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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"
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;
}
}
#if defined(__ARM_EABI__)
#if defined(COMPILER_RT_ARMHF_TARGET)
AEABI_RTABI fp_t __aeabi_ddiv(fp_t a, fp_t b) {
return __divdf3(a, b);
}
#else
AEABI_RTABI fp_t __aeabi_ddiv(fp_t a, fp_t b) COMPILER_RT_ALIAS(__divdf3);
#endif
#endif
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