fixed 1.11.0

Fixed-point numbers
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360

// Copyright © 2018–2021 Trevor Spiteri

// This library is free software: you can redistribute it and/or
// modify it under the terms of either
//
//   * the Apache License, Version 2.0 or
//   * the MIT License
//
// at your option.
//
// You should have recieved copies of the Apache License and the MIT
// License along with the library. If not, see
// <https://www.apache.org/licenses/LICENSE-2.0> and
// <https://opensource.org/licenses/MIT>.

macro_rules! shift {
    // in case of 128, split shift in two parts to avoid >> 128
    ($SRC:ident, $Fixed:ident<$Frac:ident>) => {
        $Fixed::<$Frac>::from_bits(
            (consts::$SRC.to_bits() >> (64 - $Frac::U32 / 2) >> (64 + $Frac::U32 / 2 - $Frac::U32))
                as _,
        )
    };
    ($SRC:ident, $src_frac_nbits:literal, $Fixed:ident<$Frac:ident>) => {
        $Fixed::<$Frac>::from_bits((consts::$SRC.to_bits() >> ($src_frac_nbits - $Frac::U32)) as _)
    };
}

macro_rules! fixed_const {
    (
        $Fixed:ident[$s_fixed:expr](
            $LeEqU:tt, $s_nbits:expr,
            $s_nbits_m1:expr, $s_nbits_m2:expr, $s_nbits_m3:expr, $s_nbits_m4:expr
        ),
        $LeEqU_C0:tt, $LeEqU_C1:tt, $LeEqU_C2:tt, $LeEqU_C3:tt,
        $Signedness:tt
    ) => {
        comment! {
            "This block contains constants in the range 0 < <i>x</i> < 0.5.

# Examples

```rust
use fixed::{consts, types::extra::U", $s_nbits, ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U", $s_nbits, ">;
assert_eq!(Fix::LOG10_2, Fix::from_num(consts::LOG10_2));
```
";
            impl<Frac: $LeEqU> $Fixed<Frac> {
                /// 1/τ = 0.159154…
                pub const FRAC_1_TAU: $Fixed<Frac> = shift!(FRAC_1_TAU, $Fixed<Frac>);

                /// 2/τ = 0.318309…
                pub const FRAC_2_TAU: $Fixed<Frac> = shift!(FRAC_2_TAU, $Fixed<Frac>);

                /// π/8 = 0.392699…
                pub const FRAC_PI_8: $Fixed<Frac> = shift!(FRAC_PI_8, $Fixed<Frac>);

                /// 1/π = 0.318309…
                pub const FRAC_1_PI: $Fixed<Frac> = shift!(FRAC_1_PI, $Fixed<Frac>);

                /// log<sub>10</sub> 2 = 0.301029…
                pub const LOG10_2: $Fixed<Frac> = shift!(LOG10_2, $Fixed<Frac>);

                /// log<sub>10</sub> e = 0.434294…
                pub const LOG10_E: $Fixed<Frac> = shift!(LOG10_E, $Fixed<Frac>);
            }
        }

        comment! {
            "This block contains constants in the range 0.5 ≤ <i>x</i> < 1.

",
            if_signed_else_empty_str! {
                $Signedness;
                "These constants are not representable in signed
fixed-point numbers with less than 1 integer bit.

"
            },
            "# Examples

```rust
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m1, $s_nbits),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m1, $s_nbits),
            ">;
assert_eq!(Fix::LN_2, Fix::from_num(consts::LN_2));
assert!(0.5 <= Fix::LN_2 && Fix::LN_2 < 1);
```
",
            if_signed_else_empty_str! {
                $Signedness;
                "
The following example fails to compile, since the maximum
representable value with ", $s_nbits, " fractional bits and 0 integer
bits is < 0.5.

```rust,compile_fail
use fixed::{consts, types::extra::U", $s_nbits, ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U", $s_nbits, ">;
let _ = Fix::LN_2;
```
"
            };
            impl<Frac: Unsigned> $Fixed<Frac>
            where
                Frac: IsLessOrEqual<$LeEqU_C0, Output = True>,
            {
                /// τ/8 = 0.785398…
                pub const FRAC_TAU_8: $Fixed<Frac> = shift!(FRAC_TAU_8, $Fixed<Frac>);

                /// τ/12 = 0.523598…
                pub const FRAC_TAU_12: $Fixed<Frac> = shift!(FRAC_TAU_12, $Fixed<Frac>);

                /// 4/τ = 0.636619…
                pub const FRAC_4_TAU: $Fixed<Frac> = shift!(FRAC_4_TAU, $Fixed<Frac>);

                /// π/4 = 0.785398…
                pub const FRAC_PI_4: $Fixed<Frac> = shift!(FRAC_PI_4, $Fixed<Frac>);

                /// π/6 = 0.523598…
                pub const FRAC_PI_6: $Fixed<Frac> = shift!(FRAC_PI_6, $Fixed<Frac>);

                /// 2/π = 0.636619…
                pub const FRAC_2_PI: $Fixed<Frac> = shift!(FRAC_2_PI, $Fixed<Frac>);

                /// 1/√π = 0.564189…
                pub const FRAC_1_SQRT_PI: $Fixed<Frac> = shift!(FRAC_1_SQRT_PI, $Fixed<Frac>);

                /// 1/√2 = 0.707106…
                pub const FRAC_1_SQRT_2: $Fixed<Frac> = shift!(FRAC_1_SQRT_2, $Fixed<Frac>);

                /// 1/√3 = 0.577350…
                pub const FRAC_1_SQRT_3: $Fixed<Frac> = shift!(FRAC_1_SQRT_3, $Fixed<Frac>);

                /// ln 2 = 0.693147…
                pub const LN_2: $Fixed<Frac> = shift!(LN_2, $Fixed<Frac>);

                /// The golden ratio conjugate, Φ = 1/φ = 0.618033…
                pub const FRAC_1_PHI: $Fixed<Frac> = shift!(FRAC_1_PHI, $Fixed<Frac>);

                /// The Euler-Mascheroni constant, γ = 0.577215…
                pub const GAMMA: $Fixed<Frac> = shift!(GAMMA, $Fixed<Frac>);

                /// Catalan’s constant = 0.915965…
                pub const CATALAN: $Fixed<Frac> = shift!(CATALAN, $Fixed<Frac>);
            }
        }

        comment! {
            "This block contains constants in the range 1 ≤ <i>x</i> < 2.

These constants are not representable in ",
            if_signed_unsigned!($Signedness, "signed", "unsigned"),
            " fixed-point numbers with less than ",
            if_signed_unsigned!($Signedness, "2 integer bits", "1 integer bit"),
            ".

# Examples

```rust
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m2, $s_nbits_m1),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m2, $s_nbits_m1),
            ">;
assert_eq!(Fix::LOG2_E, Fix::from_num(consts::LOG2_E));
assert!(1 <= Fix::LOG2_E && Fix::LOG2_E < 2);
```

The following example fails to compile, since the maximum
representable value with ",
            if_signed_unsigned!($Signedness, $s_nbits_m1, $s_nbits),
            " fractional bits and ",
            if_signed_unsigned!($Signedness, "1 integer bit", "0 integer bits"),
            " is < 1.

```rust,compile_fail
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m1, $s_nbits),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m1, $s_nbits),
            ">;
let _ = Fix::LOG2_E;
```
";
            impl<Frac: Unsigned> $Fixed<Frac>
            where
                Frac: IsLessOrEqual<$LeEqU_C1, Output = True>,
            {
                comment! {
                    "One.

# Examples

```rust
use fixed::{types::extra::U4, ", $s_fixed, "};
type Fix = ", $s_fixed, "<U4>;
assert_eq!(Fix::ONE, Fix::from_num(1));
```
";
                    pub const ONE: $Fixed<Frac> =
                        $Fixed::from_bits($Fixed::<Frac>::DELTA.to_bits() << Frac::U32);
                }

                /// τ/4 = 1.57079…
                pub const FRAC_TAU_4: $Fixed<Frac> = shift!(FRAC_TAU_4, 127, $Fixed<Frac>);

                /// τ/6 = 1.04719…
                pub const FRAC_TAU_6: $Fixed<Frac> = shift!(FRAC_TAU_6, 127, $Fixed<Frac>);

                /// π/2 = 1.57079…
                pub const FRAC_PI_2: $Fixed<Frac> = shift!(FRAC_PI_2, 127, $Fixed<Frac>);

                /// π/3 = 1.04719…
                pub const FRAC_PI_3: $Fixed<Frac> = shift!(FRAC_PI_3, 127, $Fixed<Frac>);

                /// √π = 1.77245…
                pub const SQRT_PI: $Fixed<Frac> = shift!(SQRT_PI, 127, $Fixed<Frac>);

                /// 2/√π = 1.12837…
                pub const FRAC_2_SQRT_PI: $Fixed<Frac> = shift!(FRAC_2_SQRT_PI, 127, $Fixed<Frac>);

                /// √2 = 1.41421…
                pub const SQRT_2: $Fixed<Frac> = shift!(SQRT_2, 127, $Fixed<Frac>);

                /// √3 = 1.73205…
                pub const SQRT_3: $Fixed<Frac> = shift!(SQRT_3, 127, $Fixed<Frac>);

                /// √e = 1.64872…
                pub const SQRT_E: $Fixed<Frac> = shift!(SQRT_E, 127, $Fixed<Frac>);

                /// log<sub>2</sub> e = 1.44269…
                pub const LOG2_E: $Fixed<Frac> = shift!(LOG2_E, 127, $Fixed<Frac>);

                /// The golden ratio, φ = 1.61803…
                pub const PHI: $Fixed<Frac> = shift!(PHI, 127, $Fixed<Frac>);

                /// √φ = 1.27201…
                pub const SQRT_PHI: $Fixed<Frac> = shift!(SQRT_PHI, 127, $Fixed<Frac>);
            }
        }

        comment! {
            "This block contains constants in the range 2 ≤ <i>x</i> < 4.

These constants are not representable in ",
            if_signed_unsigned!($Signedness, "signed", "unsigned"),
            " fixed-point numbers with less than ",
            if_signed_unsigned!($Signedness, "3", "2"),
            " integer bits.

# Examples

```rust
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m3, $s_nbits_m2),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m3, $s_nbits_m2),
            ">;
assert_eq!(Fix::E, Fix::from_num(consts::E));
assert!(2 <= Fix::E && Fix::E < 4);
```

The following example fails to compile, since the maximum
representable value with ",
            if_signed_unsigned!($Signedness, $s_nbits_m2, $s_nbits_m1),
            " fractional bits and ",
            if_signed_unsigned!($Signedness, "2 integer bits", "1 integer bit"),
            " is < 2.

```rust,compile_fail
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m2, $s_nbits_m1),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m2, $s_nbits_m1),
            ">;
let _ = Fix::E;
```
";
            impl<Frac: Unsigned> $Fixed<Frac>
            where
                Frac: IsLessOrEqual<$LeEqU_C2, Output = True>,
            {
                /// τ/2 = 3.14159…
                pub const FRAC_TAU_2: $Fixed<Frac> = shift!(FRAC_TAU_2, 126, $Fixed<Frac>);

                /// τ/3 = 2.09439…
                pub const FRAC_TAU_3: $Fixed<Frac> = shift!(FRAC_TAU_3, 126, $Fixed<Frac>);

                /// Archimedes’ constant, π = 3.14159…
                pub const PI: $Fixed<Frac> = shift!(PI, 126, $Fixed<Frac>);

                /// Euler’s number, e = 2.71828…
                pub const E: $Fixed<Frac> = shift!(E, 126, $Fixed<Frac>);

                /// log<sub>2</sub> 10 = 3.32192…
                pub const LOG2_10: $Fixed<Frac> = shift!(LOG2_10, 126, $Fixed<Frac>);

                /// ln 10 = 2.30258…
                pub const LN_10: $Fixed<Frac> = shift!(LN_10, 126, $Fixed<Frac>);
            }
        }

        comment! {
            "This block contains constants in the range 4 ≤ <i>x</i> < 8.

These constants are not representable in ",
            if_signed_unsigned!($Signedness, "signed", "unsigned"),
            " fixed-point numbers with less than ",
            if_signed_unsigned!($Signedness, "4", "3"),
            " integer bits.

# Examples

```rust
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m4, $s_nbits_m3),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m4, $s_nbits_m3),
            ">;
assert_eq!(Fix::TAU, Fix::from_num(consts::TAU));
assert!(4 <= Fix::TAU && Fix::TAU < 8);
```

The following example fails to compile, since the maximum
representable value with ",
            if_signed_unsigned!($Signedness, $s_nbits_m3, $s_nbits_m2),
            " fractional bits and ",
            if_signed_unsigned!($Signedness, "3", "2"),
            " integer bits is < 4.

```rust,compile_fail
use fixed::{consts, types::extra::U",
            if_signed_unsigned!($Signedness, $s_nbits_m3, $s_nbits_m2),
            ", ", $s_fixed, "};
type Fix = ", $s_fixed, "<U",
            if_signed_unsigned!($Signedness, $s_nbits_m3, $s_nbits_m2),
            ">;
let _ = Fix::TAU;
```
";
            impl<Frac: Unsigned> $Fixed<Frac>
            where
                Frac: IsLessOrEqual<$LeEqU_C3, Output = True>,
            {
                /// A turn, τ = 6.28318…
                pub const TAU: $Fixed<Frac> = shift!(TAU, 125, $Fixed<Frac>);
            }
        }
    };
}