lib/divsf3.c

   1 //===-- lib/divsf3.c - Single-precision division ------------------*- C -*-===//
   2 //
   3 //                     The LLVM Compiler Infrastructure
   4 //
   5 // This file is dual licensed under the MIT and the University of Illinois Open
   6 // Source Licenses. See LICENSE.TXT for details.
   7 //
   8 //===----------------------------------------------------------------------===//
   9 //
  10 // This file implements single-precision soft-float division
  11 // with the IEEE-754 default rounding (to nearest, ties to even).
  12 //
  13 // For simplicity, this implementation currently flushes denormals to zero.
  14 // It should be a fairly straightforward exercise to implement gradual
  15 // underflow with correct rounding.
  16 //
  17 //===----------------------------------------------------------------------===//
  18
  19 #define SINGLE_PRECISION
  20 #include "fp_lib.h"
  21
  22 ARM_EABI_FNALIAS(fdiv, divsf3)
  23
  24 fp_t __divsf3(fp_t a, fp_t b) {
  25
  26     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
  27     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
  28     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
  29
  30     rep_t aSignificand = toRep(a) & significandMask;
  31     rep_t bSignificand = toRep(b) & significandMask;
  32     int scale = 0;
  33
  34     // Detect if a or b is zero, denormal, infinity, or NaN.
  35     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
  36
  37         const rep_t aAbs = toRep(a) & absMask;
  38         const rep_t bAbs = toRep(b) & absMask;
  39
  40         // NaN / anything = qNaN
  41         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
  42         // anything / NaN = qNaN
  43         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
  44
  45         if (aAbs == infRep) {
  46             // infinity / infinity = NaN
  47             if (bAbs == infRep) return fromRep(qnanRep);
  48             // infinity / anything else = +/- infinity
  49             else return fromRep(aAbs | quotientSign);
  50         }
  51
  52         // anything else / infinity = +/- 0
  53         if (bAbs == infRep) return fromRep(quotientSign);
  54
  55         if (!aAbs) {
  56             // zero / zero = NaN
  57             if (!bAbs) return fromRep(qnanRep);
  58             // zero / anything else = +/- zero
  59             else return fromRep(quotientSign);
  60         }
  61         // anything else / zero = +/- infinity
  62         if (!bAbs) return fromRep(infRep | quotientSign);
  63
  64         // one or both of a or b is denormal, the other (if applicable) is a
  65         // normal number.  Renormalize one or both of a and b, and set scale to
  66         // include the necessary exponent adjustment.
  67         if (aAbs < implicitBit) scale += normalize(&aSignificand);
  68         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
  69     }
  70
  71     // Or in the implicit significand bit.  (If we fell through from the
  72     // denormal path it was already set by normalize( ), but setting it twice
  73     // won't hurt anything.)
  74     aSignificand |= implicitBit;
  75     bSignificand |= implicitBit;
  76     int quotientExponent = aExponent - bExponent + scale;
  77
  78     // Align the significand of b as a Q31 fixed-point number in the range
  79     // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
  80     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
  81     // is accurate to about 3.5 binary digits.
  82     uint32_t q31b = bSignificand << 8;
  83     uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
  84
  85     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
  86     //
  87     //     x1 = x0 * (2 - x0 * b)
  88     //
  89     // This doubles the number of correct binary digits in the approximation
  90     // with each iteration, so after three iterations, we have about 28 binary
  91     // digits of accuracy.
  92     uint32_t correction;
  93     correction = -((uint64_t)reciprocal * q31b >> 32);
  94     reciprocal = (uint64_t)reciprocal * correction >> 31;
  95     correction = -((uint64_t)reciprocal * q31b >> 32);
  96     reciprocal = (uint64_t)reciprocal * correction >> 31;
  97     correction = -((uint64_t)reciprocal * q31b >> 32);
  98     reciprocal = (uint64_t)reciprocal * correction >> 31;
  99
 100     // Exhaustive testing shows that the error in reciprocal after three steps
 101     // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
 102     // expectations.  We bump the reciprocal by a tiny value to force the error
 103     // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
 104     // be specific).  This also causes 1/1 to give a sensible approximation
 105     // instead of zero (due to overflow).
 106     reciprocal -= 2;
 107
 108     // The numerical reciprocal is accurate to within 2^-28, lies in the
 109     // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
 110     // than the true reciprocal of b.  Multiplying a by this reciprocal thus
 111     // gives a numerical q = a/b in Q24 with the following properties:
 112     //
 113     //    1. q < a/b
 114     //    2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
 115     //    3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
 116     //       from the fact that we truncate the product, and the 2^27 term
 117     //       is the error in the reciprocal of b scaled by the maximum
 118     //       possible value of a.  As a consequence of this error bound,
 119     //       either q or nextafter(q) is the correctly rounded
 120     rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
 121
 122     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
 123     // In either case, we are going to compute a residual of the form
 124     //
 125     //     r = a - q*b
 126     //
 127     // We know from the construction of q that r satisfies:
 128     //
 129     //     0 <= r < ulp(q)*b
 130     //
 131     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
 132     // already have the correct result.  The exact halfway case cannot occur.
 133     // We also take this time to right shift quotient if it falls in the [1,2)
 134     // range and adjust the exponent accordingly.
 135     rep_t residual;
 136     if (quotient < (implicitBit << 1)) {
 137         residual = (aSignificand << 24) - quotient * bSignificand;
 138         quotientExponent--;
 139     } else {
 140         quotient >>= 1;
 141         residual = (aSignificand << 23) - quotient * bSignificand;
 142     }
 143
 144     const int writtenExponent = quotientExponent + exponentBias;
 145
 146     if (writtenExponent >= maxExponent) {
 147         // If we have overflowed the exponent, return infinity.
 148         return fromRep(infRep | quotientSign);
 149     }
 150
 151     else if (writtenExponent < 1) {
 152         // Flush denormals to zero.  In the future, it would be nice to add
 153         // code to round them correctly.
 154         return fromRep(quotientSign);
 155     }
 156
 157     else {
 158         const bool round = (residual << 1) > bSignificand;
 159         // Clear the implicit bit
 160         rep_t absResult = quotient & significandMask;
 161         // Insert the exponent
 162         absResult |= (rep_t)writtenExponent << significandBits;
 163         // Round
 164         absResult += round;
 165         // Insert the sign and return
 166         return fromRep(absResult | quotientSign);
 167     }
 168 }