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 //===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===// // // The LLVM Compiler Infrastructure // // This file is dual licensed under the MIT and the University of Illinois Open // Source Licenses. See LICENSE.TXT for details. // //===----------------------------------------------------------------------===// // // This file implements double-precision soft-float division // with the IEEE-754 default rounding (to nearest, ties to even). // // For simplicity, this implementation currently flushes denormals to zero. // It should be a fairly straightforward exercise to implement gradual // underflow with correct rounding. // //===----------------------------------------------------------------------===// #define DOUBLE_PRECISION #include "fp_lib.h" ARM_EABI_FNALIAS(ddiv, divdf3) COMPILER_RT_ABI fp_t __divdf3(fp_t a, fp_t b) { const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; rep_t aSignificand = toRep(a) & significandMask; rep_t bSignificand = toRep(b) & significandMask; int scale = 0; // Detect if a or b is zero, denormal, infinity, or NaN. if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { const rep_t aAbs = toRep(a) & absMask; const rep_t bAbs = toRep(b) & absMask; // NaN / anything = qNaN if (aAbs > infRep) return fromRep(toRep(a) | quietBit); // anything / NaN = qNaN if (bAbs > infRep) return fromRep(toRep(b) | quietBit); if (aAbs == infRep) { // infinity / infinity = NaN if (bAbs == infRep) return fromRep(qnanRep); // infinity / anything else = +/- infinity else return fromRep(aAbs | quotientSign); } // anything else / infinity = +/- 0 if (bAbs == infRep) return fromRep(quotientSign); if (!aAbs) { // zero / zero = NaN if (!bAbs) return fromRep(qnanRep); // zero / anything else = +/- zero else return fromRep(quotientSign); } // anything else / zero = +/- infinity if (!bAbs) return fromRep(infRep | quotientSign); // one or both of a or b is denormal, the other (if applicable) is a // normal number. Renormalize one or both of a and b, and set scale to // include the necessary exponent adjustment. if (aAbs < implicitBit) scale += normalize(&aSignificand); if (bAbs < implicitBit) scale -= normalize(&bSignificand); } // Or in the implicit significand bit. (If we fell through from the // denormal path it was already set by normalize( ), but setting it twice // won't hurt anything.) aSignificand |= implicitBit; bSignificand |= implicitBit; int quotientExponent = aExponent - bExponent + scale; // Align the significand of b as a Q31 fixed-point number in the range // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This // is accurate to about 3.5 binary digits. const uint32_t q31b = bSignificand >> 21; uint32_t recip32 = UINT32_C(0x7504f333) - q31b; // Now refine the reciprocal estimate using a Newton-Raphson iteration: // // x1 = x0 * (2 - x0 * b) // // This doubles the number of correct binary digits in the approximation // with each iteration, so after three iterations, we have about 28 binary // digits of accuracy. uint32_t correction32; correction32 = -((uint64_t)recip32 * q31b >> 32); recip32 = (uint64_t)recip32 * correction32 >> 31; correction32 = -((uint64_t)recip32 * q31b >> 32); recip32 = (uint64_t)recip32 * correction32 >> 31; correction32 = -((uint64_t)recip32 * q31b >> 32); recip32 = (uint64_t)recip32 * correction32 >> 31; // recip32 might have overflowed to exactly zero in the preceding // computation if the high word of b is exactly 1.0. This would sabotage // the full-width final stage of the computation that follows, so we adjust // recip32 downward by one bit. recip32--; // We need to perform one more iteration to get us to 56 binary digits; // The last iteration needs to happen with extra precision. const uint32_t q63blo = bSignificand << 11; uint64_t correction, reciprocal; correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32)); uint32_t cHi = correction >> 32; uint32_t cLo = correction; reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32); // We already adjusted the 32-bit estimate, now we need to adjust the final // 64-bit reciprocal estimate downward to ensure that it is strictly smaller // than the infinitely precise exact reciprocal. Because the computation // of the Newton-Raphson step is truncating at every step, this adjustment // is small; most of the work is already done. reciprocal -= 2; // The numerical reciprocal is accurate to within 2^-56, lies in the // interval [0.5, 1.0), and is strictly smaller than the true reciprocal // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b // in Q53 with the following properties: // // 1. q < a/b // 2. q is in the interval [0.5, 2.0) // 3. the error in q is bounded away from 2^-53 (actually, we have a // couple of bits to spare, but this is all we need). // We need a 64 x 64 multiply high to compute q, which isn't a basic // operation in C, so we need to be a little bit fussy. rep_t quotient, quotientLo; wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo); // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). // In either case, we are going to compute a residual of the form // // r = a - q*b // // We know from the construction of q that r satisfies: // // 0 <= r < ulp(q)*b // // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we // already have the correct result. The exact halfway case cannot occur. // We also take this time to right shift quotient if it falls in the [1,2) // range and adjust the exponent accordingly. rep_t residual; if (quotient < (implicitBit << 1)) { residual = (aSignificand << 53) - quotient * bSignificand; quotientExponent--; } else { quotient >>= 1; residual = (aSignificand << 52) - quotient * bSignificand; } const int writtenExponent = quotientExponent + exponentBias; if (writtenExponent >= maxExponent) { // If we have overflowed the exponent, return infinity. return fromRep(infRep | quotientSign); } else if (writtenExponent < 1) { // Flush denormals to zero. In the future, it would be nice to add // code to round them correctly. return fromRep(quotientSign); } else { const bool round = (residual << 1) > bSignificand; // Clear the implicit bit rep_t absResult = quotient & significandMask; // Insert the exponent absResult |= (rep_t)writtenExponent << significandBits; // Round absResult += round; // Insert the sign and return const double result = fromRep(absResult | quotientSign); return result; } }