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1 //===-- lib/divdf3.c - Double-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 double-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 DOUBLE_PRECISION
20 #include "fp_lib.h"
21 
ARM_EABI_FNALIAS(ddiv,divdf3)22 ARM_EABI_FNALIAS(ddiv, divdf3)
23 
24 COMPILER_RT_ABI fp_t
25 __divdf3(fp_t a, fp_t b) {
26 
27     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
28     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
29     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
30 
31     rep_t aSignificand = toRep(a) & significandMask;
32     rep_t bSignificand = toRep(b) & significandMask;
33     int scale = 0;
34 
35     // Detect if a or b is zero, denormal, infinity, or NaN.
36     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
37 
38         const rep_t aAbs = toRep(a) & absMask;
39         const rep_t bAbs = toRep(b) & absMask;
40 
41         // NaN / anything = qNaN
42         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
43         // anything / NaN = qNaN
44         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
45 
46         if (aAbs == infRep) {
47             // infinity / infinity = NaN
48             if (bAbs == infRep) return fromRep(qnanRep);
49             // infinity / anything else = +/- infinity
50             else return fromRep(aAbs | quotientSign);
51         }
52 
53         // anything else / infinity = +/- 0
54         if (bAbs == infRep) return fromRep(quotientSign);
55 
56         if (!aAbs) {
57             // zero / zero = NaN
58             if (!bAbs) return fromRep(qnanRep);
59             // zero / anything else = +/- zero
60             else return fromRep(quotientSign);
61         }
62         // anything else / zero = +/- infinity
63         if (!bAbs) return fromRep(infRep | quotientSign);
64 
65         // one or both of a or b is denormal, the other (if applicable) is a
66         // normal number.  Renormalize one or both of a and b, and set scale to
67         // include the necessary exponent adjustment.
68         if (aAbs < implicitBit) scale += normalize(&aSignificand);
69         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
70     }
71 
72     // Or in the implicit significand bit.  (If we fell through from the
73     // denormal path it was already set by normalize( ), but setting it twice
74     // won't hurt anything.)
75     aSignificand |= implicitBit;
76     bSignificand |= implicitBit;
77     int quotientExponent = aExponent - bExponent + scale;
78 
79     // Align the significand of b as a Q31 fixed-point number in the range
80     // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
81     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
82     // is accurate to about 3.5 binary digits.
83     const uint32_t q31b = bSignificand >> 21;
84     uint32_t recip32 = UINT32_C(0x7504f333) - q31b;
85 
86     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
87     //
88     //     x1 = x0 * (2 - x0 * b)
89     //
90     // This doubles the number of correct binary digits in the approximation
91     // with each iteration, so after three iterations, we have about 28 binary
92     // digits of accuracy.
93     uint32_t correction32;
94     correction32 = -((uint64_t)recip32 * q31b >> 32);
95     recip32 = (uint64_t)recip32 * correction32 >> 31;
96     correction32 = -((uint64_t)recip32 * q31b >> 32);
97     recip32 = (uint64_t)recip32 * correction32 >> 31;
98     correction32 = -((uint64_t)recip32 * q31b >> 32);
99     recip32 = (uint64_t)recip32 * correction32 >> 31;
100 
101     // recip32 might have overflowed to exactly zero in the preceding
102     // computation if the high word of b is exactly 1.0.  This would sabotage
103     // the full-width final stage of the computation that follows, so we adjust
104     // recip32 downward by one bit.
105     recip32--;
106 
107     // We need to perform one more iteration to get us to 56 binary digits;
108     // The last iteration needs to happen with extra precision.
109     const uint32_t q63blo = bSignificand << 11;
110     uint64_t correction, reciprocal;
111     correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32));
112     uint32_t cHi = correction >> 32;
113     uint32_t cLo = correction;
114     reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32);
115 
116     // We already adjusted the 32-bit estimate, now we need to adjust the final
117     // 64-bit reciprocal estimate downward to ensure that it is strictly smaller
118     // than the infinitely precise exact reciprocal.  Because the computation
119     // of the Newton-Raphson step is truncating at every step, this adjustment
120     // is small; most of the work is already done.
121     reciprocal -= 2;
122 
123     // The numerical reciprocal is accurate to within 2^-56, lies in the
124     // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
125     // of b.  Multiplying a by this reciprocal thus gives a numerical q = a/b
126     // in Q53 with the following properties:
127     //
128     //    1. q < a/b
129     //    2. q is in the interval [0.5, 2.0)
130     //    3. the error in q is bounded away from 2^-53 (actually, we have a
131     //       couple of bits to spare, but this is all we need).
132 
133     // We need a 64 x 64 multiply high to compute q, which isn't a basic
134     // operation in C, so we need to be a little bit fussy.
135     rep_t quotient, quotientLo;
136     wideMultiply(aSignificand << 2, reciprocal, &quotient, &quotientLo);
137 
138     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
139     // In either case, we are going to compute a residual of the form
140     //
141     //     r = a - q*b
142     //
143     // We know from the construction of q that r satisfies:
144     //
145     //     0 <= r < ulp(q)*b
146     //
147     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
148     // already have the correct result.  The exact halfway case cannot occur.
149     // We also take this time to right shift quotient if it falls in the [1,2)
150     // range and adjust the exponent accordingly.
151     rep_t residual;
152     if (quotient < (implicitBit << 1)) {
153         residual = (aSignificand << 53) - quotient * bSignificand;
154         quotientExponent--;
155     } else {
156         quotient >>= 1;
157         residual = (aSignificand << 52) - quotient * bSignificand;
158     }
159 
160     const int writtenExponent = quotientExponent + exponentBias;
161 
162     if (writtenExponent >= maxExponent) {
163         // If we have overflowed the exponent, return infinity.
164         return fromRep(infRep | quotientSign);
165     }
166 
167     else if (writtenExponent < 1) {
168         // Flush denormals to zero.  In the future, it would be nice to add
169         // code to round them correctly.
170         return fromRep(quotientSign);
171     }
172 
173     else {
174         const bool round = (residual << 1) > bSignificand;
175         // Clear the implicit bit
176         rep_t absResult = quotient & significandMask;
177         // Insert the exponent
178         absResult |= (rep_t)writtenExponent << significandBits;
179         // Round
180         absResult += round;
181         // Insert the sign and return
182         const double result = fromRep(absResult | quotientSign);
183         return result;
184     }
185 }
186