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 } 169
