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
use crate::constants::{CircleConstants, ScalarConstants};
use crate::core::integer::{FullInt, IntConvert};
use crate::core::undefined::*;
use crate::{Circle, Integer, Scalar};
use core::ops::*;
use i256::I256;
use num_traits::{AsPrimitive, WrappingAdd, WrappingMul, WrappingNeg, WrappingSub};
/// # Scalar to Circle Conversions
///
/// This module provides implementations for creating Circle complex number values from Scalar real number values. It handles combining separate real and imaginary components while maintaining proper normalization and handling special cases.
#[allow(private_bounds)]
impl<
F: Integer
+ FullInt
+ Shl<isize, Output = F>
+ Shr<isize, Output = F>
+ Shl<F, Output = F>
+ Shr<F, Output = F>
+ Shl<E, Output = F>
+ Shr<E, Output = F>
+ WrappingNeg
+ WrappingAdd
+ WrappingMul
+ WrappingSub,
E: Integer
+ FullInt
+ Shl<isize, Output = E>
+ Shr<isize, Output = E>
+ Shl<E, Output = E>
+ Shr<E, Output = E>
+ Shl<F, Output = E>
+ Shr<F, Output = E>
+ WrappingNeg
+ WrappingAdd
+ WrappingMul
+ WrappingSub,
> Circle<F, E>
where
Circle<F, E>: CircleConstants,
Scalar<F, E>: ScalarConstants,
u8: AsPrimitive<F>,
u16: AsPrimitive<F>,
u32: AsPrimitive<F>,
u64: AsPrimitive<F>,
u128: AsPrimitive<F>,
usize: AsPrimitive<F>,
i8: AsPrimitive<F>,
i16: AsPrimitive<F>,
i32: AsPrimitive<F>,
i64: AsPrimitive<F>,
i128: AsPrimitive<F>,
isize: AsPrimitive<F>,
I256: From<F>,
u8: AsPrimitive<E>,
u16: AsPrimitive<E>,
u32: AsPrimitive<E>,
u64: AsPrimitive<E>,
u128: AsPrimitive<E>,
usize: AsPrimitive<E>,
i8: AsPrimitive<E>,
i16: AsPrimitive<E>,
i32: AsPrimitive<E>,
i64: AsPrimitive<E>,
i128: AsPrimitive<E>,
isize: AsPrimitive<E>,
I256: From<E>,
{
/// # Create a Circle from Components
///
/// Constructs a Circle from separate real and imaginary Scalar components, handling complex normalization requirements and special state preservation.
///
/// ## How Component Combination Works
///
/// This method:
/// 0. Aligns both components to share an exponent
/// 1. Preserves special states (Zero, Infinity, exploded, vanished, undefined) appropriately
/// 2. Creates a valid normalized Circle
///
/// ## Special Case Handling
///
/// | Components | Result |
/// |------------|--------|
/// | Either is undefined | First undefined encountered |
/// | Infinity & defined | Infinite Circle |
/// | Defined & Infinity | Infinite Circle |
/// | Zero & any | Returns the non-zero component's state |
/// | Any & Zero | Returns the non-zero component's state |
/// | Exploded & Zero | Preserves the exploded state |
/// | Zero & exploded | Preserves the exploded state |
/// | Vanished & normal | Normal with vanished treated as zero |
/// | Normal & vanished | Normal with vanished treated as zero |
/// | Both exploded | Undefined result (magnitude indeterminate) |
/// | Both vanished | Undefined result (magnitude indeterminate) |
///
/// ## Examples
///
/// ```ignore
/// use spirix::{Circle, Scalar, CircleF5E3, ScalarF5E3};
///
/// // Normal components (1.5 and 2.0 → magnitude = 2.5) let real = ScalarF5E3::from(1.5_f32); let imag = ScalarF5E3::from(2_i32); let z = Circle::<i32, i8>::from_ri(real, imag); assert_eq!(z.magnitude(), 2.5_f32);
///
/// // Special case: Zero & Exploded let zero = ScalarF5E3::ZERO; let huge: ScalarF5E3 = ScalarF5E3::MAX * 3_i32; // Exploded value assert!(huge.exploded());
///
/// let z2 = Circle::<i32, i8>::from_ri(zero, huge); assert!(z2.exploded()); // Exploded state is preserved assert!(z2.r().is_zero()); assert!(z2.i() > 0_i32); // Sign is also preserved
/// ```
pub(crate) fn from_ri(real: Scalar<F, E>, imaginary: Scalar<F, E>) -> Self {
// Unified AMBIG=0 encoding: Scalar and Circle share the same exp form (stored = logical k ^ E::MIN). Fraction encoding still differs (N0 vs N1), but the exp passes thru unchanged.
let to_circle_exp = |e: E| -> E { e };
// Undefined propagates: first undefined wins.
if real.is_undefined() {
return Circle {
real: real.fraction,
imaginary: real.fraction,
exponent: to_circle_exp(real.exponent),
};
}
if imaginary.is_undefined() {
return Circle {
real: imaginary.fraction,
imaginary: imaginary.fraction,
exponent: to_circle_exp(imaginary.exponent),
};
}
// Infinity beats all other classes.
if real.is_infinite() || imaginary.is_infinite() {
return Circle::<F, E>::INFINITY;
}
// Both exploded or both vanished → magnitude indeterminate.
if (real.exploded() && imaginary.exploded()) || (real.vanished() && imaginary.vanished()) {
let prefix: F = INDETERMINATE.prefix.sa();
return Circle {
real: prefix,
imaginary: prefix,
exponent: Self::ambiguous_exponent(),
};
}
// Escape patterns are identical between Scalar and Circle (escapes carry no normal sign bit), so their fractions copy thru without translation. Single exploded: preserve on its axis, other axis = 0. Circle::exploded checks !is_normal() && is_n1(), so exponent must be ambiguous.
if real.exploded() {
return Circle {
real: real.fraction,
imaginary: 0.as_(),
exponent: Self::ambiguous_exponent(),
};
}
if imaginary.exploded() {
return Circle {
real: 0.as_(),
imaginary: imaginary.fraction,
exponent: Self::ambiguous_exponent(),
};
}
// Vanished paired with normal: vanished is negligible, drop it.
if real.vanished() && imaginary.is_normal() {
let imag_c: F = (imaginary.fraction >> 1isize) ^ F::min_value();
return Circle {
real: 0.as_(),
imaginary: imag_c,
exponent: to_circle_exp(imaginary.exponent),
};
}
if imaginary.vanished() && real.is_normal() {
let real_c: F = (real.fraction >> 1isize) ^ F::min_value();
return Circle {
real: real_c,
imaginary: 0.as_(),
exponent: to_circle_exp(real.exponent),
};
}
// Vanished paired with zero: preserve vanished state.
if real.vanished() {
return Circle {
real: real.fraction,
imaginary: 0.as_(),
exponent: Self::ambiguous_exponent(),
};
}
if imaginary.vanished() {
return Circle {
real: 0.as_(),
imaginary: imaginary.fraction,
exponent: Self::ambiguous_exponent(),
};
}
// Both zero.
if real.is_zero() && imaginary.is_zero() {
return Circle::<F, E>::ZERO;
}
// Zero + normal: place normal on its axis.
if real.is_zero() {
let imag_c: F = (imaginary.fraction >> 1isize) ^ F::min_value();
return Circle {
real: 0.as_(),
imaginary: imag_c,
exponent: to_circle_exp(imaginary.exponent),
};
}
if imaginary.is_zero() {
let real_c: F = (real.fraction >> 1isize) ^ F::min_value();
return Circle {
real: real_c,
imaginary: 0.as_(),
exponent: to_circle_exp(real.exponent),
};
}
// Normal case: translate Scalar→Circle, then align exponents. Scalar fraction s carries implicit sign (~MSB); Circle carries explicit (MSB). For same value: circle = (s >> 1) XOR MSB_MASK — arithmetic shift halves the magnitude, XOR flips the sign convention.
let real_c: F = (real.fraction >> 1isize) ^ F::min_value();
let imag_c: F = (imaginary.fraction >> 1isize) ^ F::min_value();
let exp_diff = real.exponent.wrapping_sub(&imaginary.exponent);
if exp_diff == 0.as_() {
return Circle {
real: real_c,
imaginary: imag_c,
exponent: to_circle_exp(real.exponent),
};
} else if exp_diff > 0.as_() {
let shift: isize = exp_diff.as_();
if shift >= Self::fraction_bits() {
return Circle {
real: real_c,
imaginary: 0.as_(),
exponent: to_circle_exp(real.exponent),
};
}
return Circle {
real: real_c,
imaginary: imag_c >> shift,
exponent: to_circle_exp(real.exponent),
};
} else {
let zero_e: E = 0.as_();
let shift: isize = zero_e.wrapping_sub(&exp_diff).as_();
if shift >= Self::fraction_bits() {
return Circle {
real: 0.as_(),
imaginary: imag_c,
exponent: to_circle_exp(imaginary.exponent),
};
}
return Circle {
real: real_c >> shift,
imaginary: imag_c,
exponent: to_circle_exp(imaginary.exponent),
};
}
}
}