divsf3.c revision 215125
1//===-- lib/divsf3.c - Single-precision division ------------------*- C -*-===// 2// 3// The LLVM Compiler Infrastructure 4// 5// This file is distributed under the University of Illinois Open Source 6// License. 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 22fp_t __divsf3(fp_t a, fp_t b) { 23 24 const unsigned int aExponent = toRep(a) >> significandBits & maxExponent; 25 const unsigned int bExponent = toRep(b) >> significandBits & maxExponent; 26 const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit; 27 28 rep_t aSignificand = toRep(a) & significandMask; 29 rep_t bSignificand = toRep(b) & significandMask; 30 int scale = 0; 31 32 // Detect if a or b is zero, denormal, infinity, or NaN. 33 if (aExponent-1U >= maxExponent-1U || bExponent-1U >= maxExponent-1U) { 34 35 const rep_t aAbs = toRep(a) & absMask; 36 const rep_t bAbs = toRep(b) & absMask; 37 38 // NaN / anything = qNaN 39 if (aAbs > infRep) return fromRep(toRep(a) | quietBit); 40 // anything / NaN = qNaN 41 if (bAbs > infRep) return fromRep(toRep(b) | quietBit); 42 43 if (aAbs == infRep) { 44 // infinity / infinity = NaN 45 if (bAbs == infRep) return fromRep(qnanRep); 46 // infinity / anything else = +/- infinity 47 else return fromRep(aAbs | quotientSign); 48 } 49 50 // anything else / infinity = +/- 0 51 if (bAbs == infRep) return fromRep(quotientSign); 52 53 if (!aAbs) { 54 // zero / zero = NaN 55 if (!bAbs) return fromRep(qnanRep); 56 // zero / anything else = +/- zero 57 else return fromRep(quotientSign); 58 } 59 // anything else / zero = +/- infinity 60 if (!bAbs) return fromRep(infRep | quotientSign); 61 62 // one or both of a or b is denormal, the other (if applicable) is a 63 // normal number. Renormalize one or both of a and b, and set scale to 64 // include the necessary exponent adjustment. 65 if (aAbs < implicitBit) scale += normalize(&aSignificand); 66 if (bAbs < implicitBit) scale -= normalize(&bSignificand); 67 } 68 69 // Or in the implicit significand bit. (If we fell through from the 70 // denormal path it was already set by normalize( ), but setting it twice 71 // won't hurt anything.) 72 aSignificand |= implicitBit; 73 bSignificand |= implicitBit; 74 int quotientExponent = aExponent - bExponent + scale; 75 76 // Align the significand of b as a Q31 fixed-point number in the range 77 // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax 78 // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This 79 // is accurate to about 3.5 binary digits. 80 uint32_t q31b = bSignificand << 8; 81 uint32_t reciprocal = UINT32_C(0x7504f333) - q31b; 82 83 // Now refine the reciprocal estimate using a Newton-Raphson iteration: 84 // 85 // x1 = x0 * (2 - x0 * b) 86 // 87 // This doubles the number of correct binary digits in the approximation 88 // with each iteration, so after three iterations, we have about 28 binary 89 // digits of accuracy. 90 uint32_t correction; 91 correction = -((uint64_t)reciprocal * q31b >> 32); 92 reciprocal = (uint64_t)reciprocal * correction >> 31; 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 98 // Exhaustive testing shows that the error in reciprocal after three steps 99 // is in the interval [-0x1.f58108p-31, 0x1.d0e48cp-29], in line with our 100 // expectations. We bump the reciprocal by a tiny value to force the error 101 // to be strictly positive (in the range [0x1.4fdfp-37,0x1.287246p-29], to 102 // be specific). This also causes 1/1 to give a sensible approximation 103 // instead of zero (due to overflow). 104 reciprocal -= 2; 105 106 // The numerical reciprocal is accurate to within 2^-28, lies in the 107 // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller 108 // than the true reciprocal of b. Multiplying a by this reciprocal thus 109 // gives a numerical q = a/b in Q24 with the following properties: 110 // 111 // 1. q < a/b 112 // 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0) 113 // 3. the error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes 114 // from the fact that we truncate the product, and the 2^27 term 115 // is the error in the reciprocal of b scaled by the maximum 116 // possible value of a. As a consequence of this error bound, 117 // either q or nextafter(q) is the correctly rounded 118 rep_t quotient = (uint64_t)reciprocal*(aSignificand << 1) >> 32; 119 120 // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0). 121 // In either case, we are going to compute a residual of the form 122 // 123 // r = a - q*b 124 // 125 // We know from the construction of q that r satisfies: 126 // 127 // 0 <= r < ulp(q)*b 128 // 129 // if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we 130 // already have the correct result. The exact halfway case cannot occur. 131 // We also take this time to right shift quotient if it falls in the [1,2) 132 // range and adjust the exponent accordingly. 133 rep_t residual; 134 if (quotient < (implicitBit << 1)) { 135 residual = (aSignificand << 24) - quotient * bSignificand; 136 quotientExponent--; 137 } else { 138 quotient >>= 1; 139 residual = (aSignificand << 23) - quotient * bSignificand; 140 } 141 142 const int writtenExponent = quotientExponent + exponentBias; 143 144 if (writtenExponent >= maxExponent) { 145 // If we have overflowed the exponent, return infinity. 146 return fromRep(infRep | quotientSign); 147 } 148 149 else if (writtenExponent < 1) { 150 // Flush denormals to zero. In the future, it would be nice to add 151 // code to round them correctly. 152 return fromRep(quotientSign); 153 } 154 155 else { 156 const bool round = (residual << 1) > bSignificand; 157 // Clear the implicit bit 158 rep_t absResult = quotient & significandMask; 159 // Insert the exponent 160 absResult |= (rep_t)writtenExponent << significandBits; 161 // Round 162 absResult += round; 163 // Insert the sign and return 164 return fromRep(absResult | quotientSign); 165 } 166} 167