2 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
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.
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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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
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21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
27 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD$");
34 #include "math_private.h"
37 * A struct dd represents a floating-point number with twice the precision
38 * of a double. We maintain the invariant that "hi" stores the 53 high-order
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
51 static inline struct dd
52 dd_add(double a, double b)
59 ret.lo = (a - (ret.hi - s)) + (b - s);
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.,
71 * J. Coonen. An Implementation Guide to a Proposed Standard for
72 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
75 add_adjusted(double a, double b)
78 uint64_t hibits, lobits;
82 EXTRACT_WORD64(hibits, sum.hi);
83 if ((hibits & 1) == 0) {
84 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
85 EXTRACT_WORD64(lobits, sum.lo);
86 hibits += 1 - ((hibits ^ lobits) >> 62);
87 INSERT_WORD64(sum.hi, hibits);
94 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
95 * that the result will be subnormal, and care is taken to ensure that
96 * double rounding does not occur.
99 add_and_denormalize(double a, double b, int scale)
102 uint64_t hibits, lobits;
108 * If we are losing at least two bits of accuracy to denormalization,
109 * then the first lost bit becomes a round bit, and we adjust the
110 * lowest bit of sum.hi to make it a sticky bit summarizing all the
111 * bits in sum.lo. With the sticky bit adjusted, the hardware will
112 * break any ties in the correct direction.
114 * If we are losing only one bit to denormalization, however, we must
115 * break the ties manually.
118 EXTRACT_WORD64(hibits, sum.hi);
119 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
120 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
121 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
122 EXTRACT_WORD64(lobits, sum.lo);
123 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
124 INSERT_WORD64(sum.hi, hibits);
127 return (ldexp(sum.hi, scale));
131 * Compute a*b exactly, returning the exact result in a struct dd. We assume
132 * that both a and b are normalized, so no underflow or overflow will occur.
133 * The current rounding mode must be round-to-nearest.
135 static inline struct dd
136 dd_mul(double a, double b)
138 static const double split = 0x1p27 + 1.0;
140 double ha, hb, la, lb, p, q;
153 q = ha * lb + la * hb;
156 ret.lo = p - ret.hi + q + la * lb;
161 * Fused multiply-add: Compute x * y + z with a single rounding error.
163 * We use scaling to avoid overflow/underflow, along with the
164 * canonical precision-doubling technique adapted from:
166 * Dekker, T. A Floating-Point Technique for Extending the
167 * Available Precision. Numer. Math. 18, 224-242 (1971).
169 * This algorithm is sensitive to the rounding precision. FPUs such
170 * as the i387 must be set in double-precision mode if variables are
171 * to be stored in FP registers in order to avoid incorrect results.
172 * This is the default on FreeBSD, but not on many other systems.
174 * Hardware instructions should be used on architectures that support it,
175 * since this implementation will likely be several times slower.
178 fma(double x, double y, double z)
180 double xs, ys, zs, adj;
187 * Handle special cases. The order of operations and the particular
188 * return values here are crucial in handling special cases involving
189 * infinities, NaNs, overflows, and signed zeroes correctly.
191 if (x == 0.0 || y == 0.0)
195 if (!isfinite(x) || !isfinite(y))
203 oround = fegetround();
204 spread = ex + ey - ez;
207 * If x * y and z are many orders of magnitude apart, the scaling
208 * will overflow, so we handle these cases specially. Rounding
209 * modes other than FE_TONEAREST are painful.
211 if (spread < -DBL_MANT_DIG) {
212 feraiseexcept(FE_INEXACT);
214 feraiseexcept(FE_UNDERFLOW);
219 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
222 return (nextafter(z, 0));
224 if (x > 0.0 ^ y < 0.0)
227 return (nextafter(z, -INFINITY));
228 default: /* FE_UPWARD */
229 if (x > 0.0 ^ y < 0.0)
230 return (nextafter(z, INFINITY));
235 if (spread <= DBL_MANT_DIG * 2)
236 zs = ldexp(zs, -spread);
238 zs = copysign(DBL_MIN, zs);
240 fesetround(FE_TONEAREST);
241 /* work around clang bug 8100 */
242 volatile double vxs = xs;
245 * Basic approach for round-to-nearest:
247 * (xy.hi, xy.lo) = x * y (exact)
248 * (r.hi, r.lo) = xy.hi + z (exact)
249 * adj = xy.lo + r.lo (inexact; low bit is sticky)
250 * result = r.hi + adj (correctly rounded)
252 xy = dd_mul(vxs, ys);
253 r = dd_add(xy.hi, zs);
259 * When the addends cancel to 0, ensure that the result has
263 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
264 return (xy.hi + vzs + ldexp(xy.lo, spread));
267 if (oround != FE_TONEAREST) {
269 * There is no need to worry about double rounding in directed
273 /* work around clang bug 8100 */
274 volatile double vrlo = r.lo;
276 return (ldexp(r.hi + adj, spread));
279 adj = add_adjusted(r.lo, xy.lo);
280 if (spread + ilogb(r.hi) > -1023)
281 return (ldexp(r.hi + adj, spread));
283 return (add_and_denormalize(r.hi, adj, spread));
286 #if (LDBL_MANT_DIG == 53)
287 __weak_reference(fma, fmal);