1214152Sed//===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===// 2214152Sed// 3214152Sed// The LLVM Compiler Infrastructure 4214152Sed// 5222656Sed// This file is dual licensed under the MIT and the University of Illinois Open 6222656Sed// Source Licenses. See LICENSE.TXT for details. 7214152Sed// 8214152Sed//===----------------------------------------------------------------------===// 9214152Sed// 10214152Sed// This file implements double-precision soft-float division 11214152Sed// with the IEEE-754 default rounding (to nearest, ties to even). 12214152Sed// 13214152Sed// For simplicity, this implementation currently flushes denormals to zero. 14214152Sed// It should be a fairly straightforward exercise to implement gradual 15214152Sed// underflow with correct rounding. 16214152Sed// 17214152Sed//===----------------------------------------------------------------------===// 18214152Sed 19214152Sed#define DOUBLE_PRECISION 20214152Sed#include "fp_lib.h" 21214152Sed 22239138SandrewARM_EABI_FNALIAS(ddiv, divdf3) 23222656Sed 24214152Sedfp_t __divdf3(fp_t a, fp_t b) { 25214152Sed 26214152Sed const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 27214152Sed const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 28214152Sed const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 29214152Sed 30214152Sed rep_t aSignificand = toRep(a) & significandMask; 31214152Sed rep_t bSignificand = toRep(b) & significandMask; 32214152Sed int scale = 0; 33214152Sed 34214152Sed // Detect if a or b is zero, denormal, infinity, or NaN. 35214152Sed if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { 36214152Sed 37214152Sed const rep_t aAbs = toRep(a) & absMask; 38214152Sed const rep_t bAbs = toRep(b) & absMask; 39214152Sed 40214152Sed // NaN / anything = qNaN 41214152Sed if (aAbs > infRep) return fromRep(toRep(a) | quietBit); 42214152Sed // anything / NaN = qNaN 43214152Sed if (bAbs > infRep) return fromRep(toRep(b) | quietBit); 44214152Sed 45214152Sed if (aAbs == infRep) { 46214152Sed // infinity / infinity = NaN 47214152Sed if (bAbs == infRep) return fromRep(qnanRep); 48214152Sed // infinity / anything else = +/- infinity 49214152Sed else return fromRep(aAbs | quotientSign); 50214152Sed } 51214152Sed 52214152Sed // anything else / infinity = +/- 0 53214152Sed if (bAbs == infRep) return fromRep(quotientSign); 54214152Sed 55214152Sed if (!aAbs) { 56214152Sed // zero / zero = NaN 57214152Sed if (!bAbs) return fromRep(qnanRep); 58214152Sed // zero / anything else = +/- zero 59214152Sed else return fromRep(quotientSign); 60214152Sed } 61214152Sed // anything else / zero = +/- infinity 62214152Sed if (!bAbs) return fromRep(infRep | quotientSign); 63214152Sed 64214152Sed // one or both of a or b is denormal, the other (if applicable) is a 65214152Sed // normal number. Renormalize one or both of a and b, and set scale to 66214152Sed // include the necessary exponent adjustment. 67214152Sed if (aAbs < implicitBit) scale += normalize(&aSignificand); 68214152Sed if (bAbs < implicitBit) scale -= normalize(&bSignificand); 69214152Sed } 70214152Sed 71214152Sed // Or in the implicit significand bit. (If we fell through from the 72214152Sed // denormal path it was already set by normalize( ), but setting it twice 73214152Sed // won't hurt anything.) 74214152Sed aSignificand |= implicitBit; 75214152Sed bSignificand |= implicitBit; 76214152Sed int quotientExponent = aExponent - bExponent + scale; 77214152Sed 78214152Sed // Align the significand of b as a Q31 fixed-point number in the range 79214152Sed // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax 80214152Sed // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This 81214152Sed // is accurate to about 3.5 binary digits. 82214152Sed const uint32_t q31b = bSignificand >> 21; 83214152Sed uint32_t recip32 = UINT32_C(0x7504f333) - q31b; 84214152Sed 85214152Sed // Now refine the reciprocal estimate using a Newton-Raphson iteration: 86214152Sed // 87214152Sed // x1 = x0 * (2 - x0 * b) 88214152Sed // 89214152Sed // This doubles the number of correct binary digits in the approximation 90214152Sed // with each iteration, so after three iterations, we have about 28 binary 91214152Sed // digits of accuracy. 92214152Sed uint32_t correction32; 93214152Sed correction32 = -((uint64_t)recip32 * q31b >> 32); 94214152Sed recip32 = (uint64_t)recip32 * correction32 >> 31; 95214152Sed correction32 = -((uint64_t)recip32 * q31b >> 32); 96214152Sed recip32 = (uint64_t)recip32 * correction32 >> 31; 97214152Sed correction32 = -((uint64_t)recip32 * q31b >> 32); 98214152Sed recip32 = (uint64_t)recip32 * correction32 >> 31; 99214152Sed 100214152Sed // recip32 might have overflowed to exactly zero in the preceeding 101214152Sed // computation if the high word of b is exactly 1.0. This would sabotage 102214152Sed // the full-width final stage of the computation that follows, so we adjust 103214152Sed // recip32 downward by one bit. 104214152Sed recip32--; 105214152Sed 106214152Sed // We need to perform one more iteration to get us to 56 binary digits; 107214152Sed // The last iteration needs to happen with extra precision. 108214152Sed const uint32_t q63blo = bSignificand << 11; 109214152Sed uint64_t correction, reciprocal; 110214152Sed correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32)); 111214152Sed uint32_t cHi = correction >> 32; 112214152Sed uint32_t cLo = correction; 113214152Sed reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32); 114214152Sed 115214152Sed // We already adjusted the 32-bit estimate, now we need to adjust the final 116214152Sed // 64-bit reciprocal estimate downward to ensure that it is strictly smaller 117214152Sed // than the infinitely precise exact reciprocal. Because the computation 118214152Sed // of the Newton-Raphson step is truncating at every step, this adjustment 119214152Sed // is small; most of the work is already done. 120214152Sed reciprocal -= 2; 121214152Sed 122214152Sed // The numerical reciprocal is accurate to within 2^-56, lies in the 123214152Sed // interval [0.5, 1.0), and is strictly smaller than the true reciprocal 124214152Sed // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b 125214152Sed // in Q53 with the following properties: 126214152Sed // 127214152Sed // 1. q < a/b 128214152Sed // 2. q is in the interval [0.5, 2.0) 129214152Sed // 3. the error in q is bounded away from 2^-53 (actually, we have a 130214152Sed // couple of bits to spare, but this is all we need). 131214152Sed 132214152Sed // We need a 64 x 64 multiply high to compute q, which isn't a basic 133214152Sed // operation in C, so we need to be a little bit fussy. 134214152Sed rep_t quotient, quotientLo; 135214152Sed wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo); 136214152Sed 137214152Sed // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). 138214152Sed // In either case, we are going to compute a residual of the form 139214152Sed // 140214152Sed // r = a - q*b 141214152Sed // 142214152Sed // We know from the construction of q that r satisfies: 143214152Sed // 144214152Sed // 0 <= r < ulp(q)*b 145214152Sed // 146214152Sed // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we 147214152Sed // already have the correct result. The exact halfway case cannot occur. 148214152Sed // We also take this time to right shift quotient if it falls in the [1,2) 149214152Sed // range and adjust the exponent accordingly. 150214152Sed rep_t residual; 151214152Sed if (quotient < (implicitBit << 1)) { 152214152Sed residual = (aSignificand << 53) - quotient * bSignificand; 153214152Sed quotientExponent--; 154214152Sed } else { 155214152Sed quotient >>= 1; 156214152Sed residual = (aSignificand << 52) - quotient * bSignificand; 157214152Sed } 158214152Sed 159214152Sed const int writtenExponent = quotientExponent + exponentBias; 160214152Sed 161214152Sed if (writtenExponent >= maxExponent) { 162214152Sed // If we have overflowed the exponent, return infinity. 163214152Sed return fromRep(infRep | quotientSign); 164214152Sed } 165214152Sed 166214152Sed else if (writtenExponent < 1) { 167214152Sed // Flush denormals to zero. In the future, it would be nice to add 168214152Sed // code to round them correctly. 169214152Sed return fromRep(quotientSign); 170214152Sed } 171214152Sed 172214152Sed else { 173214152Sed const bool round = (residual << 1) > bSignificand; 174214152Sed // Clear the implicit bit 175214152Sed rep_t absResult = quotient & significandMask; 176214152Sed // Insert the exponent 177214152Sed absResult |= (rep_t)writtenExponent << significandBits; 178214152Sed // Round 179214152Sed absResult += round; 180214152Sed // Insert the sign and return 181214152Sed const double result = fromRep(absResult | quotientSign); 182214152Sed return result; 183214152Sed } 184214152Sed} 185