divdf3.c revision 215125
1//===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===// 2// 3// The LLVM Compiler Infrastructure 4// 5// This file is distributed under the University of Illinois Open Source 6// License. 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 22fp_t __divdf3(fp_t a, fp_t b) { 23 24 const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 25 const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 26 const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 27 28 rep_t aSignificand = toRep(a) & significandMask; 29 rep_t bSignificand = toRep(b) & significandMask; 30 int scale = 0; 31 32 // Detect if a or b is zero, denormal, infinity, or NaN. 33 if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { 34 35 const rep_t aAbs = toRep(a) & absMask; 36 const rep_t bAbs = toRep(b) & absMask; 37 38 // NaN / anything = qNaN 39 if (aAbs > infRep) return fromRep(toRep(a) | quietBit); 40 // anything / NaN = qNaN 41 if (bAbs > infRep) return fromRep(toRep(b) | quietBit); 42 43 if (aAbs == infRep) { 44 // infinity / infinity = NaN 45 if (bAbs == infRep) return fromRep(qnanRep); 46 // infinity / anything else = +/- infinity 47 else return fromRep(aAbs | quotientSign); 48 } 49 50 // anything else / infinity = +/- 0 51 if (bAbs == infRep) return fromRep(quotientSign); 52 53 if (!aAbs) { 54 // zero / zero = NaN 55 if (!bAbs) return fromRep(qnanRep); 56 // zero / anything else = +/- zero 57 else return fromRep(quotientSign); 58 } 59 // anything else / zero = +/- infinity 60 if (!bAbs) return fromRep(infRep | quotientSign); 61 62 // one or both of a or b is denormal, the other (if applicable) is a 63 // normal number. Renormalize one or both of a and b, and set scale to 64 // include the necessary exponent adjustment. 65 if (aAbs < implicitBit) scale += normalize(&aSignificand); 66 if (bAbs < implicitBit) scale -= normalize(&bSignificand); 67 } 68 69 // Or in the implicit significand bit. (If we fell through from the 70 // denormal path it was already set by normalize( ), but setting it twice 71 // won't hurt anything.) 72 aSignificand |= implicitBit; 73 bSignificand |= implicitBit; 74 int quotientExponent = aExponent - bExponent + scale; 75 76 // Align the significand of b as a Q31 fixed-point number in the range 77 // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax 78 // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This 79 // is accurate to about 3.5 binary digits. 80 const uint32_t q31b = bSignificand >> 21; 81 uint32_t recip32 = UINT32_C(0x7504f333) - q31b; 82 83 // Now refine the reciprocal estimate using a Newton-Raphson iteration: 84 // 85 // x1 = x0 * (2 - x0 * b) 86 // 87 // This doubles the number of correct binary digits in the approximation 88 // with each iteration, so after three iterations, we have about 28 binary 89 // digits of accuracy. 90 uint32_t correction32; 91 correction32 = -((uint64_t)recip32 * q31b >> 32); 92 recip32 = (uint64_t)recip32 * correction32 >> 31; 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 98 // recip32 might have overflowed to exactly zero in the preceeding 99 // computation if the high word of b is exactly 1.0. This would sabotage 100 // the full-width final stage of the computation that follows, so we adjust 101 // recip32 downward by one bit. 102 recip32--; 103 104 // We need to perform one more iteration to get us to 56 binary digits; 105 // The last iteration needs to happen with extra precision. 106 const uint32_t q63blo = bSignificand << 11; 107 uint64_t correction, reciprocal; 108 correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32)); 109 uint32_t cHi = correction >> 32; 110 uint32_t cLo = correction; 111 reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32); 112 113 // We already adjusted the 32-bit estimate, now we need to adjust the final 114 // 64-bit reciprocal estimate downward to ensure that it is strictly smaller 115 // than the infinitely precise exact reciprocal. Because the computation 116 // of the Newton-Raphson step is truncating at every step, this adjustment 117 // is small; most of the work is already done. 118 reciprocal -= 2; 119 120 // The numerical reciprocal is accurate to within 2^-56, lies in the 121 // interval [0.5, 1.0), and is strictly smaller than the true reciprocal 122 // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b 123 // in Q53 with the following properties: 124 // 125 // 1. q < a/b 126 // 2. q is in the interval [0.5, 2.0) 127 // 3. the error in q is bounded away from 2^-53 (actually, we have a 128 // couple of bits to spare, but this is all we need). 129 130 // We need a 64 x 64 multiply high to compute q, which isn't a basic 131 // operation in C, so we need to be a little bit fussy. 132 rep_t quotient, quotientLo; 133 wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo); 134 135 // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). 136 // In either case, we are going to compute a residual of the form 137 // 138 // r = a - q*b 139 // 140 // We know from the construction of q that r satisfies: 141 // 142 // 0 <= r < ulp(q)*b 143 // 144 // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we 145 // already have the correct result. The exact halfway case cannot occur. 146 // We also take this time to right shift quotient if it falls in the [1,2) 147 // range and adjust the exponent accordingly. 148 rep_t residual; 149 if (quotient < (implicitBit << 1)) { 150 residual = (aSignificand << 53) - quotient * bSignificand; 151 quotientExponent--; 152 } else { 153 quotient >>= 1; 154 residual = (aSignificand << 52) - quotient * bSignificand; 155 } 156 157 const int writtenExponent = quotientExponent + exponentBias; 158 159 if (writtenExponent >= maxExponent) { 160 // If we have overflowed the exponent, return infinity. 161 return fromRep(infRep | quotientSign); 162 } 163 164 else if (writtenExponent < 1) { 165 // Flush denormals to zero. In the future, it would be nice to add 166 // code to round them correctly. 167 return fromRep(quotientSign); 168 } 169 170 else { 171 const bool round = (residual << 1) > bSignificand; 172 // Clear the implicit bit 173 rep_t absResult = quotient & significandMask; 174 // Insert the exponent 175 absResult |= (rep_t)writtenExponent << significandBits; 176 // Round 177 absResult += round; 178 // Insert the sign and return 179 const double result = fromRep(absResult | quotientSign); 180 return result; 181 } 182} 183