2 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
5 * Redistribution and use in source and binary forms, with or without
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23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * FreeBSD SVN: 226601 (2011-10-21)
33 #include "math_private.h"
36 * A struct dd represents a floating-point number with twice the precision
37 * of a double. We maintain the invariant that "hi" stores the 53 high-order
46 * Compute a+b exactly, returning the exact result in a struct dd. We assume
47 * that both a and b are finite, but make no assumptions about their relative
50 static inline struct dd
51 dd_add(double a, double b)
58 ret.lo = (a - (ret.hi - s)) + (b - s);
63 * Compute a+b, with a small tweak: The least significant bit of the
64 * result is adjusted into a sticky bit summarizing all the bits that
65 * were lost to rounding. This adjustment negates the effects of double
66 * rounding when the result is added to another number with a higher
67 * exponent. For an explanation of round and sticky bits, see any reference
68 * on FPU design, e.g.,
70 * J. Coonen. An Implementation Guide to a Proposed Standard for
71 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
74 add_adjusted(double a, double b)
77 uint64_t hibits, lobits;
81 EXTRACT_WORD64(hibits, sum.hi);
82 if ((hibits & 1) == 0) {
83 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
84 EXTRACT_WORD64(lobits, sum.lo);
85 hibits += 1 - ((hibits ^ lobits) >> 62);
86 INSERT_WORD64(sum.hi, hibits);
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.
98 add_and_denormalize(double a, double b, int scale)
101 uint64_t hibits, lobits;
107 * If we are losing at least two bits of accuracy to denormalization,
108 * then the first lost bit becomes a round bit, and we adjust the
109 * lowest bit of sum.hi to make it a sticky bit summarizing all the
110 * bits in sum.lo. With the sticky bit adjusted, the hardware will
111 * break any ties in the correct direction.
113 * If we are losing only one bit to denormalization, however, we must
114 * break the ties manually.
117 EXTRACT_WORD64(hibits, sum.hi);
118 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
119 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
120 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
121 EXTRACT_WORD64(lobits, sum.lo);
122 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
123 INSERT_WORD64(sum.hi, hibits);
126 return (ldexp(sum.hi, scale));
130 * Compute a*b exactly, returning the exact result in a struct dd. We assume
131 * that both a and b are normalized, so no underflow or overflow will occur.
132 * The current rounding mode must be round-to-nearest.
134 static inline struct dd
135 dd_mul(double a, double b)
137 static const double split = 0x1p27 + 1.0;
139 double ha, hb, la, lb, p, q;
152 q = ha * lb + la * hb;
155 ret.lo = p - ret.hi + q + la * lb;
160 * Fused multiply-add: Compute x * y + z with a single rounding error.
162 * We use scaling to avoid overflow/underflow, along with the
163 * canonical precision-doubling technique adapted from:
165 * Dekker, T. A Floating-Point Technique for Extending the
166 * Available Precision. Numer. Math. 18, 224-242 (1971).
168 * This algorithm is sensitive to the rounding precision. FPUs such
169 * as the i387 must be set in double-precision mode if variables are
170 * to be stored in FP registers in order to avoid incorrect results.
171 * This is the default on FreeBSD, but not on many other systems.
173 * Hardware instructions should be used on architectures that support it,
174 * since this implementation will likely be several times slower.
177 fma(double x, double y, double z)
179 double xs, ys, zs, adj;
186 * Handle special cases. The order of operations and the particular
187 * return values here are crucial in handling special cases involving
188 * infinities, NaNs, overflows, and signed zeroes correctly.
190 if (x == 0.0 || y == 0.0)
194 if (!isfinite(x) || !isfinite(y))
202 oround = fegetround();
203 spread = ex + ey - ez;
206 * If x * y and z are many orders of magnitude apart, the scaling
207 * will overflow, so we handle these cases specially. Rounding
208 * modes other than FE_TONEAREST are painful.
210 if (spread < -DBL_MANT_DIG) {
211 feraiseexcept(FE_INEXACT);
213 feraiseexcept(FE_UNDERFLOW);
218 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
221 return (nextafter(z, 0));
223 if ((x > 0.0) ^ (y < 0.0))
226 return (nextafter(z, -INFINITY));
227 default: /* FE_UPWARD */
228 if ((x > 0.0) ^ (y < 0.0))
229 return (nextafter(z, INFINITY));
234 if (spread <= DBL_MANT_DIG * 2)
235 zs = ldexp(zs, -spread);
237 zs = copysign(DBL_MIN, zs);
239 fesetround(FE_TONEAREST);
242 * Basic approach for round-to-nearest:
244 * (xy.hi, xy.lo) = x * y (exact)
245 * (r.hi, r.lo) = xy.hi + z (exact)
246 * adj = xy.lo + r.lo (inexact; low bit is sticky)
247 * result = r.hi + adj (correctly rounded)
250 r = dd_add(xy.hi, zs);
256 * When the addends cancel to 0, ensure that the result has
260 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
261 return (xy.hi + vzs + ldexp(xy.lo, spread));
264 if (oround != FE_TONEAREST) {
266 * There is no need to worry about double rounding in directed
271 return (ldexp(r.hi + adj, spread));
274 adj = add_adjusted(r.lo, xy.lo);
275 if (spread + ilogb(r.hi) > -1023)
276 return (ldexp(r.hi + adj, spread));
278 return (add_and_denormalize(r.hi, adj, spread));
281 #if (LDBL_MANT_DIG == 53)
282 __weak_reference(fma, fmal);