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      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 }
    185