lib/builtins/divdf3.c

//===-- 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, &quotient, &quotientLo);

  // 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