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, "ient, "ientLo); 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
