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1 /* Remainder.
2 Copyright (C) 2011-2021 Free Software Foundation, Inc.
3
4 This file is free software: you can redistribute it and/or modify
5 it under the terms of the GNU Lesser General Public License as
6 published by the Free Software Foundation; either version 3 of the
7 License, or (at your option) any later version.
8
9 This file is distributed in the hope that it will be useful,
10 but WITHOUT ANY WARRANTY; without even the implied warranty of
11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
12 GNU Lesser General Public License for more details.
13
14 You should have received a copy of the GNU Lesser General Public License
15 along with this program. If not, see <https://www.gnu.org/licenses/>. */
16
17 #if ! defined USE_LONG_DOUBLE
18 # include <config.h>
19 #endif
20
21 /* Specification. */
22 #include <math.h>
23
24 #include <float.h>
25 #include <stdlib.h>
26
27 #ifdef USE_LONG_DOUBLE
28 # define FMOD fmodl
29 # define DOUBLE long double
30 # define MANT_DIG LDBL_MANT_DIG
31 # define L_(literal) literal##L
32 # define FREXP frexpl
33 # define LDEXP ldexpl
34 # define FABS fabsl
35 # define TRUNC truncl
36 # define ISNAN isnanl
37 #else
38 # define FMOD fmod
39 # define DOUBLE double
40 # define MANT_DIG DBL_MANT_DIG
41 # define L_(literal) literal
42 # define FREXP frexp
43 # define LDEXP ldexp
44 # define FABS fabs
45 # define TRUNC trunc
46 # define ISNAN isnand
47 #endif
48
49 /* MSVC with option -fp:strict refuses to compile constant initializers that
50 contain floating-point operations. Pacify this compiler. */
51 #if defined _MSC_VER && !defined __clang__
52 # pragma fenv_access (off)
53 #endif
54
55 #undef NAN
56 #if defined _MSC_VER
57 static DOUBLE zero;
58 # define NAN (zero / zero)
59 #else
60 # define NAN (L_(0.0) / L_(0.0))
61 #endif
62
63 /* To avoid rounding errors during the computation of x - q * y,
64 there are three possibilities:
65 - Use fma (- q, y, x).
66 - Have q be a single bit at a time, and compute x - q * y
67 through a subtraction.
68 - Have q be at most MANT_DIG/2 bits at a time, and compute
69 x - q * y by splitting y into two halves:
70 y = y1 * 2^(MANT_DIG/2) + y0
71 x - q * y = (x - q * y1 * 2^MANT_DIG/2) - q * y0.
72 The latter is probably the most efficient. */
73
74 /* Number of bits in a limb. */
75 #define LIMB_BITS (MANT_DIG / 2)
76
77 /* 2^LIMB_BITS. */
78 static const DOUBLE TWO_LIMB_BITS =
79 /* Assume LIMB_BITS <= 3 * 31.
80 Use the identity
81 n = floor(n/3) + floor((n+1)/3) + floor((n+2)/3). */
82 (DOUBLE) (1U << (LIMB_BITS / 3))
83 * (DOUBLE) (1U << ((LIMB_BITS + 1) / 3))
84 * (DOUBLE) (1U << ((LIMB_BITS + 2) / 3));
85
86 /* 2^-LIMB_BITS. */
87 static const DOUBLE TWO_LIMB_BITS_INVERSE =
88 /* Assume LIMB_BITS <= 3 * 31.
89 Use the identity
90 n = floor(n/3) + floor((n+1)/3) + floor((n+2)/3). */
91 L_(1.0) / ((DOUBLE) (1U << (LIMB_BITS / 3))
92 * (DOUBLE) (1U << ((LIMB_BITS + 1) / 3))
93 * (DOUBLE) (1U << ((LIMB_BITS + 2) / 3)));
94
95 DOUBLE
96 FMOD (DOUBLE x, DOUBLE y)
/* ![[first]](../icons/n_first.png)
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*/
97 {
98 if (isfinite (x) && isfinite (y) && y != L_(0.0))
99 {
100 if (x == L_(0.0))
101 /* Return x, regardless of the sign of y. */
102 return x;
103
104 {
105 int negate = ((!signbit (x)) ^ (!signbit (y)));
106
107 /* Take the absolute value of x and y. */
108 x = FABS (x);
109 y = FABS (y);
110
111 /* Trivial case that requires no computation. */
112 if (x < y)
113 return (negate ? - x : x);
114
115 {
116 int yexp;
117 DOUBLE ym;
118 DOUBLE y1;
119 DOUBLE y0;
120 int k;
121 DOUBLE x2;
122 DOUBLE x1;
123 DOUBLE x0;
124
125 /* Write y = 2^yexp * (y1 * 2^-LIMB_BITS + y0 * 2^-(2*LIMB_BITS))
126 where y1 is an integer, 2^(LIMB_BITS-1) <= y1 < 2^LIMB_BITS,
127 y1 has at most LIMB_BITS bits,
128 0 <= y0 < 2^LIMB_BITS,
129 y0 has at most (MANT_DIG + 1) / 2 bits. */
130 ym = FREXP (y, &yexp);
131 ym = ym * TWO_LIMB_BITS;
132 y1 = TRUNC (ym);
133 y0 = (ym - y1) * TWO_LIMB_BITS;
134
135 /* Write
136 x = 2^(yexp+(k-3)*LIMB_BITS)
137 * (x2 * 2^(2*LIMB_BITS) + x1 * 2^LIMB_BITS + x0)
138 where x2, x1, x0 are each integers >= 0, < 2^LIMB_BITS. */
139 {
140 int xexp;
141 DOUBLE xm = FREXP (x, &xexp);
142 /* Since we know x >= y, we know xexp >= yexp. */
143 xexp -= yexp;
144 /* Compute k = ceil(xexp / LIMB_BITS). */
145 k = (xexp + LIMB_BITS - 1) / LIMB_BITS;
146 /* Note: (k - 1) * LIMB_BITS + 1 <= xexp <= k * LIMB_BITS. */
147 /* Note: 0.5 <= xm < 1.0. */
148 xm = LDEXP (xm, xexp - (k - 1) * LIMB_BITS);
149 /* Note: Now xm < 2^(xexp - (k - 1) * LIMB_BITS) <= 2^LIMB_BITS
150 and xm >= 0.5 * 2^(xexp - (k - 1) * LIMB_BITS) >= 1.0
151 and xm has at most MANT_DIG <= 2*LIMB_BITS+1 bits. */
152 x2 = TRUNC (xm);
153 x1 = (xm - x2) * TWO_LIMB_BITS;
154 /* Split off x0 from x1 later. */
155 }
156
157 /* Test whether [x2,x1,0] >= 2^LIMB_BITS * [y1,y0]. */
158 if (x2 > y1 || (x2 == y1 && x1 >= y0))
159 {
160 /* Subtract 2^LIMB_BITS * [y1,y0] from [x2,x1,0]. */
161 x2 -= y1;
162 x1 -= y0;
163 if (x1 < L_(0.0))
164 {
165 if (!(x2 >= L_(1.0)))
166 abort ();
167 x2 -= L_(1.0);
168 x1 += TWO_LIMB_BITS;
169 }
170 }
171
172 /* Split off x0 from x1. */
173 {
174 DOUBLE x1int = TRUNC (x1);
175 x0 = TRUNC ((x1 - x1int) * TWO_LIMB_BITS);
176 x1 = x1int;
177 }
178
179 for (; k > 0; k--)
180 {
181 /* Multiprecision division of the limb sequence [x2,x1,x0]
182 by [y1,y0]. */
183 /* Here [x2,x1,x0] < 2^LIMB_BITS * [y1,y0]. */
184 /* The first guess takes into account only [x2,x1] and [y1].
185
186 By Knuth's theorem, we know that
187 q* = min (floor ([x2,x1] / [y1]), 2^LIMB_BITS - 1)
188 and
189 q = floor ([x2,x1,x0] / [y1,y0])
190 are not far away:
191 q* - 2 <= q <= q* + 1.
192
193 Proof:
194 a) q* * y1 <= floor ([x2,x1] / [y1]) * y1 <= [x2,x1].
195 Hence
196 [x2,x1,x0] - q* * [y1,y0]
197 = 2^LIMB_BITS * ([x2,x1] - q* * [y1]) + x0 - q* * y0
198 >= x0 - q* * y0
199 >= - q* * y0
200 > - 2^(2*LIMB_BITS)
201 >= - 2 * [y1,y0]
202 So
203 [x2,x1,x0] > (q* - 2) * [y1,y0].
204 b) If q* = floor ([x2,x1] / [y1]), then
205 [x2,x1] < (q* + 1) * y1
206 Hence
207 [x2,x1,x0] - q* * [y1,y0]
208 = 2^LIMB_BITS * ([x2,x1] - q* * [y1]) + x0 - q* * y0
209 <= 2^LIMB_BITS * (y1 - 1) + x0 - q* * y0
210 <= 2^LIMB_BITS * (2^LIMB_BITS-2) + (2^LIMB_BITS-1) - 0
211 < 2^(2*LIMB_BITS)
212 <= 2 * [y1,y0]
213 So
214 [x2,x1,x0] < (q* + 2) * [y1,y0].
215 and so
216 q < q* + 2
217 which implies
218 q <= q* + 1.
219 In the other case, q* = 2^LIMB_BITS - 1. Then trivially
220 q < 2^LIMB_BITS = q* + 1.
221
222 We know that floor ([x2,x1] / [y1]) >= 2^LIMB_BITS if and
223 only if x2 >= y1. */
224 DOUBLE q =
225 (x2 >= y1
226 ? TWO_LIMB_BITS - L_(1.0)
227 : TRUNC ((x2 * TWO_LIMB_BITS + x1) / y1));
228 if (q > L_(0.0))
229 {
230 /* Compute
231 [x2,x1,x0] - q* * [y1,y0]
232 = 2^LIMB_BITS * ([x2,x1] - q* * [y1]) + x0 - q* * y0. */
233 DOUBLE q_y1 = q * y1; /* exact, at most 2*LIMB_BITS bits */
234 DOUBLE q_y1_1 = TRUNC (q_y1 * TWO_LIMB_BITS_INVERSE);
235 DOUBLE q_y1_0 = q_y1 - q_y1_1 * TWO_LIMB_BITS;
236 DOUBLE q_y0 = q * y0; /* exact, at most MANT_DIG bits */
237 DOUBLE q_y0_1 = TRUNC (q_y0 * TWO_LIMB_BITS_INVERSE);
238 DOUBLE q_y0_0 = q_y0 - q_y0_1 * TWO_LIMB_BITS;
239 x2 -= q_y1_1;
240 x1 -= q_y1_0;
241 x1 -= q_y0_1;
242 x0 -= q_y0_0;
243 /* Move negative carry from x0 to x1 and from x1 to x2. */
244 if (x0 < L_(0.0))
245 {
246 x0 += TWO_LIMB_BITS;
247 x1 -= L_(1.0);
248 }
249 if (x1 < L_(0.0))
250 {
251 x1 += TWO_LIMB_BITS;
252 x2 -= L_(1.0);
253 if (x1 < L_(0.0)) /* not sure this can happen */
254 {
255 x1 += TWO_LIMB_BITS;
256 x2 -= L_(1.0);
257 }
258 }
259 if (x2 < L_(0.0))
260 {
261 /* Reduce q by 1. */
262 x1 += y1;
263 x0 += y0;
264 /* Move overflow from x0 to x1 and from x1 to x0. */
265 if (x0 >= TWO_LIMB_BITS)
266 {
267 x0 -= TWO_LIMB_BITS;
268 x1 += L_(1.0);
269 }
270 if (x1 >= TWO_LIMB_BITS)
271 {
272 x1 -= TWO_LIMB_BITS;
273 x2 += L_(1.0);
274 }
275 if (x2 < L_(0.0))
276 {
277 /* Reduce q by 1 again. */
278 x1 += y1;
279 x0 += y0;
280 /* Move overflow from x0 to x1 and from x1 to x0. */
281 if (x0 >= TWO_LIMB_BITS)
282 {
283 x0 -= TWO_LIMB_BITS;
284 x1 += L_(1.0);
285 }
286 if (x1 >= TWO_LIMB_BITS)
287 {
288 x1 -= TWO_LIMB_BITS;
289 x2 += L_(1.0);
290 }
291 if (x2 < L_(0.0))
292 /* Shouldn't happen, because we proved that
293 q >= q* - 2. */
294 abort ();
295 }
296 }
297 }
298 if (x2 > L_(0.0)
299 || x1 > y1
300 || (x1 == y1 && x0 >= y0))
301 {
302 /* Increase q by 1. */
303 x1 -= y1;
304 x0 -= y0;
305 /* Move negative carry from x0 to x1 and from x1 to x2. */
306 if (x0 < L_(0.0))
307 {
308 x0 += TWO_LIMB_BITS;
309 x1 -= L_(1.0);
310 }
311 if (x1 < L_(0.0))
312 {
313 x1 += TWO_LIMB_BITS;
314 x2 -= L_(1.0);
315 }
316 if (x2 < L_(0.0))
317 abort ();
318 if (x2 > L_(0.0)
319 || x1 > y1
320 || (x1 == y1 && x0 >= y0))
321 /* Shouldn't happen, because we proved that
322 q <= q* + 1. */
323 abort ();
324 }
325 /* Here [x2,x1,x0] < [y1,y0]. */
326 /* Next round. */
327 x2 = x1;
328 #if (MANT_DIG + 1) / 2 > LIMB_BITS /* y0 can have a fractional bit */
329 x1 = TRUNC (x0);
330 x0 = (x0 - x1) * TWO_LIMB_BITS;
331 #else
332 x1 = x0;
333 x0 = L_(0.0);
334 #endif
335 /* Here [x2,x1,x0] < 2^LIMB_BITS * [y1,y0]. */
336 }
337 /* Here k = 0.
338 The result is
339 2^(yexp-3*LIMB_BITS)
340 * (x2 * 2^(2*LIMB_BITS) + x1 * 2^LIMB_BITS + x0). */
341 {
342 DOUBLE r =
343 LDEXP ((x2 * TWO_LIMB_BITS + x1) * TWO_LIMB_BITS + x0,
344 yexp - 3 * LIMB_BITS);
345 return (negate ? - r : r);
346 }
347 }
348 }
349 }
350 else
351 {
352 if (ISNAN (x) || ISNAN (y))
353 return x + y; /* NaN */
354 else if (isinf (y))
355 return x;
356 else
357 /* x infinite or y zero */
358 return NAN;
359 }
360 }
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*/