204 lines
8.0 KiB
C
204 lines
8.0 KiB
C
//===-- lib/divtf3.c - Quad-precision division --------------------*- C -*-===//
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//
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// The LLVM Compiler Infrastructure
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//
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// This file is dual licensed under the MIT and the University of Illinois Open
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// Source Licenses. See LICENSE.TXT for details.
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//
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//===----------------------------------------------------------------------===//
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//
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// This file implements quad-precision soft-float division
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// with the IEEE-754 default rounding (to nearest, ties to even).
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//
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// For simplicity, this implementation currently flushes denormals to zero.
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// It should be a fairly straightforward exercise to implement gradual
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// underflow with correct rounding.
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//
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//===----------------------------------------------------------------------===//
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#define QUAD_PRECISION
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#include "fp_lib.h"
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#if defined(CRT_HAS_128BIT) && defined(CRT_LDBL_128BIT)
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COMPILER_RT_ABI fp_t __divtf3(fp_t a, fp_t b) {
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const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
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const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
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const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
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rep_t aSignificand = toRep(a) & significandMask;
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rep_t bSignificand = toRep(b) & significandMask;
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int scale = 0;
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// Detect if a or b is zero, denormal, infinity, or NaN.
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if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
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const rep_t aAbs = toRep(a) & absMask;
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const rep_t bAbs = toRep(b) & absMask;
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// NaN / anything = qNaN
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if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
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// anything / NaN = qNaN
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if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
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if (aAbs == infRep) {
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// infinity / infinity = NaN
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if (bAbs == infRep) return fromRep(qnanRep);
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// infinity / anything else = +/- infinity
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else return fromRep(aAbs | quotientSign);
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}
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// anything else / infinity = +/- 0
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if (bAbs == infRep) return fromRep(quotientSign);
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if (!aAbs) {
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// zero / zero = NaN
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if (!bAbs) return fromRep(qnanRep);
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// zero / anything else = +/- zero
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else return fromRep(quotientSign);
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}
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// anything else / zero = +/- infinity
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if (!bAbs) return fromRep(infRep | quotientSign);
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// one or both of a or b is denormal, the other (if applicable) is a
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// normal number. Renormalize one or both of a and b, and set scale to
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// include the necessary exponent adjustment.
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if (aAbs < implicitBit) scale += normalize(&aSignificand);
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if (bAbs < implicitBit) scale -= normalize(&bSignificand);
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}
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// Or in the implicit significand bit. (If we fell through from the
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// denormal path it was already set by normalize( ), but setting it twice
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// won't hurt anything.)
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aSignificand |= implicitBit;
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bSignificand |= implicitBit;
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int quotientExponent = aExponent - bExponent + scale;
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// Align the significand of b as a Q63 fixed-point number in the range
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// [1, 2.0) and get a Q64 approximate reciprocal using a small minimax
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// polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
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// is accurate to about 3.5 binary digits.
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const uint64_t q63b = bSignificand >> 49;
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uint64_t recip64 = UINT64_C(0x7504f333F9DE6484) - q63b;
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// 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
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// Now refine the reciprocal estimate using a Newton-Raphson iteration:
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//
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// x1 = x0 * (2 - x0 * b)
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//
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// This doubles the number of correct binary digits in the approximation
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// with each iteration.
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uint64_t correction64;
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correction64 = -((rep_t)recip64 * q63b >> 64);
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recip64 = (rep_t)recip64 * correction64 >> 63;
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correction64 = -((rep_t)recip64 * q63b >> 64);
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recip64 = (rep_t)recip64 * correction64 >> 63;
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correction64 = -((rep_t)recip64 * q63b >> 64);
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recip64 = (rep_t)recip64 * correction64 >> 63;
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correction64 = -((rep_t)recip64 * q63b >> 64);
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recip64 = (rep_t)recip64 * correction64 >> 63;
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correction64 = -((rep_t)recip64 * q63b >> 64);
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recip64 = (rep_t)recip64 * correction64 >> 63;
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// recip64 might have overflowed to exactly zero in the preceeding
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// computation if the high word of b is exactly 1.0. This would sabotage
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// the full-width final stage of the computation that follows, so we adjust
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// recip64 downward by one bit.
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recip64--;
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// We need to perform one more iteration to get us to 112 binary digits;
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// The last iteration needs to happen with extra precision.
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const uint64_t q127blo = bSignificand << 15;
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rep_t correction, reciprocal;
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// NOTE: This operation is equivalent to __multi3, which is not implemented
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// in some architechure
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rep_t r64q63, r64q127, r64cH, r64cL, dummy;
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wideMultiply((rep_t)recip64, (rep_t)q63b, &dummy, &r64q63);
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wideMultiply((rep_t)recip64, (rep_t)q127blo, &dummy, &r64q127);
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correction = -(r64q63 + (r64q127 >> 64));
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uint64_t cHi = correction >> 64;
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uint64_t cLo = correction;
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wideMultiply((rep_t)recip64, (rep_t)cHi, &dummy, &r64cH);
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wideMultiply((rep_t)recip64, (rep_t)cLo, &dummy, &r64cL);
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reciprocal = r64cH + (r64cL >> 64);
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// We already adjusted the 64-bit estimate, now we need to adjust the final
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// 128-bit reciprocal estimate downward to ensure that it is strictly smaller
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// than the infinitely precise exact reciprocal. Because the computation
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// of the Newton-Raphson step is truncating at every step, this adjustment
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// is small; most of the work is already done.
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reciprocal -= 2;
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// The numerical reciprocal is accurate to within 2^-112, lies in the
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// interval [0.5, 1.0), and is strictly smaller than the true reciprocal
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// of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
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// in Q127 with the following properties:
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//
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// 1. q < a/b
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// 2. q is in the interval [0.5, 2.0)
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// 3. the error in q is bounded away from 2^-113 (actually, we have a
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// couple of bits to spare, but this is all we need).
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// We need a 128 x 128 multiply high to compute q, which isn't a basic
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// operation in C, so we need to be a little bit fussy.
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rep_t quotient, quotientLo;
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wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo);
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// Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
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// In either case, we are going to compute a residual of the form
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//
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// r = a - q*b
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//
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// We know from the construction of q that r satisfies:
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//
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// 0 <= r < ulp(q)*b
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//
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// if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
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// already have the correct result. The exact halfway case cannot occur.
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// We also take this time to right shift quotient if it falls in the [1,2)
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// range and adjust the exponent accordingly.
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rep_t residual;
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rep_t qb;
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if (quotient < (implicitBit << 1)) {
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wideMultiply(quotient, bSignificand, &dummy, &qb);
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residual = (aSignificand << 113) - qb;
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quotientExponent--;
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} else {
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quotient >>= 1;
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wideMultiply(quotient, bSignificand, &dummy, &qb);
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residual = (aSignificand << 112) - qb;
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}
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const int writtenExponent = quotientExponent + exponentBias;
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if (writtenExponent >= maxExponent) {
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// If we have overflowed the exponent, return infinity.
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return fromRep(infRep | quotientSign);
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}
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else if (writtenExponent < 1) {
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// Flush denormals to zero. In the future, it would be nice to add
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// code to round them correctly.
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return fromRep(quotientSign);
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}
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else {
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const bool round = (residual << 1) >= bSignificand;
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// Clear the implicit bit
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rep_t absResult = quotient & significandMask;
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// Insert the exponent
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absResult |= (rep_t)writtenExponent << significandBits;
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// Round
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absResult += round;
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// Insert the sign and return
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const long double result = fromRep(absResult | quotientSign);
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return result;
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}
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}
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#endif
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