvalidCharacter;
19 }
20
21 var i: usize = 0;
22 const negative = s[i] == '-';
23 if (s[i] == '-' or s[i] == '+') {
24 i += 1;
25 }
26 if (s.len == i) {
27 return error.InvalidCharacter;
28 }
29
30 const n = parse.parseNumber(T, s[i..], negative) orelse {
31 return parse.parseInfOrNan(T, s[i..], negative) orelse error.InvalidCharacter;
32 };
33
34 if (n.hex) {
35 return convertHex(T, n);
36 }
37
38 if (optimize) {
39 if (convertFast(T, n)) |f| {
40 return f;
41 }
42
43 if (T == f16 or T == f32 or T == f64) {
44 // If significant digits were truncated, then we can have rounding error
45 // only if `mantissa + 1` produces a different result. We also avoid
46 // redundantly using the Eisel-Lemire algorithm if it was unable to
47 // correctly round on the first pass.
48 if (convertEiselLemire(T, n.exponent, n.mantissa)) |bf| {
49 if (!n.many_digits) {
50 return bf.toFloat(T, n.negative);
51 }
52 if (convertEiselLemire(T, n.exponent, n.mantissa + 1)) |bf2| {
53 if (bf.eql(bf2)) {
54 return bf.toFloat(T, n.negative);
55 }
56 }
57 }
58 }
59 }
60
61 // Unable to correctly round the float using the Eisel-Lemire algorithm.
62 // Fallback to a slower, but always correct algorithm.
63 return convertSlow(T, s[i..]).toFloat(T, negative);
64}
lib/std/fmt/parse_hex_float.zig deleted-347
......@@ -1,347 +0,0 @@
1// The rounding logic is inspired by LLVM's APFloat and Go's atofHex
2// implementation.
3
4const std = @import("std");
5const ascii = std.ascii;
6const fmt = std.fmt;
7const math = std.math;
8const testing = std.testing;
9
10const assert = std.debug.assert;
11
12pub fn parseHexFloat(comptime T: type, s: []const u8) !T {
13 assert(@typeInfo(T) == .Float);
14
15 const TBits = std.meta.Int(.unsigned, @typeInfo(T).Float.bits);
16
17 const mantissa_bits = math.floatMantissaBits(T);
18 const exponent_bits = math.floatExponentBits(T);
19 const exponent_min = math.floatExponentMin(T);
20 const exponent_max = math.floatExponentMax(T);
21
22 const exponent_bias = exponent_max;
23 const sign_shift = mantissa_bits + exponent_bits;
24
25 if (s.len == 0)
26 return error.InvalidCharacter;
27
28 if (ascii.eqlIgnoreCase(s, "nan")) {
29 return math.nan(T);
30 } else if (ascii.eqlIgnoreCase(s, "inf") or ascii.eqlIgnoreCase(s, "+inf")) {
31 return math.inf(T);
32 } else if (ascii.eqlIgnoreCase(s, "-inf")) {
33 return -math.inf(T);
34 }
35
36 var negative: bool = false;
37 var exp_negative: bool = false;
38
39 var mantissa: u128 = 0;
40 var exponent: i16 = 0;
41 var frac_scale: i16 = 0;
42
43 const State = enum {
44 MaybeSign,
45 Prefix,
46 LeadingIntegerDigit,
47 IntegerDigit,
48 MaybeDot,
49 LeadingFractionDigit,
50 FractionDigit,
51 ExpPrefix,
52 MaybeExpSign,
53 ExpDigit,
54 };
55
56 var state = State.MaybeSign;
57
58 var i: usize = 0;
59 while (i < s.len) {
60 const c = s[i];
61
62 switch (state) {
63 .MaybeSign => {
64 state = .Prefix;
65
66 if (c == '+') {
67 i += 1;
68 } else if (c == '-') {
69 negative = true;
70 i += 1;
71 }
72 },
73 .Prefix => {
74 state = .LeadingIntegerDigit;
75
76 // Match both 0x and 0X.
77 if (i + 2 > s.len or s[i] != '0' or s[i + 1] | 32 != 'x')
78 return error.InvalidCharacter;
79 i += 2;
80 },
81 .LeadingIntegerDigit => {
82 if (c == '0') {
83 // Skip leading zeros.
84 i += 1;
85 } else if (c == '_') {
86 return error.InvalidCharacter;
87 } else {
88 state = .IntegerDigit;
89 }
90 },
91 .IntegerDigit => {
92 if (ascii.isXDigit(c)) {
93 if (mantissa >= math.maxInt(u128) / 16)
94 return error.Overflow;
95 mantissa *%= 16;
96 mantissa += try fmt.charToDigit(c, 16);
97 i += 1;
98 } else if (c == '_') {
99 i += 1;
100 } else {
101 state = .MaybeDot;
102 }
103 },
104 .MaybeDot => {
105 if (c == '.') {
106 state = .LeadingFractionDigit;
107 i += 1;
108 } else state = .ExpPrefix;
109 },
110 .LeadingFractionDigit => {
111 if (c == '_') {
112 return error.InvalidCharacter;
113 } else state = .FractionDigit;
114 },
115 .FractionDigit => {
116 if (ascii.isXDigit(c)) {
117 if (mantissa < math.maxInt(u128) / 16) {
118 mantissa *%= 16;
119 mantissa +%= try fmt.charToDigit(c, 16);
120 frac_scale += 1;
121 } else if (c != '0') {
122 return error.Overflow;
123 }
124 i += 1;
125 } else if (c == '_') {
126 i += 1;
127 } else {
128 state = .ExpPrefix;
129 }
130 },
131 .ExpPrefix => {
132 state = .MaybeExpSign;
133 // Match both p and P.
134 if (c | 32 != 'p')
135 return error.InvalidCharacter;
136 i += 1;
137 },
138 .MaybeExpSign => {
139 state = .ExpDigit;
140
141 if (c == '+') {
142 i += 1;
143 } else if (c == '-') {
144 exp_negative = true;
145 i += 1;
146 }
147 },
148 .ExpDigit => {
149 if (ascii.isXDigit(c)) {
150 if (exponent >= math.maxInt(i16) / 10)
151 return error.Overflow;
152 exponent *%= 10;
153 exponent +%= try fmt.charToDigit(c, 10);
154 i += 1;
155 } else if (c == '_') {
156 i += 1;
157 } else {
158 return error.InvalidCharacter;
159 }
160 },
161 }
162 }
163
164 if (exp_negative)
165 exponent *= -1;
166
167 // Bring the decimal part to the left side of the decimal dot.
168 exponent -= frac_scale * 4;
169
170 if (mantissa == 0) {
171 // Signed zero.
172 return if (negative) -0.0 else 0.0;
173 }
174
175 // Divide by 2^mantissa_bits to right-align the mantissa in the fractional
176 // part.
177 exponent += mantissa_bits;
178
179 // Keep around two extra bits to correctly round any value that doesn't fit
180 // the available mantissa bits. The result LSB serves as Guard bit, the
181 // following one is the Round bit and the last one is the Sticky bit,
182 // computed by OR-ing all the dropped bits.
183
184 // Normalize by aligning the implicit one bit.
185 while (mantissa >> (mantissa_bits + 2) == 0) {
186 mantissa <<= 1;
187 exponent -= 1;
188 }
189
190 // Normalize again by dropping the excess precision.
191 // Note that the discarded bits are folded into the Sticky bit.
192 while (mantissa >> (mantissa_bits + 2 + 1) != 0) {
193 mantissa = mantissa >> 1 | (mantissa & 1);
194 exponent += 1;
195 }
196
197 // Very small numbers can be possibly represented as denormals, reduce the
198 // exponent as much as possible.
199 while (mantissa != 0 and exponent < exponent_min - 2) {
200 mantissa = mantissa >> 1 | (mantissa & 1);
201 exponent += 1;
202 }
203
204 // Whenever the guard bit is one (G=1) and:
205 // - we've truncated more than 0.5ULP (R=S=1)
206 // - we've truncated exactly 0.5ULP (R=1 S=0)
207 // Were are going to increase the mantissa (round up)
208 const guard_bit_and_half_or_more = (mantissa & 0b110) == 0b110;
209 mantissa >>= 2;
210 exponent += 2;
211
212 if (guard_bit_and_half_or_more) {
213 mantissa += 1;
214 }
215
216 if (mantissa == (1 << (mantissa_bits + 1))) {
217 // Renormalize, if the exponent overflows we'll catch that below.
218 mantissa >>= 1;
219 exponent += 1;
220 }
221
222 if (mantissa >> mantissa_bits == 0) {
223 // This is a denormal number, the biased exponent is zero.
224 exponent = -exponent_bias;
225 }
226
227 if (exponent > exponent_max) {
228 // Overflow, return +inf.
229 return math.inf(T);
230 }
231
232 // Remove the implicit bit.
233 mantissa &= @as(u128, (1 << mantissa_bits) - 1);
234
235 const raw: TBits =
236 (if (negative) @as(TBits, 1) << sign_shift else 0) |
237 @as(TBits, @bitCast(u16, exponent + exponent_bias)) << mantissa_bits |
238 @truncate(TBits, mantissa);
239
240 return @bitCast(T, raw);
241}
242
243test "special" {
244 try testing.expect(math.isNan(try parseHexFloat(f32, "nAn")));
245 try testing.expect(math.isPositiveInf(try parseHexFloat(f32, "iNf")));
246 try testing.expect(math.isPositiveInf(try parseHexFloat(f32, "+Inf")));
247 try testing.expect(math.isNegativeInf(try parseHexFloat(f32, "-iNf")));
248}
249test "zero" {
250 try testing.expectEqual(@as(f32, 0.0), try parseHexFloat(f32, "0x0"));
251 try testing.expectEqual(@as(f32, 0.0), try parseHexFloat(f32, "-0x0"));
252 try testing.expectEqual(@as(f32, 0.0), try parseHexFloat(f32, "0x0p42"));
253 try testing.expectEqual(@as(f32, 0.0), try parseHexFloat(f32, "-0x0.00000p42"));
254 try testing.expectEqual(@as(f32, 0.0), try parseHexFloat(f32, "0x0.00000p666"));
255}
256
257test "f16" {
258 const Case = struct { s: []const u8, v: f16 };
259 const cases: []const Case = &[_]Case{
260 .{ .s = "0x1p0", .v = 1.0 },
261 .{ .s = "-0x1p-1", .v = -0.5 },
262 .{ .s = "0x10p+10", .v = 16384.0 },
263 .{ .s = "0x10p-10", .v = 0.015625 },
264 // Max normalized value.
265 .{ .s = "0x1.ffcp+15", .v = math.floatMax(f16) },
266 .{ .s = "-0x1.ffcp+15", .v = -math.floatMax(f16) },
267 // Min normalized value.
268 .{ .s = "0x1p-14", .v = math.floatMin(f16) },
269 .{ .s = "-0x1p-14", .v = -math.floatMin(f16) },
270 // Min denormal value.
271 .{ .s = "0x1p-24", .v = math.floatTrueMin(f16) },
272 .{ .s = "-0x1p-24", .v = -math.floatTrueMin(f16) },
273 };
274
275 for (cases) |case| {
276 try testing.expectEqual(case.v, try parseHexFloat(f16, case.s));
277 }
278}
279test "f32" {
280 const Case = struct { s: []const u8, v: f32 };
281 const cases: []const Case = &[_]Case{
282 .{ .s = "0x1p0", .v = 1.0 },
283 .{ .s = "-0x1p-1", .v = -0.5 },
284 .{ .s = "0x10p+10", .v = 16384.0 },
285 .{ .s = "0x10p-10", .v = 0.015625 },
286 .{ .s = "0x0.ffffffp128", .v = 0x0.ffffffp128 },
287 .{ .s = "0x0.1234570p-125", .v = 0x0.1234570p-125 },
288 // Max normalized value.
289 .{ .s = "0x1.fffffeP+127", .v = math.floatMax(f32) },
290 .{ .s = "-0x1.fffffeP+127", .v = -math.floatMax(f32) },
291 // Min normalized value.
292 .{ .s = "0x1p-126", .v = math.floatMin(f32) },
293 .{ .s = "-0x1p-126", .v = -math.floatMin(f32) },
294 // Min denormal value.
295 .{ .s = "0x1P-149", .v = math.floatTrueMin(f32) },
296 .{ .s = "-0x1P-149", .v = -math.floatTrueMin(f32) },
297 };
298
299 for (cases) |case| {
300 try testing.expectEqual(case.v, try parseHexFloat(f32, case.s));
301 }
302}
303test "f64" {
304 const Case = struct { s: []const u8, v: f64 };
305 const cases: []const Case = &[_]Case{
306 .{ .s = "0x1p0", .v = 1.0 },
307 .{ .s = "-0x1p-1", .v = -0.5 },
308 .{ .s = "0x10p+10", .v = 16384.0 },
309 .{ .s = "0x10p-10", .v = 0.015625 },
310 // Max normalized value.
311 .{ .s = "0x1.fffffffffffffp+1023", .v = math.floatMax(f64) },
312 .{ .s = "-0x1.fffffffffffffp1023", .v = -math.floatMax(f64) },
313 // Min normalized value.
314 .{ .s = "0x1p-1022", .v = math.floatMin(f64) },
315 .{ .s = "-0x1p-1022", .v = -math.floatMin(f64) },
316 // Min denormalized value.
317 .{ .s = "0x1p-1074", .v = math.floatTrueMin(f64) },
318 .{ .s = "-0x1p-1074", .v = -math.floatTrueMin(f64) },
319 };
320
321 for (cases) |case| {
322 try testing.expectEqual(case.v, try parseHexFloat(f64, case.s));
323 }
324}
325test "f128" {
326 const Case = struct { s: []const u8, v: f128 };
327 const cases: []const Case = &[_]Case{
328 .{ .s = "0x1p0", .v = 1.0 },
329 .{ .s = "-0x1p-1", .v = -0.5 },
330 .{ .s = "0x10p+10", .v = 16384.0 },
331 .{ .s = "0x10p-10", .v = 0.015625 },
332 // Max normalized value.
333 .{ .s = "0xf.fffffffffffffffffffffffffff8p+16380", .v = math.floatMax(f128) },
334 .{ .s = "-0xf.fffffffffffffffffffffffffff8p+16380", .v = -math.floatMax(f128) },
335 // Min normalized value.
336 .{ .s = "0x1p-16382", .v = math.floatMin(f128) },
337 .{ .s = "-0x1p-16382", .v = -math.floatMin(f128) },
338 // // Min denormalized value.
339 .{ .s = "0x1p-16494", .v = math.floatTrueMin(f128) },
340 .{ .s = "-0x1p-16494", .v = -math.floatTrueMin(f128) },
341 .{ .s = "0x1.edcb34a235253948765432134674fp-1", .v = 0x1.edcb34a235253948765432134674fp-1 },
342 };
343
344 for (cases) |case| {
345 try testing.expectEqual(@bitCast(u128, case.v), @bitCast(u128, try parseHexFloat(f128, case.s)));
346 }
347}
lib/std/special/compiler_rt.zig+1
......@@ -813,6 +813,7 @@ comptime {
813813 @export(__floatditf, .{ .name = "__floatdikf", .linkage = linkage });
814814 @export(__floatunditf, .{ .name = "__floatundikf", .linkage = linkage });
815815 @export(__floatunsitf, .{ .name = "__floatunsikf", .linkage = linkage });
816 @export(__floatuntitf, .{ .name = "__floatuntikf", .linkage = linkage });
816817
817818 @export(__letf2, .{ .name = "__eqkf2", .linkage = linkage });
818819 @export(__letf2, .{ .name = "__nekf2", .linkage = linkage });
src/AstGen.zig+1-7
......@@ -6715,13 +6715,7 @@ fn floatLiteral(gz: *GenZir, rl: ResultLoc, node: Ast.Node.Index, sign: Sign) In
67156715
67166716 const main_token = main_tokens[node];
67176717 const bytes = tree.tokenSlice(main_token);
6718 const unsigned_float_number: f128 = if (bytes.len > 2 and bytes[1] == 'x') hex: {
6719 assert(bytes[0] == '0'); // validated by tokenizer
6720 break :hex std.fmt.parseHexFloat(f128, bytes) catch |err| switch (err) {
6721 error.InvalidCharacter => unreachable, // validated by tokenizer
6722 error.Overflow => return astgen.failNode(node, "number literal cannot be represented in a 128-bit floating point", .{}),
6723 };
6724 } else std.fmt.parseFloat(f128, bytes) catch |err| switch (err) {
6718 const unsigned_float_number = std.fmt.parseFloat(f128, bytes) catch |err| switch (err) {
67256719 error.InvalidCharacter => unreachable, // validated by tokenizer
67266720 };
67276721 const float_number = switch (sign) {
test/behavior/math.zig+8-2
......@@ -778,8 +778,14 @@ test "quad hex float literal parsing accurate" {
778778 try expect(@bitCast(u128, f) == 0x40042eab345678439abcdefea5678234);
779779 }
780780 {
781 var f: f128 = 0x1.edcb34a235253948765432134674fp-1;
782 try expect(@bitCast(u128, f) == 0x3ffeedcb34a235253948765432134674);
781 // TODO: modify stage1/parse_f128.c to use round-to-even
782 if (builtin.zig_backend == .stage1) {
783 var f: f128 = 0x1.edcb34a235253948765432134674fp-1;
784 try expect(@bitCast(u128, f) == 0x3ffeedcb34a235253948765432134674); // round-down
785 } else {
786 var f: f128 = 0x1.edcb34a235253948765432134674fp-1;
787 try expect(@bitCast(u128, f) == 0x3ffeedcb34a235253948765432134675); // round-to-even
788 }
783789 }
784790 {
785791 var f: f128 = 0x1.353e45674d89abacc3a2ebf3ff4ffp-50;