1 /*-
2 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27 //__FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.7 2011/10/21 06:30:43 das Exp $");
28
29 /*
30 * Denorms usually have an exponent biased by 1 so that they flow
31 * smoothly into the smallest normal value with an exponent of
32 * 1. However, m68k 80-bit long doubles includes exponent of zero for
33 * normal values, so denorms use the same value, eliminating the
34 * bias. That is set in s_fmal.c.
35 */
36
37 #ifndef FLOAT_DENORM_BIAS
38 #define FLOAT_DENORM_BIAS 1
39 #endif
40
41 /*
42 * A struct dd represents a floating-point number with twice the precision
43 * of a FLOAT_T. We maintain the invariant that "hi" stores the high-order
44 * bits of the result.
45 */
46 struct dd {
47 FLOAT_T hi;
48 FLOAT_T lo;
49 };
50
51 /*
52 * Compute a+b exactly, returning the exact result in a struct dd. We assume
53 * that both a and b are finite, but make no assumptions about their relative
54 * magnitudes.
55 */
56 static inline struct dd
dd_add(FLOAT_T a,FLOAT_T b)57 dd_add(FLOAT_T a, FLOAT_T b)
58 {
59 struct dd ret;
60 FLOAT_T s;
61
62 ret.hi = a + b;
63 s = ret.hi - a;
64 ret.lo = (a - (ret.hi - s)) + (b - s);
65 return (ret);
66 }
67
68 /*
69 * Compute a+b, with a small tweak: The least significant bit of the
70 * result is adjusted into a sticky bit summarizing all the bits that
71 * were lost to rounding. This adjustment negates the effects of double
72 * rounding when the result is added to another number with a higher
73 * exponent. For an explanation of round and sticky bits, see any reference
74 * on FPU design, e.g.,
75 *
76 * J. Coonen. An Implementation Guide to a Proposed Standard for
77 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
78 */
79 static inline FLOAT_T
add_adjusted(FLOAT_T a,FLOAT_T b)80 add_adjusted(FLOAT_T a, FLOAT_T b)
81 {
82 struct dd sum;
83
84 sum = dd_add(a, b);
85 if (sum.lo != 0) {
86 if (!odd_mant(sum.hi))
87 sum.hi = NEXTAFTER(sum.hi, (FLOAT_T)INFINITY * sum.lo);
88 }
89 return (sum.hi);
90 }
91
92 /*
93 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
94 * that the result will be subnormal, and care is taken to ensure that
95 * double rounding does not occur.
96 */
97 static inline FLOAT_T
add_and_denormalize(FLOAT_T a,FLOAT_T b,int scale)98 add_and_denormalize(FLOAT_T a, FLOAT_T b, int scale)
99 {
100 struct dd sum;
101 int bits_lost;
102
103 sum = dd_add(a, b);
104
105 /*
106 * If we are losing at least two bits of accuracy to denormalization,
107 * then the first lost bit becomes a round bit, and we adjust the
108 * lowest bit of sum.hi to make it a sticky bit summarizing all the
109 * bits in sum.lo. With the sticky bit adjusted, the hardware will
110 * break any ties in the correct direction.
111 *
112 * If we are losing only one bit to denormalization, however, we must
113 * break the ties manually.
114 */
115 if (sum.lo != 0) {
116 bits_lost = -EXPONENT(sum.hi) - scale + FLOAT_DENORM_BIAS;
117 if ((bits_lost != 1) ^ (int)odd_mant(sum.hi))
118 sum.hi = NEXTAFTER(sum.hi, (FLOAT_T)INFINITY * sum.lo);
119 }
120 return (LDEXP(sum.hi, scale));
121 }
122
123 /*
124 * Compute a*b exactly, returning the exact result in a struct dd. We assume
125 * that both a and b are normalized, so no underflow or overflow will occur.
126 * The current rounding mode must be round-to-nearest.
127 */
128 static inline struct dd
dd_mul(FLOAT_T a,FLOAT_T b)129 dd_mul(FLOAT_T a, FLOAT_T b)
130 {
131 static const FLOAT_T split = SPLIT;
132 struct dd ret;
133 FLOAT_T ha, hb, la, lb, p, q;
134
135 p = a * split;
136 ha = a - p;
137 ha += p;
138 la = a - ha;
139
140 p = b * split;
141 hb = b - p;
142 hb += p;
143 lb = b - hb;
144
145 p = ha * hb;
146 q = ha * lb + la * hb;
147
148 ret.hi = p + q;
149 ret.lo = p - ret.hi + q + la * lb;
150 return (ret);
151 }
152
153 #ifdef _WANT_MATH_ERRNO
154 static FLOAT_T
_scalbn_no_errno(FLOAT_T x,int n)155 _scalbn_no_errno(FLOAT_T x, int n)
156 {
157 int save_errno = errno;
158 x = SCALBN(x, n);
159 errno = save_errno;
160 return x;
161 }
162 #else
163 #define _scalbn_no_errno(a,b) SCALBN(a,b)
164 #endif
165
166 #ifdef __clang__
167 #pragma STDC FP_CONTRACT OFF
168 #endif
169
170 #if defined(FE_UPWARD) || defined(FE_DOWNWARD) || defined(FE_TOWARDZERO)
171 #define HAS_ROUNDING
172 #endif
173
174 /*
175 * Fused multiply-add: Compute x * y + z with a single rounding error.
176 *
177 * We use scaling to avoid overflow/underflow, along with the
178 * canonical precision-doubling technique adapted from:
179 *
180 * Dekker, T. A Floating-Point Technique for Extending the
181 * Available Precision. Numer. Math. 18, 224-242 (1971).
182 */
183 FLOAT_T
FMA(FLOAT_T x,FLOAT_T y,FLOAT_T z)184 FMA(FLOAT_T x, FLOAT_T y, FLOAT_T z)
185 {
186 FLOAT_T xs, ys, zs, adj;
187 struct dd xy, r;
188 int ex, ey, ez;
189 int spread;
190
191 /*
192 * Handle special cases. The order of operations and the particular
193 * return values here are crucial in handling special cases involving
194 * infinities, NaNs, overflows, and signed zeroes correctly.
195 */
196 if (!isfinite(z) && isfinite(x) && isfinite(y))
197 return z + z;
198 if (!isfinite(x) || !isfinite(y) || !isfinite(z))
199 return (x * y + z);
200 if (x == (FLOAT_T) 0.0 || y == (FLOAT_T) 0.0)
201 return (x * y + z);
202 if (z == (FLOAT_T) 0.0)
203 return (x * y);
204
205 xs = FREXP(x, &ex);
206 ys = FREXP(y, &ey);
207 zs = FREXP(z, &ez);
208 #ifdef HAS_ROUNDING
209 int oround = fegetround();
210 #endif
211 spread = ex + ey - ez;
212
213 /*
214 * If x * y and z are many orders of magnitude apart, the scaling
215 * will overflow, so we handle these cases specially. Rounding
216 * modes other than FE_TONEAREST are painful.
217 */
218 if (spread < -FLOAT_MANT_DIG) {
219 #ifdef FE_INEXACT
220 feraiseexcept(FE_INEXACT);
221 #endif
222 #ifdef FE_UNDERFLOW
223 if (!isnormal(z))
224 feraiseexcept(FE_UNDERFLOW);
225 #endif
226 #ifdef HAS_ROUNDING
227 switch (oround) {
228 default:
229 break;
230 #ifdef FE_TOWARDZERO
231 case FE_TOWARDZERO:
232 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0) ^ (z < (FLOAT_T) 0.0))
233 break;
234 else
235 return (NEXTAFTER(z, 0));
236 #endif
237 #ifdef FE_DOWNWARD
238 case FE_DOWNWARD:
239 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0))
240 break;
241 else
242 return (NEXTAFTER(z, -(FLOAT_T)INFINITY));
243 #endif
244 #ifdef FE_UPWARD
245 case FE_UPWARD:
246 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0))
247 return (NEXTAFTER(z, (FLOAT_T)INFINITY));
248 break;
249 #endif
250 }
251 #endif
252 return (z);
253 }
254 if (spread <= FLOAT_MANT_DIG * 2)
255 zs = _scalbn_no_errno(zs, -spread);
256 else
257 zs = COPYSIGN(FLOAT_MIN, zs);
258
259 #ifdef HAS_ROUNDING
260 fesetround(FE_TONEAREST);
261 #endif
262
263 /*
264 * Basic approach for round-to-nearest:
265 *
266 * (xy.hi, xy.lo) = x * y (exact)
267 * (r.hi, r.lo) = xy.hi + z (exact)
268 * adj = xy.lo + r.lo (inexact; low bit is sticky)
269 * result = r.hi + adj (correctly rounded)
270 */
271 xy = dd_mul(xs, ys);
272 r = dd_add(xy.hi, zs);
273
274 spread = ex + ey;
275
276 if (r.hi == (FLOAT_T) 0.0) {
277 /*
278 * When the addends cancel to 0, ensure that the result has
279 * the correct sign.
280 */
281 #ifdef HAS_ROUNDING
282 fesetround(oround);
283 #endif
284 volatile FLOAT_T vzs = zs; /* XXX gcc CSE bug workaround */
285 return (xy.hi + vzs + _scalbn_no_errno(xy.lo, spread));
286 }
287
288 #ifdef HAS_ROUNDING
289 if (oround != FE_TONEAREST) {
290 /*
291 * There is no need to worry about double rounding in directed
292 * rounding modes.
293 */
294 fesetround(oround);
295 adj = r.lo + xy.lo;
296 return (_scalbn_no_errno(r.hi + adj, spread));
297 }
298 #endif
299
300 adj = add_adjusted(r.lo, xy.lo);
301 if (spread + ILOGB(r.hi) > -(FLOAT_MAX_EXP - FLOAT_DENORM_BIAS))
302 return (_scalbn_no_errno(r.hi + adj, spread));
303 else
304 return (add_and_denormalize(r.hi, adj, spread));
305 }
306
