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}
185