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