2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
4 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
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29 #include <sys/cdefs.h>
30 __FBSDID("$FreeBSD$");
36 #include "math_private.h"
39 * A struct dd represents a floating-point number with twice the precision
40 * of a double. We maintain the invariant that "hi" stores the 53 high-order
49 * Compute a+b exactly, returning the exact result in a struct dd. We assume
50 * that both a and b are finite, but make no assumptions about their relative
53 static inline struct dd
54 dd_add(double a, double b)
61 ret.lo = (a - (ret.hi - s)) + (b - s);
66 * Compute a+b, with a small tweak: The least significant bit of the
67 * result is adjusted into a sticky bit summarizing all the bits that
68 * were lost to rounding. This adjustment negates the effects of double
69 * rounding when the result is added to another number with a higher
70 * exponent. For an explanation of round and sticky bits, see any reference
71 * on FPU design, e.g.,
73 * J. Coonen. An Implementation Guide to a Proposed Standard for
74 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
77 add_adjusted(double a, double b)
80 uint64_t hibits, lobits;
84 EXTRACT_WORD64(hibits, sum.hi);
85 if ((hibits & 1) == 0) {
86 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
87 EXTRACT_WORD64(lobits, sum.lo);
88 hibits += 1 - ((hibits ^ lobits) >> 62);
89 INSERT_WORD64(sum.hi, hibits);
96 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
97 * that the result will be subnormal, and care is taken to ensure that
98 * double rounding does not occur.
101 add_and_denormalize(double a, double b, int scale)
104 uint64_t hibits, lobits;
110 * If we are losing at least two bits of accuracy to denormalization,
111 * then the first lost bit becomes a round bit, and we adjust the
112 * lowest bit of sum.hi to make it a sticky bit summarizing all the
113 * bits in sum.lo. With the sticky bit adjusted, the hardware will
114 * break any ties in the correct direction.
116 * If we are losing only one bit to denormalization, however, we must
117 * break the ties manually.
120 EXTRACT_WORD64(hibits, sum.hi);
121 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
122 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
123 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
124 EXTRACT_WORD64(lobits, sum.lo);
125 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
126 INSERT_WORD64(sum.hi, hibits);
129 return (ldexp(sum.hi, scale));
133 * Compute a*b exactly, returning the exact result in a struct dd. We assume
134 * that both a and b are normalized, so no underflow or overflow will occur.
135 * The current rounding mode must be round-to-nearest.
137 static inline struct dd
138 dd_mul(double a, double b)
140 static const double split = 0x1p27 + 1.0;
142 double ha, hb, la, lb, p, q;
155 q = ha * lb + la * hb;
158 ret.lo = p - ret.hi + q + la * lb;
163 * Fused multiply-add: Compute x * y + z with a single rounding error.
165 * We use scaling to avoid overflow/underflow, along with the
166 * canonical precision-doubling technique adapted from:
168 * Dekker, T. A Floating-Point Technique for Extending the
169 * Available Precision. Numer. Math. 18, 224-242 (1971).
171 * This algorithm is sensitive to the rounding precision. FPUs such
172 * as the i387 must be set in double-precision mode if variables are
173 * to be stored in FP registers in order to avoid incorrect results.
174 * This is the default on FreeBSD, but not on many other systems.
176 * Hardware instructions should be used on architectures that support it,
177 * since this implementation will likely be several times slower.
180 fma(double x, double y, double z)
182 double xs, ys, zs, adj;
189 * Handle special cases. The order of operations and the particular
190 * return values here are crucial in handling special cases involving
191 * infinities, NaNs, overflows, and signed zeroes correctly.
193 if (x == 0.0 || y == 0.0)
197 if (!isfinite(x) || !isfinite(y))
205 oround = fegetround();
206 spread = ex + ey - ez;
209 * If x * y and z are many orders of magnitude apart, the scaling
210 * will overflow, so we handle these cases specially. Rounding
211 * modes other than FE_TONEAREST are painful.
213 if (spread < -DBL_MANT_DIG) {
214 feraiseexcept(FE_INEXACT);
216 feraiseexcept(FE_UNDERFLOW);
221 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
224 return (nextafter(z, 0));
226 if (x > 0.0 ^ y < 0.0)
229 return (nextafter(z, -INFINITY));
230 default: /* FE_UPWARD */
231 if (x > 0.0 ^ y < 0.0)
232 return (nextafter(z, INFINITY));
237 if (spread <= DBL_MANT_DIG * 2)
238 zs = ldexp(zs, -spread);
240 zs = copysign(DBL_MIN, zs);
242 fesetround(FE_TONEAREST);
243 /* work around clang bug 8100 */
244 volatile double vxs = xs;
247 * Basic approach for round-to-nearest:
249 * (xy.hi, xy.lo) = x * y (exact)
250 * (r.hi, r.lo) = xy.hi + z (exact)
251 * adj = xy.lo + r.lo (inexact; low bit is sticky)
252 * result = r.hi + adj (correctly rounded)
254 xy = dd_mul(vxs, ys);
255 r = dd_add(xy.hi, zs);
261 * When the addends cancel to 0, ensure that the result has
265 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
266 return (xy.hi + vzs + ldexp(xy.lo, spread));
269 if (oround != FE_TONEAREST) {
271 * There is no need to worry about double rounding in directed
275 /* work around clang bug 8100 */
276 volatile double vrlo = r.lo;
278 return (ldexp(r.hi + adj, spread));
281 adj = add_adjusted(r.lo, xy.lo);
282 if (spread + ilogb(r.hi) > -1023)
283 return (ldexp(r.hi + adj, spread));
285 return (add_and_denormalize(r.hi, adj, spread));
288 #if (LDBL_MANT_DIG == 53)
289 __weak_reference(fma, fmal);