divtf3.c revision 5d71de26cedae3dafc17449fe0182045c0bd20e8 
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)
23COMPILER_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