112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon//===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===// 212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// The LLVM Compiler Infrastructure 412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 59ad441ffec97db647fee3725b3424284fb913e14Howard Hinnant// This file is dual licensed under the MIT and the University of Illinois Open 69ad441ffec97db647fee3725b3424284fb913e14Howard Hinnant// Source Licenses. See LICENSE.TXT for details. 712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon//===----------------------------------------------------------------------===// 912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 1012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// This file implements double-precision soft-float division 1112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// with the IEEE-754 default rounding (to nearest, ties to even). 1212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 1312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// For simplicity, this implementation currently flushes denormals to zero. 1412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// It should be a fairly straightforward exercise to implement gradual 1512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// underflow with correct rounding. 1612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon// 1712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon//===----------------------------------------------------------------------===// 1812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 1912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon#define DOUBLE_PRECISION 2012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon#include "fp_lib.h" 2112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 220193b74976719b8aea4cb8874ba36b75836a8d6eChandler CarruthARM_EABI_FNALIAS(ddiv, divdf3) 2337b97d1cf4501b94347e0b4e880f4b25825a289fAnton Korobeynikov 242d1fdb26e458c4ddc04155c1d421bced3ba90cd0Stephen HinesCOMPILER_RT_ABI fp_t 252d1fdb26e458c4ddc04155c1d421bced3ba90cd0Stephen Hines__divdf3(fp_t a, fp_t b) { 2612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 2712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 2812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 2912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 3012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 3112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon rep_t aSignificand = toRep(a) & significandMask; 3212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon rep_t bSignificand = toRep(b) & significandMask; 3312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon int scale = 0; 3412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 3512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Detect if a or b is zero, denormal, infinity, or NaN. 3612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { 3712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 3812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const rep_t aAbs = toRep(a) & absMask; 3912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const rep_t bAbs = toRep(b) & absMask; 4012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 4112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // NaN / anything = qNaN 4212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (aAbs > infRep) return fromRep(toRep(a) | quietBit); 4312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // anything / NaN = qNaN 4412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (bAbs > infRep) return fromRep(toRep(b) | quietBit); 4512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 4612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (aAbs == infRep) { 4712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // infinity / infinity = NaN 4812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (bAbs == infRep) return fromRep(qnanRep); 4912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // infinity / anything else = +/- infinity 5012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon else return fromRep(aAbs | quotientSign); 5112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 5212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 5312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // anything else / infinity = +/- 0 5412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (bAbs == infRep) return fromRep(quotientSign); 5512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 5612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (!aAbs) { 5712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // zero / zero = NaN 5812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (!bAbs) return fromRep(qnanRep); 5912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // zero / anything else = +/- zero 6012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon else return fromRep(quotientSign); 6112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 6212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // anything else / zero = +/- infinity 6312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (!bAbs) return fromRep(infRep | quotientSign); 6412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 6512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // one or both of a or b is denormal, the other (if applicable) is a 6612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // normal number. Renormalize one or both of a and b, and set scale to 6712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // include the necessary exponent adjustment. 6812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (aAbs < implicitBit) scale += normalize(&aSignificand); 6912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (bAbs < implicitBit) scale -= normalize(&bSignificand); 7012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 7112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 7212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Or in the implicit significand bit. (If we fell through from the 7312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // denormal path it was already set by normalize( ), but setting it twice 7412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // won't hurt anything.) 7512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon aSignificand |= implicitBit; 7612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon bSignificand |= implicitBit; 7712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon int quotientExponent = aExponent - bExponent + scale; 7812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 7912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Align the significand of b as a Q31 fixed-point number in the range 8012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax 8112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This 8212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // is accurate to about 3.5 binary digits. 8312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const uint32_t q31b = bSignificand >> 21; 8412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon uint32_t recip32 = UINT32_C(0x7504f333) - q31b; 8512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 8612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Now refine the reciprocal estimate using a Newton-Raphson iteration: 8712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 8812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // x1 = x0 * (2 - x0 * b) 8912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 9012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // This doubles the number of correct binary digits in the approximation 9112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // with each iteration, so after three iterations, we have about 28 binary 9212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // digits of accuracy. 9312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon uint32_t correction32; 9412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon correction32 = -((uint64_t)recip32 * q31b >> 32); 9512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon recip32 = (uint64_t)recip32 * correction32 >> 31; 9612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon correction32 = -((uint64_t)recip32 * q31b >> 32); 9712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon recip32 = (uint64_t)recip32 * correction32 >> 31; 9812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon correction32 = -((uint64_t)recip32 * q31b >> 32); 9912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon recip32 = (uint64_t)recip32 * correction32 >> 31; 10012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 1012d1fdb26e458c4ddc04155c1d421bced3ba90cd0Stephen Hines // recip32 might have overflowed to exactly zero in the preceding 10212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // computation if the high word of b is exactly 1.0. This would sabotage 10312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // the full-width final stage of the computation that follows, so we adjust 10412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // recip32 downward by one bit. 10512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon recip32--; 10612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 10712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // We need to perform one more iteration to get us to 56 binary digits; 10812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // The last iteration needs to happen with extra precision. 10912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const uint32_t q63blo = bSignificand << 11; 11012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon uint64_t correction, reciprocal; 11112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon correction = -((uint64_t)recip32*q31b + ((uint64_t)recip32*q63blo >> 32)); 11212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon uint32_t cHi = correction >> 32; 11312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon uint32_t cLo = correction; 11412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon reciprocal = (uint64_t)recip32*cHi + ((uint64_t)recip32*cLo >> 32); 11512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 11612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // We already adjusted the 32-bit estimate, now we need to adjust the final 11712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 64-bit reciprocal estimate downward to ensure that it is strictly smaller 11812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // than the infinitely precise exact reciprocal. Because the computation 11912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // of the Newton-Raphson step is truncating at every step, this adjustment 12012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // is small; most of the work is already done. 12112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon reciprocal -= 2; 12212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 12312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // The numerical reciprocal is accurate to within 2^-56, lies in the 12412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // interval [0.5, 1.0), and is strictly smaller than the true reciprocal 12512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b 12612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // in Q53 with the following properties: 12712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 12812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 1. q < a/b 12912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 2. q is in the interval [0.5, 2.0) 13012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 3. the error in q is bounded away from 2^-53 (actually, we have a 13112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // couple of bits to spare, but this is all we need). 13212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 13312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // We need a 64 x 64 multiply high to compute q, which isn't a basic 13412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // operation in C, so we need to be a little bit fussy. 13512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon rep_t quotient, quotientLo; 13612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo); 13712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 13812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). 13912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // In either case, we are going to compute a residual of the form 14012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 14112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // r = a - q*b 14212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 14312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // We know from the construction of q that r satisfies: 14412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 14512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 0 <= r < ulp(q)*b 14612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // 14712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we 14812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // already have the correct result. The exact halfway case cannot occur. 14912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // We also take this time to right shift quotient if it falls in the [1,2) 15012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // range and adjust the exponent accordingly. 15112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon rep_t residual; 15212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (quotient < (implicitBit << 1)) { 15312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon residual = (aSignificand << 53) - quotient * bSignificand; 15412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon quotientExponent--; 15512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } else { 15612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon quotient >>= 1; 15712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon residual = (aSignificand << 52) - quotient * bSignificand; 15812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 15912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 16012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const int writtenExponent = quotientExponent + exponentBias; 16112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 16212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon if (writtenExponent >= maxExponent) { 16312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // If we have overflowed the exponent, return infinity. 16412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon return fromRep(infRep | quotientSign); 16512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 16612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 16712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon else if (writtenExponent < 1) { 16812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Flush denormals to zero. In the future, it would be nice to add 16912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // code to round them correctly. 17012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon return fromRep(quotientSign); 17112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 17212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon 17312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon else { 17412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const bool round = (residual << 1) > bSignificand; 17512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Clear the implicit bit 17612a7d094b185795f478308f4fc27b43abebdde07Stephen Canon rep_t absResult = quotient & significandMask; 17712a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Insert the exponent 17812a7d094b185795f478308f4fc27b43abebdde07Stephen Canon absResult |= (rep_t)writtenExponent << significandBits; 17912a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Round 18012a7d094b185795f478308f4fc27b43abebdde07Stephen Canon absResult += round; 18112a7d094b185795f478308f4fc27b43abebdde07Stephen Canon // Insert the sign and return 18212a7d094b185795f478308f4fc27b43abebdde07Stephen Canon const double result = fromRep(absResult | quotientSign); 18312a7d094b185795f478308f4fc27b43abebdde07Stephen Canon return result; 18412a7d094b185795f478308f4fc27b43abebdde07Stephen Canon } 18512a7d094b185795f478308f4fc27b43abebdde07Stephen Canon} 186