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
ARM_EABI_FNALIAS(fdiv,divsf3)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