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