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, "ient, "ientLo); 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 204
