lib/divdf3.c

   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
  22 ARM_EABI_FNALIAS(ddiv, divdf3)
  23
  24 fp_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 }