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, "ient, "ientLo);
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