1/* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */ 2/*- 3 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28 29#include "libm.h" 30#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 31long double fmal(long double x, long double y, long double z) 32{ 33 return fma(x, y, z); 34} 35#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384 36#include <fenv.h> 37#if LDBL_MANT_DIG == 64 38#define LASTBIT(u) (u.i.m & 1) 39#define SPLIT (0x1p32L + 1) 40#elif LDBL_MANT_DIG == 113 41#define LASTBIT(u) (u.i.lo & 1) 42#define SPLIT (0x1p57L + 1) 43#endif 44 45/* 46 * A struct dd represents a floating-point number with twice the precision 47 * of a long double. We maintain the invariant that "hi" stores the high-order 48 * bits of the result. 49 */ 50struct dd { 51 long double hi; 52 long double lo; 53}; 54 55/* 56 * Compute a+b exactly, returning the exact result in a struct dd. We assume 57 * that both a and b are finite, but make no assumptions about their relative 58 * magnitudes. 59 */ 60static inline struct dd dd_add(long double a, long double b) 61{ 62 struct dd ret; 63 long double s; 64 65 ret.hi = a + b; 66 s = ret.hi - a; 67 ret.lo = (a - (ret.hi - s)) + (b - s); 68 return (ret); 69} 70 71/* 72 * Compute a+b, with a small tweak: The least significant bit of the 73 * result is adjusted into a sticky bit summarizing all the bits that 74 * were lost to rounding. This adjustment negates the effects of double 75 * rounding when the result is added to another number with a higher 76 * exponent. For an explanation of round and sticky bits, see any reference 77 * on FPU design, e.g., 78 * 79 * J. Coonen. An Implementation Guide to a Proposed Standard for 80 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980. 81 */ 82static inline long double add_adjusted(long double a, long double b) 83{ 84 struct dd sum; 85 union ldshape u; 86 87 sum = dd_add(a, b); 88 if (sum.lo != 0) { 89 u.f = sum.hi; 90 if (!LASTBIT(u)) 91 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); 92 } 93 return (sum.hi); 94} 95 96/* 97 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed 98 * that the result will be subnormal, and care is taken to ensure that 99 * double rounding does not occur. 100 */ 101static inline long double add_and_denormalize(long double a, long double b, int scale) 102{ 103 struct dd sum; 104 int bits_lost; 105 union ldshape u; 106 107 sum = dd_add(a, b); 108 109 /* 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. 115 * 116 * If we are losing only one bit to denormalization, however, we must 117 * break the ties manually. 118 */ 119 if (sum.lo != 0) { 120 u.f = sum.hi; 121 bits_lost = -u.i.se - scale + 1; 122 if ((bits_lost != 1) ^ LASTBIT(u)) 123 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); 124 } 125 return scalbnl(sum.hi, scale); 126} 127 128/* 129 * Compute a*b exactly, returning the exact result in a struct dd. We assume 130 * that both a and b are normalized, so no underflow or overflow will occur. 131 * The current rounding mode must be round-to-nearest. 132 */ 133static inline struct dd dd_mul(long double a, long double b) 134{ 135 struct dd ret; 136 long double ha, hb, la, lb, p, q; 137 138 p = a * SPLIT; 139 ha = a - p; 140 ha += p; 141 la = a - ha; 142 143 p = b * SPLIT; 144 hb = b - p; 145 hb += p; 146 lb = b - hb; 147 148 p = ha * hb; 149 q = ha * lb + la * hb; 150 151 ret.hi = p + q; 152 ret.lo = p - ret.hi + q + la * lb; 153 return (ret); 154} 155 156/* 157 * Fused multiply-add: Compute x * y + z with a single rounding error. 158 * 159 * We use scaling to avoid overflow/underflow, along with the 160 * canonical precision-doubling technique adapted from: 161 * 162 * Dekker, T. A Floating-Point Technique for Extending the 163 * Available Precision. Numer. Math. 18, 224-242 (1971). 164 */ 165long double fmal(long double x, long double y, long double z) 166{ 167 #pragma STDC FENV_ACCESS ON 168 long double xs, ys, zs, adj; 169 struct dd xy, r; 170 int oround; 171 int ex, ey, ez; 172 int spread; 173 174 /* 175 * Handle special cases. The order of operations and the particular 176 * return values here are crucial in handling special cases involving 177 * infinities, NaNs, overflows, and signed zeroes correctly. 178 */ 179 if (!isfinite(x) || !isfinite(y)) 180 return (x * y + z); 181 if (!isfinite(z)) 182 return (z); 183 if (x == 0.0 || y == 0.0) 184 return (x * y + z); 185 if (z == 0.0) 186 return (x * y); 187 188 xs = frexpl(x, &ex); 189 ys = frexpl(y, &ey); 190 zs = frexpl(z, &ez); 191 oround = fegetround(); 192 spread = ex + ey - ez; 193 194 /* 195 * If x * y and z are many orders of magnitude apart, the scaling 196 * will overflow, so we handle these cases specially. Rounding 197 * modes other than FE_TONEAREST are painful. 198 */ 199 if (spread < -LDBL_MANT_DIG) { 200#ifdef FE_INEXACT 201 feraiseexcept(FE_INEXACT); 202#endif 203#ifdef FE_UNDERFLOW 204 if (!isnormal(z)) 205 feraiseexcept(FE_UNDERFLOW); 206#endif 207 switch (oround) { 208 default: /* FE_TONEAREST */ 209 return (z); 210#ifdef FE_TOWARDZERO 211 case FE_TOWARDZERO: 212 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 213 return (z); 214 else 215 return (nextafterl(z, 0)); 216#endif 217#ifdef FE_DOWNWARD 218 case FE_DOWNWARD: 219 if (x > 0.0 ^ y < 0.0) 220 return (z); 221 else 222 return (nextafterl(z, -INFINITY)); 223#endif 224#ifdef FE_UPWARD 225 case FE_UPWARD: 226 if (x > 0.0 ^ y < 0.0) 227 return (nextafterl(z, INFINITY)); 228 else 229 return (z); 230#endif 231 } 232 } 233 if (spread <= LDBL_MANT_DIG * 2) 234 zs = scalbnl(zs, -spread); 235 else 236 zs = copysignl(LDBL_MIN, zs); 237 238 fesetround(FE_TONEAREST); 239 240 /* 241 * Basic approach for round-to-nearest: 242 * 243 * (xy.hi, xy.lo) = x * y (exact) 244 * (r.hi, r.lo) = xy.hi + z (exact) 245 * adj = xy.lo + r.lo (inexact; low bit is sticky) 246 * result = r.hi + adj (correctly rounded) 247 */ 248 xy = dd_mul(xs, ys); 249 r = dd_add(xy.hi, zs); 250 251 spread = ex + ey; 252 253 if (r.hi == 0.0) { 254 /* 255 * When the addends cancel to 0, ensure that the result has 256 * the correct sign. 257 */ 258 fesetround(oround); 259 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */ 260 return xy.hi + vzs + scalbnl(xy.lo, spread); 261 } 262 263 if (oround != FE_TONEAREST) { 264 /* 265 * There is no need to worry about double rounding in directed 266 * rounding modes. 267 * But underflow may not be raised correctly, example in downward rounding: 268 * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L) 269 */ 270 long double ret; 271#if defined(FE_INEXACT) && defined(FE_UNDERFLOW) 272 int e = fetestexcept(FE_INEXACT); 273 feclearexcept(FE_INEXACT); 274#endif 275 fesetround(oround); 276 adj = r.lo + xy.lo; 277 ret = scalbnl(r.hi + adj, spread); 278#if defined(FE_INEXACT) && defined(FE_UNDERFLOW) 279 if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT)) 280 feraiseexcept(FE_UNDERFLOW); 281 else if (e) 282 feraiseexcept(FE_INEXACT); 283#endif 284 return ret; 285 } 286 287 adj = add_adjusted(r.lo, xy.lo); 288 if (spread + ilogbl(r.hi) > -16383) 289 return scalbnl(r.hi + adj, spread); 290 else 291 return add_and_denormalize(r.hi, adj, spread); 292} 293#endif 294