1 /* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */
3 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
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30 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
31 long double fmal(long double x, long double y, long double z)
35 #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
39 * A struct dd represents a floating-point number with twice the precision
40 * of a long double. We maintain the invariant that "hi" stores the 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 dd_add(long double a, long double b)
60 ret.lo = (a - (ret.hi - s)) + (b - s);
65 * Compute a+b, with a small tweak: The least significant bit of the
66 * result is adjusted into a sticky bit summarizing all the bits that
67 * were lost to rounding. This adjustment negates the effects of double
68 * rounding when the result is added to another number with a higher
69 * exponent. For an explanation of round and sticky bits, see any reference
70 * on FPU design, e.g.,
72 * J. Coonen. An Implementation Guide to a Proposed Standard for
73 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
75 static inline long double add_adjusted(long double a, long double b)
83 if ((u.bits.manl & 1) == 0)
84 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
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.
94 static inline long double 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 scalbnl(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 dd_mul(long double a, long double b)
128 #if LDBL_MANT_DIG == 64
129 static const long double split = 0x1p32L + 1.0;
130 #elif LDBL_MANT_DIG == 113
131 static const long double split = 0x1p57L + 1.0;
134 long double ha, hb, la, lb, p, q;
147 q = ha * lb + la * hb;
150 ret.lo = p - ret.hi + q + la * lb;
155 * Fused multiply-add: Compute x * y + z with a single rounding error.
157 * We use scaling to avoid overflow/underflow, along with the
158 * canonical precision-doubling technique adapted from:
160 * Dekker, T. A Floating-Point Technique for Extending the
161 * Available Precision. Numer. Math. 18, 224-242 (1971).
163 long double fmal(long double x, long double y, long double z)
165 long double xs, ys, zs, adj;
172 * Handle special cases. The order of operations and the particular
173 * return values here are crucial in handling special cases involving
174 * infinities, NaNs, overflows, and signed zeroes correctly.
176 if (!isfinite(x) || !isfinite(y))
180 if (x == 0.0 || y == 0.0)
188 oround = fegetround();
189 spread = ex + ey - ez;
192 * If x * y and z are many orders of magnitude apart, the scaling
193 * will overflow, so we handle these cases specially. Rounding
194 * modes other than FE_TONEAREST are painful.
196 if (spread < -LDBL_MANT_DIG) {
198 feraiseexcept(FE_INEXACT);
202 feraiseexcept(FE_UNDERFLOW);
205 default: /* FE_TONEAREST */
209 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
212 return (nextafterl(z, 0));
216 if (x > 0.0 ^ y < 0.0)
219 return (nextafterl(z, -INFINITY));
223 if (x > 0.0 ^ y < 0.0)
224 return (nextafterl(z, INFINITY));
230 if (spread <= LDBL_MANT_DIG * 2)
231 zs = scalbnl(zs, -spread);
233 zs = copysignl(LDBL_MIN, zs);
235 fesetround(FE_TONEAREST);
238 * Basic approach for round-to-nearest:
240 * (xy.hi, xy.lo) = x * y (exact)
241 * (r.hi, r.lo) = xy.hi + z (exact)
242 * adj = xy.lo + r.lo (inexact; low bit is sticky)
243 * result = r.hi + adj (correctly rounded)
246 r = dd_add(xy.hi, zs);
252 * When the addends cancel to 0, ensure that the result has
256 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
257 return xy.hi + vzs + scalbnl(xy.lo, spread);
260 if (oround != FE_TONEAREST) {
262 * There is no need to worry about double rounding in directed
267 return scalbnl(r.hi + adj, spread);
270 adj = add_adjusted(r.lo, xy.lo);
271 if (spread + ilogbl(r.hi) > -16383)
272 return scalbnl(r.hi + adj, spread);
274 return add_and_denormalize(r.hi, adj, spread);