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 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD$");
29
30 #include <fenv.h>
31 #include <float.h>
32 #include <math.h>
33
34 #include "fpmath.h"
35
36 /*
37 * A struct dd represents a floating-point number with twice the precision
38 * of a long double. We maintain the invariant that "hi" stores the high-order
39 * bits of the result.
40 */
41 struct dd {
42 long double hi;
43 long double lo;
44 };
45
46 /*
47 * Compute a+b exactly, returning the exact result in a struct dd. We assume
48 * that both a and b are finite, but make no assumptions about their relative
49 * magnitudes.
50 */
51 static inline struct dd
dd_add(long double a,long double b)52 dd_add(long double a, long double b)
53 {
54 struct dd ret;
55 long double s;
56
57 ret.hi = a + b;
58 s = ret.hi - a;
59 ret.lo = (a - (ret.hi - s)) + (b - s);
60 return (ret);
61 }
62
63 /*
64 * Compute a+b, with a small tweak: The least significant bit of the
65 * result is adjusted into a sticky bit summarizing all the bits that
66 * were lost to rounding. This adjustment negates the effects of double
67 * rounding when the result is added to another number with a higher
68 * exponent. For an explanation of round and sticky bits, see any reference
69 * on FPU design, e.g.,
70 *
71 * J. Coonen. An Implementation Guide to a Proposed Standard for
72 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
73 */
74 static inline long double
add_adjusted(long double a,long double b)75 add_adjusted(long double a, long double b)
76 {
77 struct dd sum;
78 union IEEEl2bits u;
79
80 sum = dd_add(a, b);
81 if (sum.lo != 0) {
82 u.e = sum.hi;
83 if ((u.bits.manl & 1) == 0)
84 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
85 }
86 return (sum.hi);
87 }
88
89 /*
90 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
91 * that the result will be subnormal, and care is taken to ensure that
92 * double rounding does not occur.
93 */
94 static inline long double
add_and_denormalize(long double a,long double b,int scale)95 add_and_denormalize(long double a, long double b, int scale)
96 {
97 struct dd sum;
98 int bits_lost;
99 union IEEEl2bits u;
100
101 sum = dd_add(a, b);
102
103 /*
104 * If we are losing at least two bits of accuracy to denormalization,
105 * then the first lost bit becomes a round bit, and we adjust the
106 * lowest bit of sum.hi to make it a sticky bit summarizing all the
107 * bits in sum.lo. With the sticky bit adjusted, the hardware will
108 * break any ties in the correct direction.
109 *
110 * If we are losing only one bit to denormalization, however, we must
111 * break the ties manually.
112 */
113 if (sum.lo != 0) {
114 u.e = sum.hi;
115 bits_lost = -u.bits.exp - scale + 1;
116 if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
117 sum.hi = nextafterl(sum.hi, 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(long double a,long double b)128 dd_mul(long double a, long double b)
129 {
130 #if LDBL_MANT_DIG == 64
131 static const long double split = 0x1p32L + 1.0;
132 #elif LDBL_MANT_DIG == 113
133 static const long double split = 0x1p57L + 1.0;
134 #endif
135 struct dd ret;
136 long double ha, hb, la, lb, p, q;
137
138 p = a * split;
139 ha = a - p;
140 ha += p;
141 la = a - ha;
142
143 p = b * split;
144 hb = b - p;
145 hb += p;
146 lb = b - hb;
147
148 p = ha * hb;
149 q = ha * lb + la * hb;
150
151 ret.hi = p + q;
152 ret.lo = p - ret.hi + q + la * lb;
153 return (ret);
154 }
155
156 /*
157 * Fused multiply-add: Compute x * y + z with a single rounding error.
158 *
159 * We use scaling to avoid overflow/underflow, along with the
160 * canonical precision-doubling technique adapted from:
161 *
162 * Dekker, T. A Floating-Point Technique for Extending the
163 * Available Precision. Numer. Math. 18, 224-242 (1971).
164 */
165 long double
fmal(long double x,long double y,long double z)166 fmal(long double x, long double y, long double z)
167 {
168 long double xs, ys, zs, adj;
169 struct dd xy, r;
170 int oround;
171 int ex, ey, ez;
172 int spread;
173
174 /*
175 * Handle special cases. The order of operations and the particular
176 * return values here are crucial in handling special cases involving
177 * infinities, NaNs, overflows, and signed zeroes correctly.
178 */
179 if (x == 0.0 || y == 0.0)
180 return (x * y + z);
181 if (z == 0.0)
182 return (x * y);
183 if (!isfinite(x) || !isfinite(y))
184 return (x * y + z);
185 if (!isfinite(z))
186 return (z);
187
188 xs = frexpl(x, &ex);
189 ys = frexpl(y, &ey);
190 zs = frexpl(z, &ez);
191 oround = fegetround();
192 spread = ex + ey - ez;
193
194 /*
195 * If x * y and z are many orders of magnitude apart, the scaling
196 * will overflow, so we handle these cases specially. Rounding
197 * modes other than FE_TONEAREST are painful.
198 */
199 if (spread < -LDBL_MANT_DIG) {
200 feraiseexcept(FE_INEXACT);
201 if (!isnormal(z))
202 feraiseexcept(FE_UNDERFLOW);
203 switch (oround) {
204 case FE_TONEAREST:
205 return (z);
206 case FE_TOWARDZERO:
207 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
208 return (z);
209 else
210 return (nextafterl(z, 0));
211 case FE_DOWNWARD:
212 if (x > 0.0 ^ y < 0.0)
213 return (z);
214 else
215 return (nextafterl(z, -INFINITY));
216 default: /* FE_UPWARD */
217 if (x > 0.0 ^ y < 0.0)
218 return (nextafterl(z, INFINITY));
219 else
220 return (z);
221 }
222 }
223 if (spread <= LDBL_MANT_DIG * 2)
224 zs = ldexpl(zs, -spread);
225 else
226 zs = copysignl(LDBL_MIN, zs);
227
228 fesetround(FE_TONEAREST);
229 /* work around clang bug 8100 */
230 volatile long double vxs = xs;
231
232 /*
233 * Basic approach for round-to-nearest:
234 *
235 * (xy.hi, xy.lo) = x * y (exact)
236 * (r.hi, r.lo) = xy.hi + z (exact)
237 * adj = xy.lo + r.lo (inexact; low bit is sticky)
238 * result = r.hi + adj (correctly rounded)
239 */
240 xy = dd_mul(vxs, ys);
241 r = dd_add(xy.hi, zs);
242
243 spread = ex + ey;
244
245 if (r.hi == 0.0) {
246 /*
247 * When the addends cancel to 0, ensure that the result has
248 * the correct sign.
249 */
250 fesetround(oround);
251 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
252 return (xy.hi + vzs + ldexpl(xy.lo, spread));
253 }
254
255 if (oround != FE_TONEAREST) {
256 /*
257 * There is no need to worry about double rounding in directed
258 * rounding modes.
259 */
260 fesetround(oround);
261 /* work around clang bug 8100 */
262 volatile long double vrlo = r.lo;
263 adj = vrlo + xy.lo;
264 return (ldexpl(r.hi + adj, spread));
265 }
266
267 adj = add_adjusted(r.lo, xy.lo);
268 if (spread + ilogbl(r.hi) > -16383)
269 return (ldexpl(r.hi + adj, spread));
270 else
271 return (add_and_denormalize(r.hi, adj, spread));
272 }
273