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
28 /*
29 * Denorms usually have an exponent biased by 1 so that they flow
30 * smoothly into the smallest normal value with an exponent of
31 * 1. However, m68k 80-bit long doubles includes exponent of zero for
32 * normal values, so denorms use the same value, eliminating the
33 * bias. That is set in s_fmal.c.
34 */
35
36 #ifndef FLOAT_DENORM_BIAS
37 #define FLOAT_DENORM_BIAS 1
38 #endif
39
40 /*
41 * A struct dd represents a floating-point number with twice the precision
42 * of a FLOAT_T. We maintain the invariant that "hi" stores the high-order
43 * bits of the result.
44 */
45 struct dd {
46 FLOAT_T hi;
47 FLOAT_T lo;
48 };
49
50 /*
51 * Compute a+b exactly, returning the exact result in a struct dd. We assume
52 * that both a and b are finite, but make no assumptions about their relative
53 * magnitudes.
54 */
55 static inline struct dd
dd_add(FLOAT_T a,FLOAT_T b)56 dd_add(FLOAT_T a, FLOAT_T b)
57 {
58 struct dd ret;
59 FLOAT_T s;
60
61 ret.hi = a + b;
62 s = ret.hi - a;
63 ret.lo = (a - (ret.hi - s)) + (b - s);
64 return (ret);
65 }
66
67 /*
68 * Compute a+b, with a small tweak: The least significant bit of the
69 * result is adjusted into a sticky bit summarizing all the bits that
70 * were lost to rounding. This adjustment negates the effects of double
71 * rounding when the result is added to another number with a higher
72 * exponent. For an explanation of round and sticky bits, see any reference
73 * on FPU design, e.g.,
74 *
75 * J. Coonen. An Implementation Guide to a Proposed Standard for
76 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
77 */
78 static inline FLOAT_T
add_adjusted(FLOAT_T a,FLOAT_T b)79 add_adjusted(FLOAT_T a, FLOAT_T b)
80 {
81 struct dd sum;
82
83 sum = dd_add(a, b);
84 if (sum.lo != 0) {
85 if (!odd_mant(sum.hi))
86 sum.hi = NEXTAFTER(sum.hi, (FLOAT_T)INFINITY * sum.lo);
87 }
88 return (sum.hi);
89 }
90
91 /*
92 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
93 * that the result will be subnormal, and care is taken to ensure that
94 * double rounding does not occur.
95 */
96 static inline FLOAT_T
add_and_denormalize(FLOAT_T a,FLOAT_T b,int scale)97 add_and_denormalize(FLOAT_T a, FLOAT_T b, int scale)
98 {
99 struct dd sum;
100 int bits_lost;
101
102 sum = dd_add(a, b);
103
104 /*
105 * If we are losing at least two bits of accuracy to denormalization,
106 * then the first lost bit becomes a round bit, and we adjust the
107 * lowest bit of sum.hi to make it a sticky bit summarizing all the
108 * bits in sum.lo. With the sticky bit adjusted, the hardware will
109 * break any ties in the correct direction.
110 *
111 * If we are losing only one bit to denormalization, however, we must
112 * break the ties manually.
113 */
114 if (sum.lo != 0) {
115 bits_lost = -EXPONENT(sum.hi) - scale + FLOAT_DENORM_BIAS;
116 if ((bits_lost != 1) ^ (int)odd_mant(sum.hi))
117 sum.hi = NEXTAFTER(sum.hi, (FLOAT_T)INFINITY * sum.lo);
118 }
119 return (LDEXP(sum.hi, scale));
120 }
121
122 /*
123 * Compute a*b exactly, returning the exact result in a struct dd. We assume
124 * that both a and b are normalized, so no underflow or overflow will occur.
125 * The current rounding mode must be round-to-nearest.
126 */
127 static inline struct dd
dd_mul(FLOAT_T a,FLOAT_T b)128 dd_mul(FLOAT_T a, FLOAT_T b)
129 {
130 static const FLOAT_T split = SPLIT;
131 struct dd ret;
132 FLOAT_T ha, hb, la, lb, p, q;
133
134 p = a * split;
135 ha = a - p;
136 ha += p;
137 la = a - ha;
138
139 p = b * split;
140 hb = b - p;
141 hb += p;
142 lb = b - hb;
143
144 p = ha * hb;
145 q = ha * lb + la * hb;
146
147 ret.hi = p + q;
148 ret.lo = p - ret.hi + q + la * lb;
149 return (ret);
150 }
151
152 #ifdef _WANT_MATH_ERRNO
153 static FLOAT_T
_scalbn_no_errno(FLOAT_T x,int n)154 _scalbn_no_errno(FLOAT_T x, int n)
155 {
156 int save_errno = errno;
157 x = SCALBN(x, n);
158 errno = save_errno;
159 return x;
160 }
161 #else
162 #define _scalbn_no_errno(a,b) SCALBN(a,b)
163 #endif
164
165 #ifdef __clang__
166 #pragma STDC FP_CONTRACT OFF
167 #endif
168
169 #if defined(FE_UPWARD) || defined(FE_DOWNWARD) || defined(FE_TOWARDZERO)
170 #define HAS_ROUNDING
171 #endif
172
173 /*
174 * Fused multiply-add: Compute x * y + z with a single rounding error.
175 *
176 * We use scaling to avoid overflow/underflow, along with the
177 * canonical precision-doubling technique adapted from:
178 *
179 * Dekker, T. A Floating-Point Technique for Extending the
180 * Available Precision. Numer. Math. 18, 224-242 (1971).
181 */
182 FLOAT_T
FMA(FLOAT_T x,FLOAT_T y,FLOAT_T z)183 FMA(FLOAT_T x, FLOAT_T y, FLOAT_T z)
184 {
185 FLOAT_T xs, ys, zs, adj;
186 struct dd xy, r;
187 int ex, ey, ez;
188 int spread;
189
190 /*
191 * Handle special cases. The order of operations and the particular
192 * return values here are crucial in handling special cases involving
193 * infinities, NaNs, overflows, and signed zeroes correctly.
194 */
195 if (!isfinite(z) && isfinite(x) && isfinite(y))
196 return z + z;
197 if (!isfinite(x) || !isfinite(y) || !isfinite(z))
198 return (x * y + z);
199 if (x == (FLOAT_T) 0.0 || y == (FLOAT_T) 0.0)
200 return (x * y + z);
201 if (z == (FLOAT_T) 0.0)
202 return (x * y);
203
204 xs = FREXP(x, &ex);
205 ys = FREXP(y, &ey);
206 zs = FREXP(z, &ez);
207 #ifdef HAS_ROUNDING
208 int oround = fegetround();
209 #endif
210 spread = ex + ey - ez;
211
212 /*
213 * If x * y and z are many orders of magnitude apart, the scaling
214 * will overflow, so we handle these cases specially. Rounding
215 * modes other than FE_TONEAREST are painful.
216 */
217 if (spread < -FLOAT_MANT_DIG) {
218 #ifdef FE_INEXACT
219 feraiseexcept(FE_INEXACT);
220 #endif
221 #ifdef FE_UNDERFLOW
222 if (!isnormal(z))
223 feraiseexcept(FE_UNDERFLOW);
224 #endif
225 #ifdef HAS_ROUNDING
226 switch (oround) {
227 default:
228 break;
229 #ifdef FE_TOWARDZERO
230 case FE_TOWARDZERO:
231 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0) ^ (z < (FLOAT_T) 0.0))
232 break;
233 else
234 return (NEXTAFTER(z, 0));
235 #endif
236 #ifdef FE_DOWNWARD
237 case FE_DOWNWARD:
238 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0))
239 break;
240 else
241 return (NEXTAFTER(z, -(FLOAT_T)INFINITY));
242 #endif
243 #ifdef FE_UPWARD
244 case FE_UPWARD:
245 if ((x > (FLOAT_T) 0.0) ^ (y < (FLOAT_T) 0.0))
246 return (NEXTAFTER(z, (FLOAT_T)INFINITY));
247 break;
248 #endif
249 }
250 #endif
251 return (z);
252 }
253 if (spread <= FLOAT_MANT_DIG * 2)
254 zs = _scalbn_no_errno(zs, -spread);
255 else
256 zs = COPYSIGN(FLOAT_MIN, zs);
257
258 #ifdef HAS_ROUNDING
259 fesetround(FE_TONEAREST);
260 #endif
261
262 /*
263 * Basic approach for round-to-nearest:
264 *
265 * (xy.hi, xy.lo) = x * y (exact)
266 * (r.hi, r.lo) = xy.hi + z (exact)
267 * adj = xy.lo + r.lo (inexact; low bit is sticky)
268 * result = r.hi + adj (correctly rounded)
269 */
270 xy = dd_mul(xs, ys);
271 r = dd_add(xy.hi, zs);
272
273 spread = ex + ey;
274
275 if (r.hi == (FLOAT_T) 0.0) {
276 /*
277 * When the addends cancel to 0, ensure that the result has
278 * the correct sign.
279 */
280 #ifdef HAS_ROUNDING
281 fesetround(oround);
282 #endif
283 volatile FLOAT_T vzs = zs; /* XXX gcc CSE bug workaround */
284 return (xy.hi + vzs + _scalbn_no_errno(xy.lo, spread));
285 }
286
287 #ifdef HAS_ROUNDING
288 if (oround != FE_TONEAREST) {
289 /*
290 * There is no need to worry about double rounding in directed
291 * rounding modes.
292 */
293 fesetround(oround);
294 adj = r.lo + xy.lo;
295 return (_scalbn_no_errno(r.hi + adj, spread));
296 }
297 #endif
298
299 adj = add_adjusted(r.lo, xy.lo);
300 if (spread + ILOGB(r.hi) > -(FLOAT_MAX_EXP - FLOAT_DENORM_BIAS))
301 return (_scalbn_no_errno(r.hi + adj, spread));
302 else
303 return (add_and_denormalize(r.hi, adj, spread));
304 }
305
