2 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
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26 * $FreeBSD: head/lib/msun/src/s_fmal.c 252170 2013-06-24 19:12:17Z eadler $
36 * A struct dd represents a floating-point number with twice the precision
37 * of a long double. We maintain the invariant that "hi" stores the 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(long double a, long 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.
73 static inline long double
74 add_adjusted(long double a, long double b)
82 if ((u.bits.manl & 1) == 0)
83 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
89 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
90 * that the result will be subnormal, and care is taken to ensure that
91 * double rounding does not occur.
93 static inline long double
94 add_and_denormalize(long double a, long double b, int scale)
103 * If we are losing at least two bits of accuracy to denormalization,
104 * then the first lost bit becomes a round bit, and we adjust the
105 * lowest bit of sum.hi to make it a sticky bit summarizing all the
106 * bits in sum.lo. With the sticky bit adjusted, the hardware will
107 * break any ties in the correct direction.
109 * If we are losing only one bit to denormalization, however, we must
110 * break the ties manually.
114 bits_lost = -u.bits.exp - scale + 1;
115 if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
116 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
118 return (ldexp(sum.hi, scale));
122 * Compute a*b exactly, returning the exact result in a struct dd. We assume
123 * that both a and b are normalized, so no underflow or overflow will occur.
124 * The current rounding mode must be round-to-nearest.
126 static inline struct dd
127 dd_mul(long double a, long double b)
129 #if LDBL_MANT_DIG == 64
130 static const long double split = 0x1p32L + 1.0;
131 #elif LDBL_MANT_DIG == 113
132 static const long double split = 0x1p57L + 1.0;
135 long double ha, hb, la, lb, p, q;
148 q = ha * lb + la * hb;
151 ret.lo = p - ret.hi + q + la * lb;
156 * Fused multiply-add: Compute x * y + z with a single rounding error.
158 * We use scaling to avoid overflow/underflow, along with the
159 * canonical precision-doubling technique adapted from:
161 * Dekker, T. A Floating-Point Technique for Extending the
162 * Available Precision. Numer. Math. 18, 224-242 (1971).
165 fmal(long double x, long double y, long double z)
167 long double xs, ys, zs, adj;
174 * Handle special cases. The order of operations and the particular
175 * return values here are crucial in handling special cases involving
176 * infinities, NaNs, overflows, and signed zeroes correctly.
178 if (x == 0.0 || y == 0.0)
182 if (!isfinite(x) || !isfinite(y))
190 oround = fegetround();
191 spread = ex + ey - ez;
194 * If x * y and z are many orders of magnitude apart, the scaling
195 * will overflow, so we handle these cases specially. Rounding
196 * modes other than FE_TONEAREST are painful.
198 if (spread < -LDBL_MANT_DIG) {
199 feraiseexcept(FE_INEXACT);
201 feraiseexcept(FE_UNDERFLOW);
206 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
209 return (nextafterl(z, 0));
211 if (x > 0.0 ^ y < 0.0)
214 return (nextafterl(z, -INFINITY));
215 default: /* FE_UPWARD */
216 if (x > 0.0 ^ y < 0.0)
217 return (nextafterl(z, INFINITY));
222 if (spread <= LDBL_MANT_DIG * 2)
223 zs = ldexpl(zs, -spread);
225 zs = copysignl(LDBL_MIN, zs);
227 fesetround(FE_TONEAREST);
228 /* work around clang bug 8100 */
229 volatile long double vxs = xs;
232 * Basic approach for round-to-nearest:
234 * (xy.hi, xy.lo) = x * y (exact)
235 * (r.hi, r.lo) = xy.hi + z (exact)
236 * adj = xy.lo + r.lo (inexact; low bit is sticky)
237 * result = r.hi + adj (correctly rounded)
239 xy = dd_mul(vxs, ys);
240 r = dd_add(xy.hi, zs);
246 * When the addends cancel to 0, ensure that the result has
250 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
251 return (xy.hi + vzs + ldexpl(xy.lo, spread));
254 if (oround != FE_TONEAREST) {
256 * There is no need to worry about double rounding in directed
260 /* work around clang bug 8100 */
261 volatile long double vrlo = r.lo;
263 return (ldexpl(r.hi + adj, spread));
266 adj = add_adjusted(r.lo, xy.lo);
267 if (spread + ilogbl(r.hi) > -16383)
268 return (ldexpl(r.hi + adj, spread));
270 return (add_and_denormalize(r.hi, adj, spread));