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      1 /*-
      2  * Copyright (c) 2005-2011 David Schultz <das (at) FreeBSD.ORG>
      3  * All rights reserved.
      4  *
      5  * Redistribution and use in source and binary forms, with or without
      6  * modification, are permitted provided that the following conditions
      7  * are met:
      8  * 1. Redistributions of source code must retain the above copyright
      9  *    notice, this list of conditions and the following disclaimer.
     10  * 2. Redistributions in binary form must reproduce the above copyright
     11  *    notice, this list of conditions and the following disclaimer in the
     12  *    documentation and/or other materials provided with the distribution.
     13  *
     14  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
     15  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     16  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     17  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
     18  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     19  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     20  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     21  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     22  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     23  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     24  * SUCH DAMAGE.
     25  */
     26 
     27 #include <sys/cdefs.h>
     28 __FBSDID("$FreeBSD$");
     29 
     30 #include <fenv.h>
     31 #include <float.h>
     32 #include <math.h>
     33 
     34 #include "fpmath.h"
     35 
     36 /*
     37  * A struct dd represents a floating-point number with twice the precision
     38  * of a long double.  We maintain the invariant that "hi" stores the high-order
     39  * bits of the result.
     40  */
     41 struct dd {
     42 	long double hi;
     43 	long double lo;
     44 };
     45 
     46 /*
     47  * Compute a+b exactly, returning the exact result in a struct dd.  We assume
     48  * that both a and b are finite, but make no assumptions about their relative
     49  * magnitudes.
     50  */
     51 static inline struct dd
     52 dd_add(long double a, long double b)
     53 {
     54 	struct dd ret;
     55 	long double s;
     56 
     57 	ret.hi = a + b;
     58 	s = ret.hi - a;
     59 	ret.lo = (a - (ret.hi - s)) + (b - s);
     60 	return (ret);
     61 }
     62 
     63 /*
     64  * Compute a+b, with a small tweak:  The least significant bit of the
     65  * result is adjusted into a sticky bit summarizing all the bits that
     66  * were lost to rounding.  This adjustment negates the effects of double
     67  * rounding when the result is added to another number with a higher
     68  * exponent.  For an explanation of round and sticky bits, see any reference
     69  * on FPU design, e.g.,
     70  *
     71  *     J. Coonen.  An Implementation Guide to a Proposed Standard for
     72  *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
     73  */
     74 static inline long double
     75 add_adjusted(long double a, long double b)
     76 {
     77 	struct dd sum;
     78 	union IEEEl2bits u;
     79 
     80 	sum = dd_add(a, b);
     81 	if (sum.lo != 0) {
     82 		u.e = sum.hi;
     83 		if ((u.bits.manl & 1) == 0)
     84 			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
     85 	}
     86 	return (sum.hi);
     87 }
     88 
     89 /*
     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.
     93  */
     94 static inline long double
     95 add_and_denormalize(long double a, long double b, int scale)
     96 {
     97 	struct dd sum;
     98 	int bits_lost;
     99 	union IEEEl2bits u;
    100 
    101 	sum = dd_add(a, b);
    102 
    103 	/*
    104 	 * If we are losing at least two bits of accuracy to denormalization,
    105 	 * then the first lost bit becomes a round bit, and we adjust the
    106 	 * lowest bit of sum.hi to make it a sticky bit summarizing all the
    107 	 * bits in sum.lo. With the sticky bit adjusted, the hardware will
    108 	 * break any ties in the correct direction.
    109 	 *
    110 	 * If we are losing only one bit to denormalization, however, we must
    111 	 * break the ties manually.
    112 	 */
    113 	if (sum.lo != 0) {
    114 		u.e = sum.hi;
    115 		bits_lost = -u.bits.exp - scale + 1;
    116 		if (bits_lost != 1 ^ (int)(u.bits.manl & 1))
    117 			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
    118 	}
    119 	return (ldexp(sum.hi, scale));
    120 }
    121 
    122 /*
    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.
    126  */
    127 static inline struct dd
    128 dd_mul(long double a, long double b)
    129 {
    130 #if LDBL_MANT_DIG == 64
    131 	static const long double split = 0x1p32L + 1.0;
    132 #elif LDBL_MANT_DIG == 113
    133 	static const long double split = 0x1p57L + 1.0;
    134 #endif
    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  */
    165 long double
    166 fmal(long double x, long double y, long double z)
    167 {
    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 (x == 0.0 || y == 0.0)
    180 		return (x * y + z);
    181 	if (z == 0.0)
    182 		return (x * y);
    183 	if (!isfinite(x) || !isfinite(y))
    184 		return (x * y + z);
    185 	if (!isfinite(z))
    186 		return (z);
    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 		feraiseexcept(FE_INEXACT);
    201 		if (!isnormal(z))
    202 			feraiseexcept(FE_UNDERFLOW);
    203 		switch (oround) {
    204 		case FE_TONEAREST:
    205 			return (z);
    206 		case FE_TOWARDZERO:
    207 			if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
    208 				return (z);
    209 			else
    210 				return (nextafterl(z, 0));
    211 		case FE_DOWNWARD:
    212 			if (x > 0.0 ^ y < 0.0)
    213 				return (z);
    214 			else
    215 				return (nextafterl(z, -INFINITY));
    216 		default:	/* FE_UPWARD */
    217 			if (x > 0.0 ^ y < 0.0)
    218 				return (nextafterl(z, INFINITY));
    219 			else
    220 				return (z);
    221 		}
    222 	}
    223 	if (spread <= LDBL_MANT_DIG * 2)
    224 		zs = ldexpl(zs, -spread);
    225 	else
    226 		zs = copysignl(LDBL_MIN, zs);
    227 
    228 	fesetround(FE_TONEAREST);
    229 	/* work around clang bug 8100 */
    230 	volatile long double vxs = xs;
    231 
    232 	/*
    233 	 * Basic approach for round-to-nearest:
    234 	 *
    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)
    239 	 */
    240 	xy = dd_mul(vxs, ys);
    241 	r = dd_add(xy.hi, zs);
    242 
    243 	spread = ex + ey;
    244 
    245 	if (r.hi == 0.0) {
    246 		/*
    247 		 * When the addends cancel to 0, ensure that the result has
    248 		 * the correct sign.
    249 		 */
    250 		fesetround(oround);
    251 		volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
    252 		return (xy.hi + vzs + ldexpl(xy.lo, spread));
    253 	}
    254 
    255 	if (oround != FE_TONEAREST) {
    256 		/*
    257 		 * There is no need to worry about double rounding in directed
    258 		 * rounding modes.
    259 		 */
    260 		fesetround(oround);
    261 		/* work around clang bug 8100 */
    262 		volatile long double vrlo = r.lo;
    263 		adj = vrlo + xy.lo;
    264 		return (ldexpl(r.hi + adj, spread));
    265 	}
    266 
    267 	adj = add_adjusted(r.lo, xy.lo);
    268 	if (spread + ilogbl(r.hi) > -16383)
    269 		return (ldexpl(r.hi + adj, spread));
    270 	else
    271 		return (add_and_denormalize(r.hi, adj, spread));
    272 }
    273