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 } 170
