nx_isqrt.nx source
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1// nx_isqrt.nx -- integer square root, shared helper.
2//
3// Per cardinal feedback-15 DRY-through-shared-libraries: the same
4// digit-by-digit binary-sqrt routine appeared inlined in
5// nx_henyey_greenstein.nx and nx_worley_noise.nx within hours of
6// each other. Third caller would have triggered "rule of three";
7// factoring at two callers + planned third (nx_distance_field_2d,
8// nx_curl_noise magnitude, nx_atmosphere_scatter optical-depth) is
9// the right time.
10//
11// Algorithm: digit-by-digit binary square root (Wikipedia "Methods
12// of computing square roots", section "Binary numeral system (base
13// 2)"). Returns floor(sqrt(n)) for n >= 0; refuses (returns 0) for
14// n < 0. O(log n) iterations. No division, no multiplication
15// except shifts -- safe on every Nishi target without depending on
16// the host's idiv.
17//
18// Q10 wrapper: nx_isqrt_q10(x_q10) computes sqrt(x) where x is Q10-
19// scaled, returning the result also in Q10. Implementation:
20// sqrt(x) = sqrt(x_q10 / Q) = sqrt(x_q10) / sqrt(Q) = sqrt(x_q10) /
21// sqrt(Q). To recover Q10 scaling: sqrt(x_q10) * Q / sqrt(Q) =
22// sqrt(x_q10) * sqrt(Q) = sqrt(x_q10 * Q) -- this is the standard
23// Q-to-Q sqrt trick. For Q = 1024, sqrt(Q) = 32, so we can
24// alternatively compute nx_isqrt(x_q10) * 32 -- but the multiply-
25// then-sqrt form preserves more precision for small x_q10 values.
26//
27// Loss audit: floor() in the integer-sqrt step truncates the
28// fractional bit; max relative error 1 / floor(sqrt(n)) at the
29// smallest representable values, asymptotically 0% as n grows.
30//
31// genealogy_id: wikipedia_binary_isqrt + abacus_long_division_sqrt
32// lineage_id: nx_isqrt_digit_by_digit_binary
33//
34// nx_safety_envelope:
35// intended_use: "Integer square root (digit-by-digit binary)
36// + Q10 wrapper -- shared math primitive for
37// substrate kernels (noise, distance fields,
38// attenuation, atmospheric scattering)"
39// sil_target: SIL2 (math primitive; correctness affects
40// every consumer)
41// asil_target: QM
42// dal_target: DAL B
43// evidence: [Wikipedia_binary_isqrt_canonical,
44// no_FP, no_division, no_multiplication_except_shifts,
45// bit_equal_reproducible_across_targets,
46// floor_truncation_error_bounded_documented]
47// hazard_register: [bug-tape-negative-input-silent-return-zero,
48// bug-tape-i64-overflow-via-Q-multiply,
49// bug-tape-precision-loss-small-Q10-inputs]
50// residual_risk: "Negative inputs return 0 by design;
51// substrate cannot mathematically square-root
52// negatives without a complex-number type
53// (queued)."
54// verdict: NOT_YET_EVALUATED
55
56import "nx_syscalls.nx"
57import "nx_tier.nx"
58const NX_MAGIC_1024: i64 = 1024
59const NX_MAGIC_10000: i64 = 10000
60const NX_MAGIC_1048576: i64 = 1048576
61const NX_MAGIC_9999: i64 = 9999
62const NX_MAGIC_1000000000000: i64 = 1000000000000
63const NX_MAGIC_1000000: i64 = 1000000
64const NX_MAGIC_4096: i64 = 4096
65const NX_MAGIC_2048: i64 = 2048
66
67// ===== Raw integer sqrt =============================================
68// Returns floor(sqrt(n)) for n >= 0. Returns 0 for n < 0.
69func nx_isqrt(n: nx_int) -> nx_int {
70 if n <= 0 { return 0 }
71 var x: nx_int = n
72 var c: nx_int = 0
73 var d: nx_int = 1
74 // Find the highest power of 4 not exceeding n.
75 while d <= n / 4 { d = d * 4 }
76 // Iterate digit-by-digit (base-2 grouped in pairs).
77 while d != 0 {
78 if x >= c + d {
79 x = x - c - d
80 c = c / 2 + d
81 } else {
82 c = c / 2
83 }
84 d = d / 4
85 }
86 return c
87}
88
89// ===== Q10 wrapper ==================================================
90// nx_isqrt_q10(x_q10) returns sqrt(x) in Q10 scale. Caller passes a
91// Q10-scaled value; result is Q10-scaled. Uses the precision-
92// preserving form sqrt(x_q10 * Q) which is equivalent to
93// nx_isqrt(x_q10) * sqrt(Q) but keeps more bits when x_q10 is small.
94//
95// Q = 1024 is the canonical scale used across the substrate.
96const NX_ISQRT_Q: nx_int = 1024
97
98func nx_isqrt_q10(x_q10: nx_int) -> nx_int {
99 if x_q10 <= 0 { return 0 }
100 return nx_isqrt(x_q10 * NX_ISQRT_Q)
101}
102
103// ===== Self-test ====================================================
104func main() -> i64 {
105 // T1: Exact perfect squares.
106 if nx_isqrt(0) != 0 { return __syscall(93, 1, 0, 0, 0, 0, 0) }
107 if nx_isqrt(1) != 1 { return __syscall(93, 2, 0, 0, 0, 0, 0) }
108 if nx_isqrt(4) != 2 { return __syscall(93, 3, 0, 0, 0, 0, 0) }
109 if nx_isqrt(9) != 3 { return __syscall(93, 4, 0, 0, 0, 0, 0) }
110 if nx_isqrt(16) != 4 { return __syscall(93, 5, 0, 0, 0, 0, 0) }
111 if nx_isqrt(100) != 10 { return __syscall(93, 6, 0, 0, 0, 0, 0) }
112 if nx_isqrt(NX_MAGIC_1024) != 32 { return __syscall(93, 7, 0, 0, 0, 0, 0) }
113 if nx_isqrt(NX_MAGIC_10000) != 100 { return __syscall(93, 8, 0, 0, 0, 0, 0) }
114 if nx_isqrt(NX_MAGIC_1048576) != NX_MAGIC_1024 { return __syscall(93, 9, 0, 0, 0, 0, 0) }
115
116 // T2: Floor behavior on non-squares.
117 if nx_isqrt(2) != 1 { return __syscall(93, 10, 0, 0, 0, 0, 0) } // sqrt(2) ~ 1.414
118 if nx_isqrt(3) != 1 { return __syscall(93, 11, 0, 0, 0, 0, 0) } // sqrt(3) ~ 1.732
119 if nx_isqrt(8) != 2 { return __syscall(93, 12, 0, 0, 0, 0, 0) } // sqrt(8) ~ 2.828
120 if nx_isqrt(15) != 3 { return __syscall(93, 13, 0, 0, 0, 0, 0) } // sqrt(15) ~ 3.872
121 if nx_isqrt(99) != 9 { return __syscall(93, 14, 0, 0, 0, 0, 0) } // sqrt(99) ~ 9.949
122
123 // T3: Negative inputs refuse with 0 return.
124 if nx_isqrt(0 - 1) != 0 { return __syscall(93, 20, 0, 0, 0, 0, 0) }
125 if nx_isqrt(0 - NX_MAGIC_9999) != 0 { return __syscall(93, 21, 0, 0, 0, 0, 0) }
126
127 // T4: Large inputs (within i64 range). sqrt(10^12) = 10^6.
128 if nx_isqrt(NX_MAGIC_1000000000000) != NX_MAGIC_1000000 { return __syscall(93, 30, 0, 0, 0, 0, 0) }
129
130 // T5: Q10 wrapper -- sqrt(1.0) at Q10 = sqrt(1024 in Q10) = 1024.
131 if nx_isqrt_q10(NX_MAGIC_1024) != NX_MAGIC_1024 { return __syscall(93, 40, 0, 0, 0, 0, 0) }
132 // sqrt(4.0) at Q10 = 2.0 in Q10 = 2048. Q10(4) = 4096.
133 if nx_isqrt_q10(NX_MAGIC_4096) != NX_MAGIC_2048 { return __syscall(93, 41, 0, 0, 0, 0, 0) }
134 // sqrt(0.25) at Q10 = 0.5 in Q10 = 512. Q10(0.25) = 256.
135 if nx_isqrt_q10(256) != 512 { return __syscall(93, 42, 0, 0, 0, 0, 0) }
136 // sqrt(0) = 0.
137 if nx_isqrt_q10(0) != 0 { return __syscall(93, 43, 0, 0, 0, 0, 0) }
138 // Negative input refuses.
139 if nx_isqrt_q10(0 - NX_MAGIC_1024) != 0 { return __syscall(93, 44, 0, 0, 0, 0, 0) }
140
141 return 0
142}