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 COMPILER_RT_ABI fp_t 25 __divdf3(fp_t a, fp_t b) { 26 27 const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 28 const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 29 const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 30 31 rep_t aSignificand = toRep(a) & significandMask; 32 rep_t bSignificand = toRep(b) & significandMask; 33 int scale = 0; 34 35 // Detect if a or b is zero, denormal, infinity, or NaN. 36 if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { 37 38 const rep_t aAbs = toRep(a) & absMask; 39 const rep_t bAbs = toRep(b) & absMask; 40 41 // NaN / anything = qNaN 42 if (aAbs > infRep) return fromRep(toRep(a) | quietBit); 43 // anything / NaN = qNaN 44 if (bAbs > infRep) return fromRep(toRep(b) | quietBit); 45 46 if (aAbs == infRep) { 47 // infinity / infinity = NaN 48 if (bAbs == infRep) return fromRep(qnanRep); 49 // infinity / anything else = +/- infinity 50 else return fromRep(aAbs | quotientSign); 51 } 52 53 // anything else / infinity = +/- 0 54 if (bAbs == infRep) return fromRep(quotientSign); 55 56 if (!aAbs) { 57 // zero / zero = NaN 58 if (!bAbs) return fromRep(qnanRep); 59 // zero / anything else = +/- zero 60 else return fromRep(quotientSign); 61 } 62 // anything else / zero = +/- infinity 63 if (!bAbs) return fromRep(infRep | quotientSign); 64 65 // one or both of a or b is denormal, the other (if applicable) is a 66 // normal number. Renormalize one or both of a and b, and set scale to 67 // include the necessary exponent adjustment. 68 if (aAbs < implicitBit) scale += normalize(&aSignificand); 69 if (bAbs < implicitBit) scale -= normalize(&bSignificand); 70 } 71 72 // Or in the implicit significand bit. (If we fell through from the 73 // denormal path it was already set by normalize( ), but setting it twice 74 // won't hurt anything.) 75 aSignificand |= implicitBit; 76 bSignificand |= implicitBit; 77 int quotientExponent = aExponent - bExponent + scale; 78 79 // Align the significand of b as a Q31 fixed-point number in the range 80 // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax 81 // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This 82 // is accurate to about 3.5 binary digits. 83 const uint32_t q31b = bSignificand >> 21; 84 uint32_t recip32 = UINT32_C(0x7504f333) - q31b; 85 86 // Now refine the reciprocal estimate using a Newton-Raphson iteration: 87 // 88 // x1 = x0 * (2 - x0 * b) 89 // 90 // This doubles the number of correct binary digits in the approximation 91 // with each iteration, so after three iterations, we have about 28 binary 92 // digits of accuracy. 93 uint32_t correction32; 94 correction32 = -((uint64_t)recip32 * q31b >> 32); 95 recip32 = (uint64_t)recip32 * correction32 >> 31; 96 correction32 = -((uint64_t)recip32 * q31b >> 32); 97 recip32 = (uint64_t)recip32 * correction32 >> 31; 98 correction32 = -((uint64_t)recip32 * q31b >> 32); 99 recip32 = (uint64_t)recip32 * correction32 >> 31; 100 101 // recip32 might have overflowed to exactly zero in the preceding 102 // computation if the high word of b is exactly 1.0. This would sabotage 103 // the full-width final stage of the computation that follows, so we adjust 104 // recip32 downward by one bit. 105 recip32--; 106 107 // We need to perform one more iteration to get us to 56 binary digits; 108 // The last iteration needs to happen with extra precision. 109 const uint32_t q63blo = bSignificand << 11; 110 uint64_t correction, reciprocal; 111 correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32)); 112 uint32_t cHi = correction >> 32; 113 uint32_t cLo = correction; 114 reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32); 115 116 // We already adjusted the 32-bit estimate, now we need to adjust the final 117 // 64-bit reciprocal estimate downward to ensure that it is strictly smaller 118 // than the infinitely precise exact reciprocal. Because the computation 119 // of the Newton-Raphson step is truncating at every step, this adjustment 120 // is small; most of the work is already done. 121 reciprocal -= 2; 122 123 // The numerical reciprocal is accurate to within 2^-56, lies in the 124 // interval [0.5, 1.0), and is strictly smaller than the true reciprocal 125 // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b 126 // in Q53 with the following properties: 127 // 128 // 1. q < a/b 129 // 2. q is in the interval [0.5, 2.0) 130 // 3. the error in q is bounded away from 2^-53 (actually, we have a 131 // couple of bits to spare, but this is all we need). 132 133 // We need a 64 x 64 multiply high to compute q, which isn't a basic 134 // operation in C, so we need to be a little bit fussy. 135 rep_t quotient, quotientLo; 136 wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo); 137 138 // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). 139 // In either case, we are going to compute a residual of the form 140 // 141 // r = a - q*b 142 // 143 // We know from the construction of q that r satisfies: 144 // 145 // 0 <= r < ulp(q)*b 146 // 147 // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we 148 // already have the correct result. The exact halfway case cannot occur. 149 // We also take this time to right shift quotient if it falls in the [1,2) 150 // range and adjust the exponent accordingly. 151 rep_t residual; 152 if (quotient < (implicitBit << 1)) { 153 residual = (aSignificand << 53) - quotient * bSignificand; 154 quotientExponent--; 155 } else { 156 quotient >>= 1; 157 residual = (aSignificand << 52) - quotient * bSignificand; 158 } 159 160 const int writtenExponent = quotientExponent + exponentBias; 161 162 if (writtenExponent >= maxExponent) { 163 // If we have overflowed the exponent, return infinity. 164 return fromRep(infRep | quotientSign); 165 } 166 167 else if (writtenExponent < 1) { 168 // Flush denormals to zero. In the future, it would be nice to add 169 // code to round them correctly. 170 return fromRep(quotientSign); 171 } 172 173 else { 174 const bool round = (residual << 1) > bSignificand; 175 // Clear the implicit bit 176 rep_t absResult = quotient & significandMask; 177 // Insert the exponent 178 absResult |= (rep_t)writtenExponent << significandBits; 179 // Round 180 absResult += round; 181 // Insert the sign and return 182 const double result = fromRep(absResult | quotientSign); 183 return result; 184 } 185 } 186
