text: //===-- lib/divsf3.c - Single-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 single-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 SINGLE_PRECISION
  #include "fp_lib.h"
  
  ARM_EABI_FNALIAS(fdiv, divsf3);
  
  fp_t __divsf3(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.
      uint32_t q31b = bSignificand << 8;
      uint32_t reciprocal = 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 correction;
      correction = -((uint64_t)reciprocal * q31b >> 32);
      reciprocal = (uint64_t)reciprocal * correction >> 31;
      correction = -((uint64_t)reciprocal * q31b >> 32);
      reciprocal = (uint64_t)reciprocal * correction >> 31;
      correction = -((uint64_t)reciprocal * q31b >> 32);
      reciprocal = (uint64_t)reciprocal * correction >> 31;
      
      // Exhaustive testing shows that the error in reciprocal after three steps
      // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
      // expectations.  We bump the reciprocal by a tiny value to force the error
      // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
      // be specific).  This also causes 1/1 to give a sensible approximation
      // instead of zero (due to overflow).
      reciprocal -= 2;
      
      // The numerical reciprocal is accurate to within 2^-28, lies in the
      // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
      // than the true reciprocal of b.  Multiplying a by this reciprocal thus
      // gives a numerical q = a/b in Q24 with the following properties:
      //
      //    1. q < a/b
      //    2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
      //    3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
      //       from the fact that we truncate the product, and the 2^27 term
      //       is the error in the reciprocal of b scaled by the maximum
      //       possible value of a.  As a consequence of this error bound,
      //       either q or nextafter(q) is the correctly rounded 
      rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
      
      // 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 << 24) - quotient * bSignificand;
          quotientExponent--;
      } else {
          quotient >>= 1;
          residual = (aSignificand << 23) - 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
          return fromRep(absResult | quotientSign);
      }
  }
  