lib/builtins/divtf3.c

   1 //===-- lib/divtf3.c - Quad-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 quad-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 QUAD_PRECISION
  20 #include "fp_lib.h"
  21
  22 #if defined(CRT_HAS_128BIT) && defined(CRT_LDBL_128BIT)
  23 COMPILER_RT_ABI fp_t __divtf3(fp_t a, fp_t b) {
  24
  25     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
  26     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
  27     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
  28
  29     rep_t aSignificand = toRep(a) & significandMask;
  30     rep_t bSignificand = toRep(b) & significandMask;
  31     int scale = 0;
  32
  33     // Detect if a or b is zero, denormal, infinity, or NaN.
  34     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
  35
  36         const rep_t aAbs = toRep(a) & absMask;
  37         const rep_t bAbs = toRep(b) & absMask;
  38
  39         // NaN / anything = qNaN
  40         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
  41         // anything / NaN = qNaN
  42         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
  43
  44         if (aAbs == infRep) {
  45             // infinity / infinity = NaN
  46             if (bAbs == infRep) return fromRep(qnanRep);
  47             // infinity / anything else = +/- infinity
  48             else return fromRep(aAbs | quotientSign);
  49         }
  50
  51         // anything else / infinity = +/- 0
  52         if (bAbs == infRep) return fromRep(quotientSign);
  53
  54         if (!aAbs) {
  55             // zero / zero = NaN
  56             if (!bAbs) return fromRep(qnanRep);
  57             // zero / anything else = +/- zero
  58             else return fromRep(quotientSign);
  59         }
  60         // anything else / zero = +/- infinity
  61         if (!bAbs) return fromRep(infRep | quotientSign);
  62
  63         // one or both of a or b is denormal, the other (if applicable) is a
  64         // normal number.  Renormalize one or both of a and b, and set scale to
  65         // include the necessary exponent adjustment.
  66         if (aAbs < implicitBit) scale += normalize(&aSignificand);
  67         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
  68     }
  69
  70     // Or in the implicit significand bit.  (If we fell through from the
  71     // denormal path it was already set by normalize( ), but setting it twice
  72     // won't hurt anything.)
  73     aSignificand |= implicitBit;
  74     bSignificand |= implicitBit;
  75     int quotientExponent = aExponent - bExponent + scale;
  76
  77     // Align the significand of b as a Q63 fixed-point number in the range
  78     // [1, 2.0) and get a Q64 approximate reciprocal using a small minimax
  79     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
  80     // is accurate to about 3.5 binary digits.
  81     const uint64_t q63b = bSignificand >> 49;
  82     uint64_t recip64 = UINT64_C(0x7504f333F9DE6484) - q63b;
  83     // 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
  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.
  91     uint64_t correction64;
  92     correction64 = -((rep_t)recip64 * q63b >> 64);
  93     recip64 = (rep_t)recip64 * correction64 >> 63;
  94     correction64 = -((rep_t)recip64 * q63b >> 64);
  95     recip64 = (rep_t)recip64 * correction64 >> 63;
  96     correction64 = -((rep_t)recip64 * q63b >> 64);
  97     recip64 = (rep_t)recip64 * correction64 >> 63;
  98     correction64 = -((rep_t)recip64 * q63b >> 64);
  99     recip64 = (rep_t)recip64 * correction64 >> 63;
 100     correction64 = -((rep_t)recip64 * q63b >> 64);
 101     recip64 = (rep_t)recip64 * correction64 >> 63;
 102
 103     // recip64 might have overflowed to exactly zero in the preceeding
 104     // computation if the high word of b is exactly 1.0.  This would sabotage
 105     // the full-width final stage of the computation that follows, so we adjust
 106     // recip64 downward by one bit.
 107     recip64--;
 108
 109     // We need to perform one more iteration to get us to 112 binary digits;
 110     // The last iteration needs to happen with extra precision.
 111     const uint64_t q127blo = bSignificand << 15;
 112     rep_t correction, reciprocal;
 113
 114     // NOTE: This operation is equivalent to __multi3, which is not implemented
 115     //       in some architechure
 116     rep_t r64q63, r64q127, r64cH, r64cL, dummy;
 117     wideMultiply((rep_t)recip64, (rep_t)q63b, &dummy, &r64q63);
 118     wideMultiply((rep_t)recip64, (rep_t)q127blo, &dummy, &r64q127);
 119
 120     correction = -(r64q63 + (r64q127 >> 64));
 121
 122     uint64_t cHi = correction >> 64;
 123     uint64_t cLo = correction;
 124
 125     wideMultiply((rep_t)recip64, (rep_t)cHi, &dummy, &r64cH);
 126     wideMultiply((rep_t)recip64, (rep_t)cLo, &dummy, &r64cL);
 127
 128     reciprocal = r64cH + (r64cL >> 64);
 129
 130     // We already adjusted the 64-bit estimate, now we need to adjust the final
 131     // 128-bit reciprocal estimate downward to ensure that it is strictly smaller
 132     // than the infinitely precise exact reciprocal.  Because the computation
 133     // of the Newton-Raphson step is truncating at every step, this adjustment
 134     // is small; most of the work is already done.
 135     reciprocal -= 2;
 136
 137     // The numerical reciprocal is accurate to within 2^-112, lies in the
 138     // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
 139     // of b.  Multiplying a by this reciprocal thus gives a numerical q = a/b
 140     // in Q127 with the following properties:
 141     //
 142     //    1. q < a/b
 143     //    2. q is in the interval [0.5, 2.0)
 144     //    3. the error in q is bounded away from 2^-113 (actually, we have a
 145     //       couple of bits to spare, but this is all we need).
 146
 147     // We need a 128 x 128 multiply high to compute q, which isn't a basic
 148     // operation in C, so we need to be a little bit fussy.
 149     rep_t quotient, quotientLo;
 150     wideMultiply(aSignificand << 2, reciprocal, &quotient, &quotientLo);
 151
 152     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
 153     // In either case, we are going to compute a residual of the form
 154     //
 155     //     r = a - q*b
 156     //
 157     // We know from the construction of q that r satisfies:
 158     //
 159     //     0 <= r < ulp(q)*b
 160     //
 161     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
 162     // already have the correct result.  The exact halfway case cannot occur.
 163     // We also take this time to right shift quotient if it falls in the [1,2)
 164     // range and adjust the exponent accordingly.
 165     rep_t residual;
 166     rep_t qb;
 167
 168     if (quotient < (implicitBit << 1)) {
 169         wideMultiply(quotient, bSignificand, &dummy, &qb);
 170         residual = (aSignificand << 113) - qb;
 171         quotientExponent--;
 172     } else {
 173         quotient >>= 1;
 174         wideMultiply(quotient, bSignificand, &dummy, &qb);
 175         residual = (aSignificand << 112) - qb;
 176     }
 177
 178     const int writtenExponent = quotientExponent + exponentBias;
 179
 180     if (writtenExponent >= maxExponent) {
 181         // If we have overflowed the exponent, return infinity.
 182         return fromRep(infRep | quotientSign);
 183     }
 184     else if (writtenExponent < 1) {
 185         // Flush denormals to zero.  In the future, it would be nice to add
 186         // code to round them correctly.
 187         return fromRep(quotientSign);
 188     }
 189     else {
 190         const bool round = (residual << 1) >= bSignificand;
 191         // Clear the implicit bit
 192         rep_t absResult = quotient & significandMask;
 193         // Insert the exponent
 194         absResult |= (rep_t)writtenExponent << significandBits;
 195         // Round
 196         absResult += round;
 197         // Insert the sign and return
 198         const long double result = fromRep(absResult | quotientSign);
 199         return result;
 200     }
 201 }
 202
 203 #endif