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| //===-- lib/divdf3.c - Double-precision division ------------------*- C -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// 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);
}
// Set 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;
// 0x7504F333 / 2^32 + 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.
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;
// The reciprocal may have overflowed to zero if the upper half of b is
// exactly 1.0. This would sabatoge the full-width final stage of the
// computation that follows, so we adjust the reciprocal down 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);
// 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) {
if (writtenExponent == 0) {
// Check whether the rounded result is normal.
const bool round = (residual << 1) > bSignificand;
// Clear the implicit bit.
rep_t absResult = quotient & significandMask;
// Round.
absResult += round;
if (absResult & ~significandMask) {
// The rounded result is normal; return it.
return fromRep(absResult | quotientSign);
}
}
// 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
COMPILER_RT_ALIAS(__divdf3, __aeabi_ddiv)
#endif
#endif
|