| //===-- lib/divtf3.c - Quad-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 quad-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 QUAD_PRECISION |
| #include "fp_lib.h" |
| |
| #if defined(CRT_HAS_128BIT) && defined(CRT_LDBL_128BIT) |
| COMPILER_RT_ABI fp_t __divtf3(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 Q63 fixed-point number in the range |
| // [1, 2.0) and get a Q64 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 uint64_t q63b = bSignificand >> 49; |
| uint64_t recip64 = UINT64_C(0x7504f333F9DE6484) - q63b; |
| // 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2) |
| |
| // 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. |
| uint64_t correction64; |
| correction64 = -((rep_t)recip64 * q63b >> 64); |
| recip64 = (rep_t)recip64 * correction64 >> 63; |
| correction64 = -((rep_t)recip64 * q63b >> 64); |
| recip64 = (rep_t)recip64 * correction64 >> 63; |
| correction64 = -((rep_t)recip64 * q63b >> 64); |
| recip64 = (rep_t)recip64 * correction64 >> 63; |
| correction64 = -((rep_t)recip64 * q63b >> 64); |
| recip64 = (rep_t)recip64 * correction64 >> 63; |
| correction64 = -((rep_t)recip64 * q63b >> 64); |
| recip64 = (rep_t)recip64 * correction64 >> 63; |
| |
| // recip64 might have overflowed to exactly zero in the preceeding |
| // 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 |
| // recip64 downward by one bit. |
| recip64--; |
| |
| // We need to perform one more iteration to get us to 112 binary digits; |
| // The last iteration needs to happen with extra precision. |
| const uint64_t q127blo = bSignificand << 15; |
| rep_t correction, reciprocal; |
| |
| // NOTE: This operation is equivalent to __multi3, which is not implemented |
| // in some architechure |
| rep_t r64q63, r64q127, r64cH, r64cL, dummy; |
| wideMultiply((rep_t)recip64, (rep_t)q63b, &dummy, &r64q63); |
| wideMultiply((rep_t)recip64, (rep_t)q127blo, &dummy, &r64q127); |
| |
| correction = -(r64q63 + (r64q127 >> 64)); |
| |
| uint64_t cHi = correction >> 64; |
| uint64_t cLo = correction; |
| |
| wideMultiply((rep_t)recip64, (rep_t)cHi, &dummy, &r64cH); |
| wideMultiply((rep_t)recip64, (rep_t)cLo, &dummy, &r64cL); |
| |
| reciprocal = r64cH + (r64cL >> 64); |
| |
| // We already adjusted the 64-bit estimate, now we need to adjust the final |
| // 128-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^-112, 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 Q127 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^-113 (actually, we have a |
| // couple of bits to spare, but this is all we need). |
| |
| // We need a 128 x 128 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; |
| rep_t qb; |
| |
| if (quotient < (implicitBit << 1)) { |
| wideMultiply(quotient, bSignificand, &dummy, &qb); |
| residual = (aSignificand << 113) - qb; |
| quotientExponent--; |
| } else { |
| quotient >>= 1; |
| wideMultiply(quotient, bSignificand, &dummy, &qb); |
| residual = (aSignificand << 112) - qb; |
| } |
| |
| 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 long double result = fromRep(absResult | quotientSign); |
| return result; |
| } |
| } |
| |
| #endif |
