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1//===-- Calculate square root of fixed point numbers. -----*- C++ -*-=========//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#ifndef LLVM_LIBC_SRC___SUPPORT_FIXEDPOINT_SQRT_H
10#define LLVM_LIBC_SRC___SUPPORT_FIXEDPOINT_SQRT_H
11
12#include "include/llvm-libc-macros/stdfix-macros.h"
13#include "src/__support/CPP/bit.h"
14#include "src/__support/CPP/limits.h" // CHAR_BIT
15#include "src/__support/CPP/type_traits.h"
16#include "src/__support/macros/attributes.h" // LIBC_INLINE
17#include "src/__support/macros/config.h"
18#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
19
20#include "fx_rep.h"
21
22#ifdef LIBC_COMPILER_HAS_FIXED_POINT
23
24namespace LIBC_NAMESPACE_DECL {
25namespace fixed_point {
26
27namespace internal {
28
29template <typename T> struct SqrtConfig;
30
31template <> struct SqrtConfig<unsigned short fract> {
32 using Type = unsigned short fract;
33 static constexpr int EXTRA_STEPS = 0;
34
35 // Linear approximation for the initial values, with errors bounded by:
36 // max(1.5 * 2^-11, eps)
37 // Generated with Sollya:
38 // > for i from 4 to 15 do {
39 // P = fpminimax(sqrt(x), 1, [|8, 8|], [i * 2^-4, (i + 1)*2^-4],
40 // fixed, absolute);
41 // print("{", coeff(P, 1), "uhr,", coeff(P, 0), "uhr},");
42 // };
43 static constexpr Type FIRST_APPROX[12][2] = {
44 {0x1.e8p-1uhr, 0x1.0cp-2uhr}, {0x1.bap-1uhr, 0x1.28p-2uhr},
45 {0x1.94p-1uhr, 0x1.44p-2uhr}, {0x1.74p-1uhr, 0x1.6p-2uhr},
46 {0x1.6p-1uhr, 0x1.74p-2uhr}, {0x1.4ep-1uhr, 0x1.88p-2uhr},
47 {0x1.3ep-1uhr, 0x1.9cp-2uhr}, {0x1.32p-1uhr, 0x1.acp-2uhr},
48 {0x1.22p-1uhr, 0x1.c4p-2uhr}, {0x1.18p-1uhr, 0x1.d4p-2uhr},
49 {0x1.08p-1uhr, 0x1.fp-2uhr}, {0x1.04p-1uhr, 0x1.f8p-2uhr},
50 };
51};
52
53template <> struct SqrtConfig<unsigned fract> {
54 using Type = unsigned fract;
55 static constexpr int EXTRA_STEPS = 1;
56
57 // Linear approximation for the initial values, with errors bounded by:
58 // max(1.5 * 2^-11, eps)
59 // Generated with Sollya:
60 // > for i from 4 to 14 do {
61 // P = fpminimax(sqrt(x), 1, [|16, 16|], [i * 2^-4, (i + 1)*2^-4],
62 // fixed, absolute);
63 // print("{", coeff(P, 1), "ur,", coeff(P, 0), "ur},");
64 // };
65 // For the last interval [15/16, 1), we choose the linear function Q such that
66 // Q(1) = 1 and Q(15/16) = P(15/16),
67 // where P is the polynomial generated by Sollya above for [14/16, 15/16].
68 // This is to prevent overflow in the last interval [15/16, 1).
69 static constexpr Type FIRST_APPROX[12][2] = {
70 {0x1.e378p-1ur, 0x1.0ebp-2ur}, {0x1.b512p-1ur, 0x1.2b94p-2ur},
71 {0x1.91fp-1ur, 0x1.45dcp-2ur}, {0x1.7622p-1ur, 0x1.5e24p-2ur},
72 {0x1.5f5ap-1ur, 0x1.74e4p-2ur}, {0x1.4c58p-1ur, 0x1.8a4p-2ur},
73 {0x1.3c1ep-1ur, 0x1.9e84p-2ur}, {0x1.2e0cp-1ur, 0x1.b1d8p-2ur},
74 {0x1.21aap-1ur, 0x1.c468p-2ur}, {0x1.16bap-1ur, 0x1.d62cp-2ur},
75 {0x1.0cfp-1ur, 0x1.e74cp-2ur}, {0x1.039p-1ur, 0x1.f8ep-2ur},
76 };
77};
78
79template <> struct SqrtConfig<unsigned long fract> {
80 using Type = unsigned long fract;
81 static constexpr int EXTRA_STEPS = 2;
82
83 // Linear approximation for the initial values, with errors bounded by:
84 // max(1.5 * 2^-11, eps)
85 // Generated with Sollya:
86 // > for i from 4 to 14 do {
87 // P = fpminimax(sqrt(x), 1, [|32, 32|], [i * 2^-4, (i + 1)*2^-4],
88 // fixed, absolute);
89 // print("{", coeff(P, 1), "ulr,", coeff(P, 0), "ulr},");
90 // };
91 // For the last interval [15/16, 1), we choose the linear function Q such that
92 // Q(1) = 1 and Q(15/16) = P(15/16),
93 // where P is the polynomial generated by Sollya above for [14/16, 15/16].
94 // This is to prevent overflow in the last interval [15/16, 1).
95 static constexpr Type FIRST_APPROX[12][2] = {
96 {0x1.e3779b98p-1ulr, 0x1.0eaff788p-2ulr},
97 {0x1.b5167872p-1ulr, 0x1.2b908ad4p-2ulr},
98 {0x1.91f195cap-1ulr, 0x1.45da800cp-2ulr},
99 {0x1.761ebcb4p-1ulr, 0x1.5e27004cp-2ulr},
100 {0x1.5f619986p-1ulr, 0x1.74db933cp-2ulr},
101 {0x1.4c583adep-1ulr, 0x1.8a3fbfccp-2ulr},
102 {0x1.3c1a591cp-1ulr, 0x1.9e88373cp-2ulr},
103 {0x1.2e08545ap-1ulr, 0x1.b1dd2534p-2ulr},
104 {0x1.21b05c0ap-1ulr, 0x1.c45e023p-2ulr},
105 {0x1.16becd02p-1ulr, 0x1.d624031p-2ulr},
106 {0x1.0cf49fep-1ulr, 0x1.e743b844p-2ulr},
107 {0x1.038cdfcp-1ulr, 0x1.f8e6408p-2ulr},
108 };
109};
110
111template <>
112struct SqrtConfig<unsigned short accum> : SqrtConfig<unsigned fract> {};
113
114template <>
115struct SqrtConfig<unsigned accum> : SqrtConfig<unsigned long fract> {};
116
117// Integer square root
118template <> struct SqrtConfig<unsigned short> {
119 using OutType = unsigned short accum;
120 using FracType = unsigned fract;
121 // For fast-but-less-accurate version
122 using FastFracType = unsigned short fract;
123 using HalfType = unsigned char;
124};
125
126template <> struct SqrtConfig<unsigned int> {
127 using OutType = unsigned accum;
128 using FracType = unsigned long fract;
129 // For fast-but-less-accurate version
130 using FastFracType = unsigned fract;
131 using HalfType = unsigned short;
132};
133
134// TODO: unsigned long accum type is 64-bit, and will need 64-bit fract type.
135// Probably we will use DyadicFloat<64> for intermediate computations instead.
136
137} // namespace internal
138
139// Core computation for sqrt with normalized inputs (0.25 <= x < 1).
140template <typename Config>
141LIBC_INLINE constexpr typename Config::Type
142sqrt_core(typename Config::Type x_frac) {
143 using FracType = typename Config::Type;
144 using FXRep = FXRep<FracType>;
145 using StorageType = typename FXRep::StorageType;
146 // Exact case:
147 if (x_frac == FXRep::ONE_FOURTH())
148 return FXRep::ONE_HALF();
149
150 // Use use Newton method to approximate sqrt(a):
151 // x_{n + 1} = 1/2 (x_n + a / x_n)
152 // For the initial values, we choose x_0
153
154 // Use the leading 4 bits to do look up for sqrt(x).
155 // After normalization, 0.25 <= x_frac < 1, so the leading 4 bits of x_frac
156 // are between 0b0100 and 0b1111. Hence the lookup table only needs 12
157 // entries, and we can get the index by subtracting the leading 4 bits of
158 // x_frac by 4 = 0b0100.
159 StorageType x_bit = cpp::bit_cast<StorageType>(x_frac);
160 int index = (static_cast<int>(x_bit >> (FXRep::TOTAL_LEN - 4))) - 4;
161 FracType a = Config::FIRST_APPROX[index][0];
162 FracType b = Config::FIRST_APPROX[index][1];
163
164 // Initial approximation step.
165 // Estimated error bounds: | r - sqrt(x_frac) | < max(1.5 * 2^-11, eps).
166 FracType r = a * x_frac + b;
167
168 // Further Newton-method iterations for square-root:
169 // x_{n + 1} = 0.5 * (x_n + a / x_n)
170 // We distribute and do the multiplication by 0.5 first to avoid overflow.
171 // TODO: Investigate the performance and accuracy of using division-free
172 // iterations from:
173 // Blanchard, J. D. and Chamberland, M., "Newton's Method Without Division",
174 // The American Mathematical Monthly (2023).
175 // https://chamberland.math.grinnell.edu/papers/newton.pdf
176 for (int i = 0; i < Config::EXTRA_STEPS; ++i)
177 r = (r >> 1) + (x_frac >> 1) / r;
178
179 return r;
180}
181
182template <typename T>
183LIBC_INLINE constexpr cpp::enable_if_t<cpp::is_fixed_point_v<T>, T> sqrt(T x) {
184 using BitType = typename FXRep<T>::StorageType;
185 BitType x_bit = cpp::bit_cast<BitType>(x);
186
187 if (LIBC_UNLIKELY(x_bit == 0))
188 return FXRep<T>::ZERO();
189
190 int leading_zeros = cpp::countl_zero(x_bit);
191 constexpr int STORAGE_LENGTH = sizeof(BitType) * CHAR_BIT;
192 constexpr int EXP_ADJUSTMENT = STORAGE_LENGTH - FXRep<T>::FRACTION_LEN - 1;
193 // x_exp is the real exponent of the leading bit of x.
194 int x_exp = EXP_ADJUSTMENT - leading_zeros;
195 int shift = EXP_ADJUSTMENT - 1 - (x_exp & (~1));
196 // Normalize.
197 x_bit <<= shift;
198 using FracType = typename internal::SqrtConfig<T>::Type;
199 FracType x_frac = cpp::bit_cast<FracType>(x_bit);
200
201 // Compute sqrt(x_frac) using Newton-method.
202 FracType r = sqrt_core<internal::SqrtConfig<T>>(x_frac);
203
204 // Re-scaling
205 r >>= EXP_ADJUSTMENT - (x_exp >> 1);
206
207 // Return result.
208 return cpp::bit_cast<T>(r);
209}
210
211// Integer square root - Accurate version:
212// Absolute errors < 2^(-fraction length).
213template <typename T>
214LIBC_INLINE constexpr typename internal::SqrtConfig<T>::OutType isqrt(T x) {
215 using OutType = typename internal::SqrtConfig<T>::OutType;
216 using FracType = typename internal::SqrtConfig<T>::FracType;
217
218 if (x == 0)
219 return FXRep<OutType>::ZERO();
220
221 // Normalize the leading bits to the first two bits.
222 // Shift and then Bit cast x to x_frac gives us:
223 // x = 2^(FRACTION_LEN + 1 - shift) * x_frac;
224 int leading_zeros = cpp::countl_zero(x);
225 int shift = ((leading_zeros >> 1) << 1);
226 x <<= shift;
227 // Convert to frac type and compute square root.
228 FracType x_frac = cpp::bit_cast<FracType>(x);
229 FracType r = sqrt_core<internal::SqrtConfig<FracType>>(x_frac);
230 // To rescale back to the OutType (Accum)
231 r >>= (shift >> 1);
232
233 return cpp::bit_cast<OutType>(r);
234}
235
236// Integer square root - Fast but less accurate version:
237// Relative errors < 2^(-fraction length).
238template <typename T>
239LIBC_INLINE constexpr typename internal::SqrtConfig<T>::OutType
240isqrt_fast(T x) {
241 using OutType = typename internal::SqrtConfig<T>::OutType;
242 using FracType = typename internal::SqrtConfig<T>::FastFracType;
243 using StorageType = typename FXRep<FracType>::StorageType;
244
245 if (x == 0)
246 return FXRep<OutType>::ZERO();
247
248 // Normalize the leading bits to the first two bits.
249 // Shift and then Bit cast x to x_frac gives us:
250 // x = 2^(FRACTION_LEN + 1 - shift) * x_frac;
251 int leading_zeros = cpp::countl_zero(x);
252 int shift = (leading_zeros & (~1));
253 x <<= shift;
254 // Convert to frac type and compute square root.
255 FracType x_frac = cpp::bit_cast<FracType>(
256 static_cast<StorageType>(x >> FXRep<FracType>::FRACTION_LEN));
257 OutType r =
258 static_cast<OutType>(sqrt_core<internal::SqrtConfig<FracType>>(x_frac));
259 // To rescale back to the OutType (Accum)
260 r <<= (FXRep<OutType>::INTEGRAL_LEN - (shift >> 1));
261 return cpp::bit_cast<OutType>(r);
262}
263
264} // namespace fixed_point
265} // namespace LIBC_NAMESPACE_DECL
266
267#endif // LIBC_COMPILER_HAS_FIXED_POINT
268
269#endif // LLVM_LIBC_SRC___SUPPORT_FIXEDPOINT_SQRT_H
270

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source code of libc/src/__support/fixed_point/sqrt.h