1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2007 Julien Pommier
5 // Copyright (C) 2014 Pedro Gonnet ([email protected])
6 // Copyright (C) 2016 Gael Guennebaud <[email protected]>
7 //
8 // Copyright (C) 2018 Wave Computing, Inc.
9 // Written by:
10 // Chris Larsen
11 // Alexey Frunze ([email protected])
12 //
13 // This Source Code Form is subject to the terms of the Mozilla
14 // Public License v. 2.0. If a copy of the MPL was not distributed
15 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
16
17 /* The sin, cos, exp, and log functions of this file come from
18 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
19 */
20
21 /* The tanh function of this file is an adaptation of
22 * template<typename T> T generic_fast_tanh_float(const T&)
23 * from MathFunctionsImpl.h.
24 */
25
26 #ifndef EIGEN_MATH_FUNCTIONS_MSA_H
27 #define EIGEN_MATH_FUNCTIONS_MSA_H
28
29 namespace Eigen {
30
31 namespace internal {
32
33 template <>
34 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
35 plog<Packet4f>(const Packet4f& _x) {
36 static _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f);
37 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292e-2f);
38 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, -1.1514610310e-1f);
39 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740e-1f);
40 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, -1.2420140846e-1f);
41 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, +1.4249322787e-1f);
42 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, -1.6668057665e-1f);
43 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, +2.0000714765e-1f);
44 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, -2.4999993993e-1f);
45 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, +3.3333331174e-1f);
46 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f);
47 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f);
48 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
49 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f);
50
51 // Convert negative argument into NAN (quiet negative, to be specific).
52 Packet4f zero = (Packet4f)__builtin_msa_ldi_w(0);
53 Packet4i neg_mask = __builtin_msa_fclt_w(_x, zero);
54 Packet4i zero_mask = __builtin_msa_fceq_w(_x, zero);
55 Packet4f non_neg_x_or_nan = padd(_x, (Packet4f)neg_mask); // Add 0.0 or NAN.
56 Packet4f x = non_neg_x_or_nan;
57
58 // Extract exponent from x = mantissa * 2**exponent, where 1.0 <= mantissa < 2.0.
59 // N.B. the exponent is one less of what frexpf() would return.
60 Packet4i e_int = __builtin_msa_ftint_s_w(__builtin_msa_flog2_w(x));
61 // Multiply x by 2**(-exponent-1) to get 0.5 <= x < 1.0 as from frexpf().
62 x = __builtin_msa_fexp2_w(x, (Packet4i)__builtin_msa_nori_b((v16u8)e_int, 0));
63
64 /*
65 if (x < SQRTHF) {
66 x = x + x - 1.0;
67 } else {
68 e += 1;
69 x = x - 1.0;
70 }
71 */
72 Packet4f xx = padd(x, x);
73 Packet4i ge_mask = __builtin_msa_fcle_w(p4f_cephes_SQRTHF, x);
74 e_int = psub(e_int, ge_mask);
75 x = (Packet4f)__builtin_msa_bsel_v((v16u8)ge_mask, (v16u8)xx, (v16u8)x);
76 x = psub(x, p4f_1);
77 Packet4f e = __builtin_msa_ffint_s_w(e_int);
78
79 Packet4f x2 = pmul(x, x);
80 Packet4f x3 = pmul(x2, x);
81
82 Packet4f y, y1, y2;
83 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1);
84 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4);
85 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7);
86 y = pmadd(y, x, p4f_cephes_log_p2);
87 y1 = pmadd(y1, x, p4f_cephes_log_p5);
88 y2 = pmadd(y2, x, p4f_cephes_log_p8);
89 y = pmadd(y, x3, y1);
90 y = pmadd(y, x3, y2);
91 y = pmul(y, x3);
92
93 y = pmadd(e, p4f_cephes_log_q1, y);
94 x = __builtin_msa_fmsub_w(x, x2, p4f_half);
95 x = padd(x, y);
96 x = pmadd(e, p4f_cephes_log_q2, x);
97
98 // x is now the logarithm result candidate. We still need to handle the
99 // extreme arguments of zero and positive infinity, though.
100 // N.B. if the argument is +INFINITY, x is NAN because the polynomial terms
101 // contain infinities of both signs (see the coefficients and code above).
102 // INFINITY - INFINITY is NAN.
103
104 // If the argument is +INFINITY, make it the new result candidate.
105 // To achieve that we choose the smaller of the result candidate and the
106 // argument.
107 // This is correct for all finite pairs of values (the logarithm is smaller
108 // than the argument).
109 // This is also correct in the special case when the argument is +INFINITY
110 // and the result candidate is NAN. This is because the fmin.df instruction
111 // prefers non-NANs to NANs.
112 x = __builtin_msa_fmin_w(x, non_neg_x_or_nan);
113
114 // If the argument is zero (including -0.0), the result becomes -INFINITY.
115 Packet4i neg_infs = __builtin_msa_slli_w(zero_mask, 23);
116 x = (Packet4f)__builtin_msa_bsel_v((v16u8)zero_mask, (v16u8)x, (v16u8)neg_infs);
117
118 return x;
119 }
120
121 template <>
122 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
123 pexp<Packet4f>(const Packet4f& _x) {
124 // Limiting single-precision pexp's argument to [-128, +128] lets pexp
125 // reach 0 and INFINITY naturally.
126 static _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -128.0f);
127 static _EIGEN_DECLARE_CONST_Packet4f(exp_hi, +128.0f);
128 static _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f);
129 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f);
130 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f);
131 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500e-4f);
132 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507e-3f);
133 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073e-3f);
134 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894e-2f);
135 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459e-1f);
136 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201e-1f);
137 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
138 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f);
139
140 Packet4f x = _x;
141
142 // Clamp x.
143 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(x, p4f_exp_lo), (v16u8)x,
144 (v16u8)p4f_exp_lo);
145 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(p4f_exp_hi, x), (v16u8)x,
146 (v16u8)p4f_exp_hi);
147
148 // Round to nearest integer by adding 0.5 (with x's sign) and truncating.
149 Packet4f x2_add = (Packet4f)__builtin_msa_binsli_w((v4u32)p4f_half, (v4u32)x, 0);
150 Packet4f x2 = pmadd(x, p4f_cephes_LOG2EF, x2_add);
151 Packet4i x2_int = __builtin_msa_ftrunc_s_w(x2);
152 Packet4f x2_int_f = __builtin_msa_ffint_s_w(x2_int);
153
154 x = __builtin_msa_fmsub_w(x, x2_int_f, p4f_cephes_exp_C1);
155 x = __builtin_msa_fmsub_w(x, x2_int_f, p4f_cephes_exp_C2);
156
157 Packet4f z = pmul(x, x);
158
159 Packet4f y = p4f_cephes_exp_p0;
160 y = pmadd(y, x, p4f_cephes_exp_p1);
161 y = pmadd(y, x, p4f_cephes_exp_p2);
162 y = pmadd(y, x, p4f_cephes_exp_p3);
163 y = pmadd(y, x, p4f_cephes_exp_p4);
164 y = pmadd(y, x, p4f_cephes_exp_p5);
165 y = pmadd(y, z, x);
166 y = padd(y, p4f_1);
167
168 // y *= 2**exponent.
169 y = __builtin_msa_fexp2_w(y, x2_int);
170
171 return y;
172 }
173
174 template <>
175 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
176 ptanh<Packet4f>(const Packet4f& _x) {
177 static _EIGEN_DECLARE_CONST_Packet4f(tanh_tiny, 1e-4f);
178 static _EIGEN_DECLARE_CONST_Packet4f(tanh_hi, 9.0f);
179 // The monomial coefficients of the numerator polynomial (odd).
180 static _EIGEN_DECLARE_CONST_Packet4f(alpha_1, 4.89352455891786e-3f);
181 static _EIGEN_DECLARE_CONST_Packet4f(alpha_3, 6.37261928875436e-4f);
182 static _EIGEN_DECLARE_CONST_Packet4f(alpha_5, 1.48572235717979e-5f);
183 static _EIGEN_DECLARE_CONST_Packet4f(alpha_7, 5.12229709037114e-8f);
184 static _EIGEN_DECLARE_CONST_Packet4f(alpha_9, -8.60467152213735e-11f);
185 static _EIGEN_DECLARE_CONST_Packet4f(alpha_11, 2.00018790482477e-13f);
186 static _EIGEN_DECLARE_CONST_Packet4f(alpha_13, -2.76076847742355e-16f);
187 // The monomial coefficients of the denominator polynomial (even).
188 static _EIGEN_DECLARE_CONST_Packet4f(beta_0, 4.89352518554385e-3f);
189 static _EIGEN_DECLARE_CONST_Packet4f(beta_2, 2.26843463243900e-3f);
190 static _EIGEN_DECLARE_CONST_Packet4f(beta_4, 1.18534705686654e-4f);
191 static _EIGEN_DECLARE_CONST_Packet4f(beta_6, 1.19825839466702e-6f);
192
193 Packet4f x = pabs(_x);
194 Packet4i tiny_mask = __builtin_msa_fclt_w(x, p4f_tanh_tiny);
195
196 // Clamp the inputs to the range [-9, 9] since anything outside
197 // this range is -/+1.0f in single-precision.
198 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(p4f_tanh_hi, x), (v16u8)x,
199 (v16u8)p4f_tanh_hi);
200
201 // Since the polynomials are odd/even, we need x**2.
202 Packet4f x2 = pmul(x, x);
203
204 // Evaluate the numerator polynomial p.
205 Packet4f p = pmadd(x2, p4f_alpha_13, p4f_alpha_11);
206 p = pmadd(x2, p, p4f_alpha_9);
207 p = pmadd(x2, p, p4f_alpha_7);
208 p = pmadd(x2, p, p4f_alpha_5);
209 p = pmadd(x2, p, p4f_alpha_3);
210 p = pmadd(x2, p, p4f_alpha_1);
211 p = pmul(x, p);
212
213 // Evaluate the denominator polynomial q.
214 Packet4f q = pmadd(x2, p4f_beta_6, p4f_beta_4);
215 q = pmadd(x2, q, p4f_beta_2);
216 q = pmadd(x2, q, p4f_beta_0);
217
218 // Divide the numerator by the denominator.
219 p = pdiv(p, q);
220
221 // Reinstate the sign.
222 p = (Packet4f)__builtin_msa_binsli_w((v4u32)p, (v4u32)_x, 0);
223
224 // When the argument is very small in magnitude it's more accurate to just return it.
225 p = (Packet4f)__builtin_msa_bsel_v((v16u8)tiny_mask, (v16u8)p, (v16u8)_x);
226
227 return p;
228 }
229
230 template <bool sine>
psincos_inner_msa_float(const Packet4f & _x)231 Packet4f psincos_inner_msa_float(const Packet4f& _x) {
232 static _EIGEN_DECLARE_CONST_Packet4f(sincos_max_arg, 13176795.0f); // Approx. (2**24) / (4/Pi).
233 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1, -0.78515625f);
234 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f);
235 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f);
236 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891e-4f);
237 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736e-3f);
238 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611e-1f);
239 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948e-5f);
240 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765e-3f);
241 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827e-2f);
242 static _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4/Pi.
243 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
244 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f);
245
246 Packet4f x = pabs(_x);
247
248 // Translate infinite arguments into NANs.
249 Packet4f zero_or_nan_if_inf = psub(_x, _x);
250 x = padd(x, zero_or_nan_if_inf);
251 // Prevent sin/cos from generating values larger than 1.0 in magnitude
252 // for very large arguments by setting x to 0.0.
253 Packet4i small_or_nan_mask = __builtin_msa_fcult_w(x, p4f_sincos_max_arg);
254 x = pand(x, (Packet4f)small_or_nan_mask);
255
256 // Scale x by 4/Pi to find x's octant.
257 Packet4f y = pmul(x, p4f_cephes_FOPI);
258 // Get the octant. We'll reduce x by this number of octants or by one more than it.
259 Packet4i y_int = __builtin_msa_ftrunc_s_w(y);
260 // x's from even-numbered octants will translate to octant 0: [0, +Pi/4].
261 // x's from odd-numbered octants will translate to octant -1: [-Pi/4, 0].
262 // Adjustment for odd-numbered octants: octant = (octant + 1) & (~1).
263 Packet4i y_int1 = __builtin_msa_addvi_w(y_int, 1);
264 Packet4i y_int2 = (Packet4i)__builtin_msa_bclri_w((Packet4ui)y_int1, 0); // bclri = bit-clear
265 y = __builtin_msa_ffint_s_w(y_int2);
266
267 // Compute the sign to apply to the polynomial.
268 Packet4i sign_mask = sine ? pxor(__builtin_msa_slli_w(y_int1, 29), (Packet4i)_x)
269 : __builtin_msa_slli_w(__builtin_msa_addvi_w(y_int, 3), 29);
270
271 // Get the polynomial selection mask.
272 // We'll calculate both (sin and cos) polynomials and then select from the two.
273 Packet4i poly_mask = __builtin_msa_ceqi_w(__builtin_msa_slli_w(y_int2, 30), 0);
274
275 // Reduce x by y octants to get: -Pi/4 <= x <= +Pi/4.
276 // The magic pass: "Extended precision modular arithmetic"
277 // x = ((x - y * DP1) - y * DP2) - y * DP3
278 Packet4f tmp1 = pmul(y, p4f_minus_cephes_DP1);
279 Packet4f tmp2 = pmul(y, p4f_minus_cephes_DP2);
280 Packet4f tmp3 = pmul(y, p4f_minus_cephes_DP3);
281 x = padd(x, tmp1);
282 x = padd(x, tmp2);
283 x = padd(x, tmp3);
284
285 // Evaluate the cos(x) polynomial.
286 y = p4f_coscof_p0;
287 Packet4f z = pmul(x, x);
288 y = pmadd(y, z, p4f_coscof_p1);
289 y = pmadd(y, z, p4f_coscof_p2);
290 y = pmul(y, z);
291 y = pmul(y, z);
292 y = __builtin_msa_fmsub_w(y, z, p4f_half);
293 y = padd(y, p4f_1);
294
295 // Evaluate the sin(x) polynomial.
296 Packet4f y2 = p4f_sincof_p0;
297 y2 = pmadd(y2, z, p4f_sincof_p1);
298 y2 = pmadd(y2, z, p4f_sincof_p2);
299 y2 = pmul(y2, z);
300 y2 = pmadd(y2, x, x);
301
302 // Select the correct result from the two polynomials.
303 y = sine ? (Packet4f)__builtin_msa_bsel_v((v16u8)poly_mask, (v16u8)y, (v16u8)y2)
304 : (Packet4f)__builtin_msa_bsel_v((v16u8)poly_mask, (v16u8)y2, (v16u8)y);
305
306 // Update the sign.
307 sign_mask = pxor(sign_mask, (Packet4i)y);
308 y = (Packet4f)__builtin_msa_binsli_w((v4u32)y, (v4u32)sign_mask, 0); // binsli = bit-insert-left
309 return y;
310 }
311
312 template <>
313 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
314 psin<Packet4f>(const Packet4f& x) {
315 return psincos_inner_msa_float</* sine */ true>(x);
316 }
317
318 template <>
319 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
320 pcos<Packet4f>(const Packet4f& x) {
321 return psincos_inner_msa_float</* sine */ false>(x);
322 }
323
324 template <>
325 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet2d
326 pexp<Packet2d>(const Packet2d& _x) {
327 // Limiting double-precision pexp's argument to [-1024, +1024] lets pexp
328 // reach 0 and INFINITY naturally.
329 static _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -1024.0);
330 static _EIGEN_DECLARE_CONST_Packet2d(exp_hi, +1024.0);
331 static _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599);
332 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125);
333 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6);
334 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4);
335 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2);
336 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1);
337 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6);
338 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3);
339 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1);
340 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0);
341 static _EIGEN_DECLARE_CONST_Packet2d(half, 0.5);
342 static _EIGEN_DECLARE_CONST_Packet2d(1, 1.0);
343 static _EIGEN_DECLARE_CONST_Packet2d(2, 2.0);
344
345 Packet2d x = _x;
346
347 // Clamp x.
348 x = (Packet2d)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_d(x, p2d_exp_lo), (v16u8)x,
349 (v16u8)p2d_exp_lo);
350 x = (Packet2d)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_d(p2d_exp_hi, x), (v16u8)x,
351 (v16u8)p2d_exp_hi);
352
353 // Round to nearest integer by adding 0.5 (with x's sign) and truncating.
354 Packet2d x2_add = (Packet2d)__builtin_msa_binsli_d((v2u64)p2d_half, (v2u64)x, 0);
355 Packet2d x2 = pmadd(x, p2d_cephes_LOG2EF, x2_add);
356 Packet2l x2_long = __builtin_msa_ftrunc_s_d(x2);
357 Packet2d x2_long_d = __builtin_msa_ffint_s_d(x2_long);
358
359 x = __builtin_msa_fmsub_d(x, x2_long_d, p2d_cephes_exp_C1);
360 x = __builtin_msa_fmsub_d(x, x2_long_d, p2d_cephes_exp_C2);
361
362 x2 = pmul(x, x);
363
364 Packet2d px = p2d_cephes_exp_p0;
365 px = pmadd(px, x2, p2d_cephes_exp_p1);
366 px = pmadd(px, x2, p2d_cephes_exp_p2);
367 px = pmul(px, x);
368
369 Packet2d qx = p2d_cephes_exp_q0;
370 qx = pmadd(qx, x2, p2d_cephes_exp_q1);
371 qx = pmadd(qx, x2, p2d_cephes_exp_q2);
372 qx = pmadd(qx, x2, p2d_cephes_exp_q3);
373
374 x = pdiv(px, psub(qx, px));
375 x = pmadd(p2d_2, x, p2d_1);
376
377 // x *= 2**exponent.
378 x = __builtin_msa_fexp2_d(x, x2_long);
379
380 return x;
381 }
382
383 } // end namespace internal
384
385 } // end namespace Eigen
386
387 #endif // EIGEN_MATH_FUNCTIONS_MSA_H
388