• Home
  • Line#
  • Scopes#
  • Navigate#
  • Raw
  • Download
1 //===-- lib/divtf3.c - Quad-precision division --------------------*- C -*-===//
2 //
3 //                     The LLVM Compiler Infrastructure
4 //
5 // This file is dual licensed under the MIT and the University of Illinois Open
6 // Source Licenses. See LICENSE.TXT for details.
7 //
8 //===----------------------------------------------------------------------===//
9 //
10 // This file implements quad-precision soft-float division
11 // with the IEEE-754 default rounding (to nearest, ties to even).
12 //
13 // For simplicity, this implementation currently flushes denormals to zero.
14 // It should be a fairly straightforward exercise to implement gradual
15 // underflow with correct rounding.
16 //
17 //===----------------------------------------------------------------------===//
18 
19 #define QUAD_PRECISION
20 #include "fp_lib.h"
21 
22 #if defined(CRT_HAS_128BIT) && defined(CRT_LDBL_128BIT)
__divtf3(fp_t a,fp_t b)23 COMPILER_RT_ABI fp_t __divtf3(fp_t a, fp_t b) {
24 
25     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
26     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
27     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
28 
29     rep_t aSignificand = toRep(a) & significandMask;
30     rep_t bSignificand = toRep(b) & significandMask;
31     int scale = 0;
32 
33     // Detect if a or b is zero, denormal, infinity, or NaN.
34     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
35 
36         const rep_t aAbs = toRep(a) & absMask;
37         const rep_t bAbs = toRep(b) & absMask;
38 
39         // NaN / anything = qNaN
40         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
41         // anything / NaN = qNaN
42         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
43 
44         if (aAbs == infRep) {
45             // infinity / infinity = NaN
46             if (bAbs == infRep) return fromRep(qnanRep);
47             // infinity / anything else = +/- infinity
48             else return fromRep(aAbs | quotientSign);
49         }
50 
51         // anything else / infinity = +/- 0
52         if (bAbs == infRep) return fromRep(quotientSign);
53 
54         if (!aAbs) {
55             // zero / zero = NaN
56             if (!bAbs) return fromRep(qnanRep);
57             // zero / anything else = +/- zero
58             else return fromRep(quotientSign);
59         }
60         // anything else / zero = +/- infinity
61         if (!bAbs) return fromRep(infRep | quotientSign);
62 
63         // one or both of a or b is denormal, the other (if applicable) is a
64         // normal number.  Renormalize one or both of a and b, and set scale to
65         // include the necessary exponent adjustment.
66         if (aAbs < implicitBit) scale += normalize(&aSignificand);
67         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
68     }
69 
70     // Or in the implicit significand bit.  (If we fell through from the
71     // denormal path it was already set by normalize( ), but setting it twice
72     // won't hurt anything.)
73     aSignificand |= implicitBit;
74     bSignificand |= implicitBit;
75     int quotientExponent = aExponent - bExponent + scale;
76 
77     // Align the significand of b as a Q63 fixed-point number in the range
78     // [1, 2.0) and get a Q64 approximate reciprocal using a small minimax
79     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
80     // is accurate to about 3.5 binary digits.
81     const uint64_t q63b = bSignificand >> 49;
82     uint64_t recip64 = UINT64_C(0x7504f333F9DE6484) - q63b;
83     // 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
84 
85     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
86     //
87     //     x1 = x0 * (2 - x0 * b)
88     //
89     // This doubles the number of correct binary digits in the approximation
90     // with each iteration.
91     uint64_t correction64;
92     correction64 = -((rep_t)recip64 * q63b >> 64);
93     recip64 = (rep_t)recip64 * correction64 >> 63;
94     correction64 = -((rep_t)recip64 * q63b >> 64);
95     recip64 = (rep_t)recip64 * correction64 >> 63;
96     correction64 = -((rep_t)recip64 * q63b >> 64);
97     recip64 = (rep_t)recip64 * correction64 >> 63;
98     correction64 = -((rep_t)recip64 * q63b >> 64);
99     recip64 = (rep_t)recip64 * correction64 >> 63;
100     correction64 = -((rep_t)recip64 * q63b >> 64);
101     recip64 = (rep_t)recip64 * correction64 >> 63;
102 
103     // recip64 might have overflowed to exactly zero in the preceeding
104     // computation if the high word of b is exactly 1.0.  This would sabotage
105     // the full-width final stage of the computation that follows, so we adjust
106     // recip64 downward by one bit.
107     recip64--;
108 
109     // We need to perform one more iteration to get us to 112 binary digits;
110     // The last iteration needs to happen with extra precision.
111     const uint64_t q127blo = bSignificand << 15;
112     rep_t correction, reciprocal;
113 
114     // NOTE: This operation is equivalent to __multi3, which is not implemented
115     //       in some architechure
116     rep_t r64q63, r64q127, r64cH, r64cL, dummy;
117     wideMultiply((rep_t)recip64, (rep_t)q63b, &dummy, &r64q63);
118     wideMultiply((rep_t)recip64, (rep_t)q127blo, &dummy, &r64q127);
119 
120     correction = -(r64q63 + (r64q127 >> 64));
121 
122     uint64_t cHi = correction >> 64;
123     uint64_t cLo = correction;
124 
125     wideMultiply((rep_t)recip64, (rep_t)cHi, &dummy, &r64cH);
126     wideMultiply((rep_t)recip64, (rep_t)cLo, &dummy, &r64cL);
127 
128     reciprocal = r64cH + (r64cL >> 64);
129 
130     // We already adjusted the 64-bit estimate, now we need to adjust the final
131     // 128-bit reciprocal estimate downward to ensure that it is strictly smaller
132     // than the infinitely precise exact reciprocal.  Because the computation
133     // of the Newton-Raphson step is truncating at every step, this adjustment
134     // is small; most of the work is already done.
135     reciprocal -= 2;
136 
137     // The numerical reciprocal is accurate to within 2^-112, lies in the
138     // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
139     // of b.  Multiplying a by this reciprocal thus gives a numerical q = a/b
140     // in Q127 with the following properties:
141     //
142     //    1. q < a/b
143     //    2. q is in the interval [0.5, 2.0)
144     //    3. the error in q is bounded away from 2^-113 (actually, we have a
145     //       couple of bits to spare, but this is all we need).
146 
147     // We need a 128 x 128 multiply high to compute q, which isn't a basic
148     // operation in C, so we need to be a little bit fussy.
149     rep_t quotient, quotientLo;
150     wideMultiply(aSignificand << 2, reciprocal, &quotient, &quotientLo);
151 
152     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
153     // In either case, we are going to compute a residual of the form
154     //
155     //     r = a - q*b
156     //
157     // We know from the construction of q that r satisfies:
158     //
159     //     0 <= r < ulp(q)*b
160     //
161     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
162     // already have the correct result.  The exact halfway case cannot occur.
163     // We also take this time to right shift quotient if it falls in the [1,2)
164     // range and adjust the exponent accordingly.
165     rep_t residual;
166     rep_t qb;
167 
168     if (quotient < (implicitBit << 1)) {
169         wideMultiply(quotient, bSignificand, &dummy, &qb);
170         residual = (aSignificand << 113) - qb;
171         quotientExponent--;
172     } else {
173         quotient >>= 1;
174         wideMultiply(quotient, bSignificand, &dummy, &qb);
175         residual = (aSignificand << 112) - qb;
176     }
177 
178     const int writtenExponent = quotientExponent + exponentBias;
179 
180     if (writtenExponent >= maxExponent) {
181         // If we have overflowed the exponent, return infinity.
182         return fromRep(infRep | quotientSign);
183     }
184     else if (writtenExponent < 1) {
185         // Flush denormals to zero.  In the future, it would be nice to add
186         // code to round them correctly.
187         return fromRep(quotientSign);
188     }
189     else {
190         const bool round = (residual << 1) >= bSignificand;
191         // Clear the implicit bit
192         rep_t absResult = quotient & significandMask;
193         // Insert the exponent
194         absResult |= (rep_t)writtenExponent << significandBits;
195         // Round
196         absResult += round;
197         // Insert the sign and return
198         const long double result = fromRep(absResult | quotientSign);
199         return result;
200     }
201 }
202 
203 #endif
204