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
24COMPILER_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