lib/builtins/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 COMPILER_RT_ABI fp_t
  25 __divsf3(fp_t a, fp_t b) {
  26
  27     const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
  28     const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
  29     const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
  30
  31     rep_t aSignificand = toRep(a) & significandMask;
  32     rep_t bSignificand = toRep(b) & significandMask;
  33     int scale = 0;
  34
  35     // Detect if a or b is zero, denormal, infinity, or NaN.
  36     if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) {
  37
  38         const rep_t aAbs = toRep(a) & absMask;
  39         const rep_t bAbs = toRep(b) & absMask;
  40
  41         // NaN / anything = qNaN
  42         if (aAbs > infRep) return fromRep(toRep(a) | quietBit);
  43         // anything / NaN = qNaN
  44         if (bAbs > infRep) return fromRep(toRep(b) | quietBit);
  45
  46         if (aAbs == infRep) {
  47             // infinity / infinity = NaN
  48             if (bAbs == infRep) return fromRep(qnanRep);
  49             // infinity / anything else = +/- infinity
  50             else return fromRep(aAbs | quotientSign);
  51         }
  52
  53         // anything else / infinity = +/- 0
  54         if (bAbs == infRep) return fromRep(quotientSign);
  55
  56         if (!aAbs) {
  57             // zero / zero = NaN
  58             if (!bAbs) return fromRep(qnanRep);
  59             // zero / anything else = +/- zero
  60             else return fromRep(quotientSign);
  61         }
  62         // anything else / zero = +/- infinity
  63         if (!bAbs) return fromRep(infRep | quotientSign);
  64
  65         // one or both of a or b is denormal, the other (if applicable) is a
  66         // normal number.  Renormalize one or both of a and b, and set scale to
  67         // include the necessary exponent adjustment.
  68         if (aAbs < implicitBit) scale += normalize(&aSignificand);
  69         if (bAbs < implicitBit) scale -= normalize(&bSignificand);
  70     }
  71
  72     // Or in the implicit significand bit.  (If we fell through from the
  73     // denormal path it was already set by normalize( ), but setting it twice
  74     // won't hurt anything.)
  75     aSignificand |= implicitBit;
  76     bSignificand |= implicitBit;
  77     int quotientExponent = aExponent - bExponent + scale;
  78
  79     // Align the significand of b as a Q31 fixed-point number in the range
  80     // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
  81     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
  82     // is accurate to about 3.5 binary digits.
  83     uint32_t q31b = bSignificand << 8;
  84     uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
  85
  86     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
  87     //
  88     //     x1 = x0 * (2 - x0 * b)
  89     //
  90     // This doubles the number of correct binary digits in the approximation
  91     // with each iteration, so after three iterations, we have about 28 binary
  92     // digits of accuracy.
  93     uint32_t correction;
  94     correction = -((uint64_t)reciprocal * q31b >> 32);
  95     reciprocal = (uint64_t)reciprocal * correction >> 31;
  96     correction = -((uint64_t)reciprocal * q31b >> 32);
  97     reciprocal = (uint64_t)reciprocal * correction >> 31;
  98     correction = -((uint64_t)reciprocal * q31b >> 32);
  99     reciprocal = (uint64_t)reciprocal * correction >> 31;
 100
 101     // Exhaustive testing shows that the error in reciprocal after three steps
 102     // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our
 103     // expectations.  We bump the reciprocal by a tiny value to force the error
 104     // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to
 105     // be specific).  This also causes 1/1 to give a sensible approximation
 106     // instead of zero (due to overflow).
 107     reciprocal -= 2;
 108
 109     // The numerical reciprocal is accurate to within 2^-28, lies in the
 110     // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
 111     // than the true reciprocal of b.  Multiplying a by this reciprocal thus
 112     // gives a numerical q = a/b in Q24 with the following properties:
 113     //
 114     //    1. q < a/b
 115     //    2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
 116     //    3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
 117     //       from the fact that we truncate the product, and the 2^27 term
 118     //       is the error in the reciprocal of b scaled by the maximum
 119     //       possible value of a.  As a consequence of this error bound,
 120     //       either q or nextafter(q) is the correctly rounded
 121     rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32;
 122
 123     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
 124     // In either case, we are going to compute a residual of the form
 125     //
 126     //     r = a - q*b
 127     //
 128     // We know from the construction of q that r satisfies:
 129     //
 130     //     0 <= r < ulp(q)*b
 131     //
 132     // if r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
 133     // already have the correct result.  The exact halfway case cannot occur.
 134     // We also take this time to right shift quotient if it falls in the [1,2)
 135     // range and adjust the exponent accordingly.
 136     rep_t residual;
 137     if (quotient < (implicitBit << 1)) {
 138         residual = (aSignificand << 24) - quotient * bSignificand;
 139         quotientExponent--;
 140     } else {
 141         quotient >>= 1;
 142         residual = (aSignificand << 23) - quotient * bSignificand;
 143     }
 144
 145     const int writtenExponent = quotientExponent + exponentBias;
 146
 147     if (writtenExponent >= maxExponent) {
 148         // If we have overflowed the exponent, return infinity.
 149         return fromRep(infRep | quotientSign);
 150     }
 151
 152     else if (writtenExponent < 1) {
 153         // Flush denormals to zero.  In the future, it would be nice to add
 154         // code to round them correctly.
 155         return fromRep(quotientSign);
 156     }
 157
 158     else {
 159         const bool round = (residual << 1) > bSignificand;
 160         // Clear the implicit bit
 161         rep_t absResult = quotient & significandMask;
 162         // Insert the exponent
 163         absResult |= (rep_t)writtenExponent << significandBits;
 164         // Round
 165         absResult += round;
 166         // Insert the sign and return
 167         return fromRep(absResult | quotientSign);
 168     }
 169 }