1 /* origin: FreeBSD /usr/src/lib/msun/src/s_fma.c */
3 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
6 * Redistribution and use in source and binary forms, with or without
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22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
32 * A struct dd represents a floating-point number with twice the precision
33 * of a double. We maintain the invariant that "hi" stores the 53 high-order
42 * Compute a+b exactly, returning the exact result in a struct dd. We assume
43 * that both a and b are finite, but make no assumptions about their relative
46 static inline struct dd dd_add(double a, double b)
53 ret.lo = (a - (ret.hi - s)) + (b - s);
58 * Compute a+b, with a small tweak: The least significant bit of the
59 * result is adjusted into a sticky bit summarizing all the bits that
60 * were lost to rounding. This adjustment negates the effects of double
61 * rounding when the result is added to another number with a higher
62 * exponent. For an explanation of round and sticky bits, see any reference
63 * on FPU design, e.g.,
65 * J. Coonen. An Implementation Guide to a Proposed Standard for
66 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
68 static inline double add_adjusted(double a, double b)
71 uint64_t hibits, lobits;
75 EXTRACT_WORD64(hibits, sum.hi);
76 if ((hibits & 1) == 0) {
77 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
78 EXTRACT_WORD64(lobits, sum.lo);
79 hibits += 1 - ((hibits ^ lobits) >> 62);
80 INSERT_WORD64(sum.hi, hibits);
87 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
88 * that the result will be subnormal, and care is taken to ensure that
89 * double rounding does not occur.
91 static inline double add_and_denormalize(double a, double b, int scale)
94 uint64_t hibits, lobits;
100 * If we are losing at least two bits of accuracy to denormalization,
101 * then the first lost bit becomes a round bit, and we adjust the
102 * lowest bit of sum.hi to make it a sticky bit summarizing all the
103 * bits in sum.lo. With the sticky bit adjusted, the hardware will
104 * break any ties in the correct direction.
106 * If we are losing only one bit to denormalization, however, we must
107 * break the ties manually.
110 EXTRACT_WORD64(hibits, sum.hi);
111 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
112 if (bits_lost != 1 ^ (int)(hibits & 1)) {
113 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
114 EXTRACT_WORD64(lobits, sum.lo);
115 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
116 INSERT_WORD64(sum.hi, hibits);
119 return (ldexp(sum.hi, scale));
123 * Compute a*b exactly, returning the exact result in a struct dd. We assume
124 * that both a and b are normalized, so no underflow or overflow will occur.
125 * The current rounding mode must be round-to-nearest.
127 static inline struct dd dd_mul(double a, double b)
129 static const double split = 0x1p27 + 1.0;
131 double ha, hb, la, lb, p, q;
144 q = ha * lb + la * hb;
147 ret.lo = p - ret.hi + q + la * lb;
152 * Fused multiply-add: Compute x * y + z with a single rounding error.
154 * We use scaling to avoid overflow/underflow, along with the
155 * canonical precision-doubling technique adapted from:
157 * Dekker, T. A Floating-Point Technique for Extending the
158 * Available Precision. Numer. Math. 18, 224-242 (1971).
160 * This algorithm is sensitive to the rounding precision. FPUs such
161 * as the i387 must be set in double-precision mode if variables are
162 * to be stored in FP registers in order to avoid incorrect results.
163 * This is the default on FreeBSD, but not on many other systems.
165 * Hardware instructions should be used on architectures that support it,
166 * since this implementation will likely be several times slower.
168 double fma(double x, double y, double z)
170 double xs, ys, zs, adj;
177 * Handle special cases. The order of operations and the particular
178 * return values here are crucial in handling special cases involving
179 * infinities, NaNs, overflows, and signed zeroes correctly.
181 if (x == 0.0 || y == 0.0)
185 if (!isfinite(x) || !isfinite(y))
193 oround = fegetround();
194 spread = ex + ey - ez;
197 * If x * y and z are many orders of magnitude apart, the scaling
198 * will overflow, so we handle these cases specially. Rounding
199 * modes other than FE_TONEAREST are painful.
201 if (spread < -DBL_MANT_DIG) {
202 feraiseexcept(FE_INEXACT);
204 feraiseexcept(FE_UNDERFLOW);
209 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
212 return (nextafter(z, 0));
214 if (x > 0.0 ^ y < 0.0)
217 return (nextafter(z, -INFINITY));
218 default: /* FE_UPWARD */
219 if (x > 0.0 ^ y < 0.0)
220 return (nextafter(z, INFINITY));
225 if (spread <= DBL_MANT_DIG * 2)
226 zs = ldexp(zs, -spread);
228 zs = copysign(DBL_MIN, zs);
230 fesetround(FE_TONEAREST);
233 * Basic approach for round-to-nearest:
235 * (xy.hi, xy.lo) = x * y (exact)
236 * (r.hi, r.lo) = xy.hi + z (exact)
237 * adj = xy.lo + r.lo (inexact; low bit is sticky)
238 * result = r.hi + adj (correctly rounded)
241 r = dd_add(xy.hi, zs);
247 * When the addends cancel to 0, ensure that the result has
251 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
252 return (xy.hi + vzs + ldexp(xy.lo, spread));
255 if (oround != FE_TONEAREST) {
257 * There is no need to worry about double rounding in directed
262 return (ldexp(r.hi + adj, spread));
265 adj = add_adjusted(r.lo, xy.lo);
266 if (spread + ilogb(r.hi) > -1023)
267 return (ldexp(r.hi + adj, spread));
269 return (add_and_denormalize(r.hi, adj, spread));