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, "ient, "ientLo);
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